. PART I. GENERAL PHYSICS. CHARLES GRIFFIN AND COMPANY, LTD., PUBLISHERS. AN ELEMENTARY TEXT-BOOK OF PHYSICS. BY R. WALLACE STEWART, D.Sc.(Lond.). In Four Volumes. Crown 8uo. Cloth. Each Fully Illustrated. Sold Separately. VOL. I. GENERAL PHYSICS. VOL. II. Profusely Illustrated. $1.25 net. SOUND. CONTENTS. Simple Harmonic Vibration. Production of Sound. Wave Motion. Propagation of Sound. Characteristics of Sound. Reflection and Refraction of Sound. Velocity of Sound in Air and Water. Transverse Vibration of Strings. Longitudinal Vibration of Rods and Columns of Air. INDEX. "Should supply the much-felt need of an elementary treatment of this subject . . . distinctly good." Mature. VOL. III. With 142 Illustrations. $1.50 net. LIGHT. CONTENTS. Introductory. Rectilinear Propagation of Light. Photometry. Reflection at Plane Surfaces. Reflection at Spherical Surfaces. Refraction. Refraction through Lenses. Dispersion. INDEX. "This elementary treatise resembles Part II. (Sound) in its attractiveness . . . the treatment is good . . . excellent diagrams . . . very clear." Journ. of Inst. of Teachers in Technical Institutes. VOL. IV. With 84 Illustrations. $1.50 net. HEAT. CONTENTS. Introductory. Thermometry. Expansion of Solids. Expansion of Liquids. Expansion of Gases. Calorimetry. Specific Heat. Liquefaction and Solidification. Vaporisation and Condensation. Conduction of Heat. Convection. Mechanical Equivalent of Heat. Radiation. INDEX. "A new book . . . not mere re-arrangement . . . very readable and interesting." Journ. of Inst. of Teachers in Technical Institiites. LONDON : CHARLES GRIFFIN & CO., LTD., EXETER STREET, STRAND. AN -ELEMENTARY TEXT- BOOK OF PHYSICS: / PART I. GENERAL PHYSICS. BY R. WALLACE STEWART, D.Sc.(LoND.) TKflitb 187 illustrations. LONDON: CHARLES GRIFFIN & COMPANY, LIMITED. PHILADELPHIA : J. B. LIPPINCOTT COMPANY. 1910. PREFACE. This volume, although issued subsequently to Part II. "SOUND," Part III. "LIGHT," Part IV. "HEAT," is presented as the opening one of the series forming " AN ELEMENTARY TEXT-BOOK ON PHYSICS." It is written in the same exact, simple, and straightforward manner which has commended the other volumes and made them popular with Students who are preparing for any of the usual Elementary Examinations on Physics. The presentation of the subject in separate volumes suited to the requirements of the Student was considered desirable, as it enabled the author to deal adequately with the fundamental facts and principles without the loss of interest always manifest when the whole subject is com- pressed into one small volume. Teachers and reviewers have been unanimous in their praise of the earlier volumes, both in regard to the manner in which the subjects have been treated, and the excellent print and diagrams, which are new, and not mere re- arrangements of the old stereotyped forms. A melancholy interest attaches to this present volume, from the fact that the distinguished author died suddenly soon after the completion of the work. October, 1910, 281498 CONTENTS. CHAPTER PAGE I. INTRODUCTORY, 1 II. SCALAR AND VECTOR QUANTITIES, 8 III. MEASUREMENT OF LENGTH, AREA, AND VOLUME, . . 15 IV. MEASUREMENT or TIME, 31 V. MEASUREMENT or MASS, . . . . . .37 VI. VELOCITY, 45 VII. ACCELERATION, 57 VIII. CIRCULAR MOTION AND SIMPLE HARMONIC MOTION, . 73 IX. FORCE, 87 X. WORK AND ENERGY, 122 XI. COMPOSITION AND RESOLUTION OF FORCES, . . . 153 XII. CENTRE OF GRAVITY, .172 XIII. EQUILIBRIUM OF FORCES, ....... 189 XIV. FRICTION, 207 XV. THE BALANCE, 221 XVI. GENERAL PROPERTIES OF MATTER, 235 XVII. PROPERTIES OF SOLIDS, ....... 265 XVIII. HYDROSTATICS, 294 XIX. EXPERIMENTAL DETERMINATION OF SPECIFIC GRAVITY AND DENSITY 308 XX. PROPERTIES OF LIQUIDS, 330 XXI. PROPERTIES OF LIQUIDS (continued), 337 XXII. PROPERTIES OF GASES, 355 INDEX, 411 GENERAL PHYSICS CHAPTER I. INTRODUCTORY. 1. The Scope Of Physics. The science of Physics may be said to deal with the phenomena of matter and ether. Matter is the material or stuff of which are made all bodies which occupy space, and which are perceptible by us through the senses. It possesses certain fundamental properties, such as inertia and gravitation, which are dealt with later. The ether is the medium which is assumed to fill all space, and to permeate all matter. It cannot be perceived by the senses, but there can be little doubt of its objective existence. Our knowledge of its existence and properties is entirely of an indirect character, derived from the study of the phenomena which are assumed to be associated with it, but this knowledge rests upon a very firm foundation of experimental evidence. The relation between ether arid matter is of too uncertain and speculative a character to be considered here. It may, however, be stated that it is in every way probable that matter is in some sense a modified form of ether, and that some of its properties are determined by its relation with the ether with which it is associated. The branches into which the science of Physics is usually divided are Motion, Properties of Matter, Sound, Heat (including Radiation), Light, and Magnetism and Electricity. 1 2 , GENERAL PHYSICS. The measurement of space, the measurement of time, and the measurement of quantity of matter are the fundamental mea- surements of the science. The distribution of matter in space, the specification of the relative position or configuration of any system of material bodies, and the consideration of the changes of configuration which take place in any system with time, are included under the general term motion. The study of motion, in this general sense, and the study of the general properties of matter, form the usual introduction to Physics. Sound or Acoustics deals with the vibratory motion of bodies and with wave motion in material media, with special reference to the sense of hearing. The phenomena of Heat are phenomena pertaining to matter, and include many important changes in the state and properties of matter. Radiation in a limited sense is the transverse wave motion set up in the ether by the vibratory motion of the molecules of a body, and Light is radiation within certain limits of wave length. The phenomena of Magnetism and Electricity are essentially ether phenomena produced under conditions associated with the existence of certain states of strain or motion in the ether, and with the effects attending the presence of matter in the ether under these conditions. The study of matter, in so far as it relates to the different kinds of elementary substances or elements which constitute matter, to the interaction of these elements with each other, and to the composition and properties of the compounds they form with each other, constitutes the science of Chemistry, and does not come directly within the scope of Physics. 2. Physical Quantities. Physics is one of the exact sciences. Measurement is the basis of its experimental work and mathematical reasoning is the basis of its theory. The physical quantities which are defined and measured experimentally will be explained as they arise in the exposition INTRODUCTORY. 3 of the subject. It will be understood, however, that at the very outset quantities such as length, duration of time, and mass must be measured. These lead to other quantities, such as area, volume, velocity, and acceleration; and, as the subject develops, more complex quantities such as force, work, energy, and power are introduced. In the same way in every branch of the subject all measurable quantities are denned and measured so that the theoretical principles of the subject can be established on a sure foundation of exact quantitative knowledge. 3. Units. The first essential in measuring any quantity is a suitable unit of measurement. A unit must evidently be of the same kind or denomination as the quantity to be measured, and its magnitude must be definitely specified either by direct reference to a standard in which it is realised, or by defining its relation to the standard or standards on which its value ulti- mately depends. A quantity is measured in terms of a given unit by determin- ing the number of times it contains the unit, and the magnitude of the quantity can then be expressed as n times the magnitude of the unit, or as equal to n units, where n is a number which may be a whole number or a fraction. For example, any length may be measured in terms of the yard as unit, and if a given length is found to be 5*3 times the length of a yard it is said to be 5*3 yards in length. It will be seen that the magnitude of any quantity must always be given as n units, and in order that it may be completely specified, the value of n and the name of the unit must be definitely stated. The number n which gives the number of units in any magnitude is called the measure or numeric of the magnitude. When the magnitude of the same quantity is given in terms of different units of the same kind the measures of the magni- tude must evidently be different and must vary inversely as the relative magnitudes of the units employed, A length of 2 yards 4 GENERAL PHYSICS. may, for example, be expressed as 6 feet, or 72 inches. The measures here are in the ratio 1 : 3 : 36 when the relative magnitudes of the units are in the ratio 1 : -J : -^g-. That is r the measure is multiplied oy n when the magnitude of the unit is divided by n. 4. Fundamental and Derived Units. The magnitude of the unit selected for the measurement of any quantity may be decided on purely arbitrary grounds, or it may be determined by a formal definition of the unit. If an arbitrary unit were selected for the measurement of every physical quantity, without any consideration of the inter-relations of the quantities, it would be found that endless confusion and trouble would result from the complicated relations between the units selected. For example, if any arbitrary length were selected as unit of length, and any arbitrary area as unit of area, it would be necessary, in finding the area of any regular plane figure by the rules of mensuration, to know the area of the square on the unit of length in terms of the arbitrary unit of area. For this reason the plan has been adopted of selecting the units of certain quantities as fundamental units, and then deriving the units for all other quantities from these fundamental units by means of carefully framed definitions. The units de- rived in this way from the fundamental units are known as derived units. It is found that in order to build up a system of units on this plan, the fundamental quantities need not be more than three in number, and may, in theory, be any three quantities. The units of these quantities, the fundamental units, may be selected as arbitrary units or determined by definition, but, in either case, they must be capable of exact realisation as permanent standards of reference. The quantities adopted as fundamental physical quantities are length, mass, and time, so that the units of length, mass, and INTRODUCTORY. 5 time are the fundamental units of the whole system of physical units. All the other units of the system are derived units. The unit of area, for example, is defined as the area of the square on the unit of length, and is, therefore, derived from the unit of length. Similarly the unit of volume is the volume of a cube having its edge of unit length, and is also derived from the unit of length. The unit of velocity is the unit of length per unit of time, and is, therefore, derived from the units of length and time. The unit of force is that force which, acting on unit mass, pro- duces unit change of velocity in unit time, and is thus derived from the units of length, mass, and time. In the same way any other physical unit may be derived from one or more of the three fundamental units. The quantities length, mass, and time are specially suitable for use as fundamental quantities. The units of length and mass are capable of very exact realisation as permanent and invariable standards, and the unit of time can be definitely specified in terms of the time in which the earth makes one complete revolution. The units of length and mass in general use are arbitrary units. An attempt was made by French physicists, as explained in Chapters iii. and v., to connect both these units with natural constants, and so to put them on the same basis as the unit of time. It was found, however, that the constants selected were not ascertained with sufficient accuracy'" to enable * This point may be made clearer by an example. Suppose the unit of length to be specified as the millionth part of the polar diameter of the Earth. In order to realise this unit as a permanent standard of reference it is necessary to construct a bar of platinum or some permanent metal of the specified length. This means that the length of the polar diameter must be accurately known, and that the bar must be constructed so as to be exactly equal to the millionth part of this known length. If the length of the bar is derived from an inaccurate value of the polar diameter, it must evidently be reconstructed when a more accurate value is^found, or, if the old standard is retained, the specification of the unit must be changed. 6 GENERAL PHYSICS. the reference standards to be constructed in exact accordance with the specifications of the units. A system of units built up, in this way, of units derived from certain fundamental units, is called an absolute system of units or a system of absolute units. The system of absolute units in general use in physics is the C. Gr. S. system, in which the fundamental units of length, mass, and time are the Centimetre, the Gramme, and the Second respectively. The " English " system of absolute units, sometimes called the F. P. S. system, is a system in which the Foot, the Pound, and the Second are the fundamental units of length, mass, and time respectively. It is still used in text books on theoretical mechanics, but is seldom used in physics. 5. The Metric System. The metric system is not a system of units in the sense explained in the foregoing article. It is practically a set of " measures " in which the decimal system is employed in forming the multiples and sub-multiples of the units selected. The notation of the system is the same in every table. The multiples of the unit by 10, 100, and 1,000 are designated by placing the Greek prefixes deca-, hecto-, and kilo- before the unit ; and the corresponding sub-multiples are designated by placing the Latin prefixes deci-, centi-, and milli- before the unit. The multiple by 10,000, which is sometimes used, is dis- tinguished by placing the prefix myria- before the unit. Thus we have the following scheme in each table : Myria- (unit) = 10,000 units. ( Kilo- (unit) .- 1,000 units. Multiples, < Hecto (unit) = 100 units. ( Deca- (unit) = 10 units. (Unit) 1 unit. !Deci- (unit) = f l unit. Centi- (unit) = '01 unit. Milli- (unit) = "001 unit. INTRODUCTORY. The units of the system are given below : QUANTITY MEASURED. NAME OF UNIT. Length, Metre. Area, .... The square metre. Area (Land Measure), . The are or square decametre. Volume, The cubic metre. Capacity, Litre or cubic decimetre. Mass, .... Gramme. The system takes its name, the metric system, from the metre the unit of length. CHAPTER II. SCALAR AND VECTOR QUANTITIES. 6. Scalar Quantities. A quantity which possesses magni- tude only, and is, therefore, completely specified by its magnitude, is known as a scalar or scalar quantity. Thus, area, volume, time, mass, and other quantities to be dealt with later, are scalar quantities. Scalar quantities of the same kind are added and subtracted by the ordinary arithmetical rules, or they may be assigned positive and negative signs according to some recognised con- vention, and treated as algebraic quantities. 7. Vector Quantities. A quantity which possesses direction .as well as magnitude, is known as a vector or a vector quantity. The distance of one point from another is a vector quantity, and it will be found later that other quantities, such as dis- placement, velocity, acceleration, and force, are vector quantities. In dealing with vector quantities it is necessary to be able to add and subtract them in such a way as to take account of the direction as well as the magnitude of the quantities. Ordinary arithmetical addition and subtraction take account of magnitude only. Algebraic methods make provision for the addition and subtraction of quantities which may have one of two opposite directions denoted respectively by a positive and negative sign. A special method must therefore be found for the addition and subtraction of vector quantities. 8. Composition of Vector Quantities. The usual method of adding and subtracting vector quantities is a graphical one. SCALAR AND VECTOR QUANTITIES. 9 A straight line may represent any vector quantity if the number which measures the length of the line is the same as the number which measures the magnitude of the quantity, and if the direction of the line represents the direction of the quantity. The vector quantities to be added or subtracted may thus be represented by straight lines, and addition or subtraction becomes a graphical process. The process of adding or compounding vector quantities is known as the composition of the quantities, and the sum of the added quantities is called the resultant. It is obvious that only quantities of the same kind can be compounded, and that the resultant is a quantity of the same kind as the quantities compounded. The rule for compounding two vector quantities by this method may be given in the following terms. From any point A, Fig. 1, draw two straight lines AB and AC, to represent the quantities in magnitude and direction. Then complete the parallelogram, ABDC, of which these two lines are adjacent sides, and draw the diagonal AD from the point A, This diagonal, AD, now repre- sents the resultant of the two quantities in the same way as the lines AB and AC represent the quantities themselves. That is, the number which measures the length of AD is the number which measures the magnitude of the resultant, and the direction of AD represents the direction of the resultant. This form of the rule may be called the parallelogram rule. Another form known as the triangle rule may be derived from it. It will be seen in Fig. 1 that the line BD is the same in magnitude and direction as AC. The lines AB and BD, there- fore, represent the quantities to be compounded, and AD repre- sents their resultant. Hence, if starting from any point A, we draw, one after the other, in order, two lines, AB and BD, to 10 GENERAL PHYSICS. represent the quantities to be compounded ; then AD, the third side of the triangle ABD, drawn from the starting point A to the finishing point D, represents the resultant of the two quantities. This is the triangle form of the rule for the composition of two vector quantities. It has the advantage that it can be extended in the same terms to provide a rule for the composi- tion of any number of vector quantities. Thus, if we wish to compound five vector quantities of the same kind we draw from any starting point A (Fig. 2) five lines, AB, BC, CD, DE, and EF, in successive order to represent the five quanti- Fig. 3. ties in magnitude and direction. The line AF, drawn from the starting point A to the finishing point F of this sequence of lines, then represents the resultant of the five vector quantities in the same way as the lines AB, BC, CD, DE ; and EF repre- sent the quantities themselves. This rule, known as the polygon rule, is evidently an extension of the triangle rule, for, by the triangle rule, the resultant of the quantities represented by AB and BC is represented by AC. Similarly, AD represents the resultant of the quantities represented by AC and CD, or the resultant of the three quantities represented by AB, BC, and CD. In the same way it follows that AE represents the resultant of the quantities represented by AB, BC, CD, and SCALAR AND VECTOR QUANTITIES. 11 DE, and that AF represents the resultant of the quantities represented by AB, BC, CD, DE, and EF. The method of finding the difference of two vector quantities is most conveniently derived from the triangle rule for com- pounding two quantities. Thus, in Fig. 1, since the resultant of the two quantities represented by AB and BD is represented by AD, it follows that the difference between the two quantities represented by AD and AB is represented by BD. The line BD (with the arrow from B to D) represents the difference obtained by subtracting the quantity represented by AB from the quantity represented by AD, while the line DB (with the arrow from D to B) represents the difference obtained by subtracting the quantity represented by AD from the quantity represented by AB. Hence, we have the following rule for finding the difference of any two vector quantities of the same kind. From any point A, Fig. 3, draw lines AB and AC to repre- sent in magnitude and direction the two quantities whose differ- ence is required ; then join BC. The line BC, drawn from B to C, then represents the differ- ence obtained by taking the quantity represented by AB from the quantity represented by AC. 9. Resolution of a Vector Quantity. Just as a number (such as 12) may be split up arithmetically into an infinite number of pairs of numbers (such as 9 and 3, or 8 and 4, or 10 and 2, or 5 '9 and 6'1, &c.), which, when added together, make up the number as their sum, so any vector quantity may be resolved into an infinite number of pairs of components which, when compounded together, make up the given quantity as their resultant. This may be done by direct application of the parallelogram or triangle rule. Let AB, Fig. 4, represent any vector quantity ; then, if we construct any parallelogram, ACDB, on AB as diagonal, the lines AC and AD represent two 12 GENERAL PHYSICS. quantities which, if compounded together, make up the quantity represented by AB as their resultant. That is, the quantity represented by AB is.resolved, or split up into two components represented by AC and AD. Since this is true for any parallelo- gram constructed on AB as diagonal, it is evident that the quantity represented by AB may, in this way, be resolved into any number of pairs of components. In the same way with the triangle rule, if AB, Fig. 5, repre- sent any vector quantity, and ACB be any triangle constructed on AB as base, the lines AC and CB represent quantities which, if compounded together, make up the quantity represented by AB as resultant. That is, the quantity represented by AB is resolved into two components represented by AC and CB. Fig 4. Fig. 5. If it is required to resolve any given quantity into two com- ponents, of which one is given, it is evidently sufficient to find the difference between the two given quantities, as explained above. Thus, if AB, Fig. 5, represent the given quantity, and AC the known component, then CB represents the other com- ponent, for the quantities represented by AC and CB have the quantity represented by AB as their resultant. If it is required to resolve a given quantity into two com- ponents in given directions, it will be found most convenient to apply the parallelogram rule. Thus, let AB, Fig. 6, represent the given quantity, and AX and AY the given directions for the required components. Through B draw the lines BC and BD, parallel respectively to AY and AX, and cutting these lines at the points C and D. Then the lines AC and AD obviously represent the required components. SCALAR AND VECTOR QUANTITIES. 13 An important case of the resolution of a vector quantity into components, is that in which the given quantity is to be resolved into two components at right angles, one of which is required to be in a given direction, and the other at right angles to it. For example, let it be required to resolve the given quantity, repre- sented by AB in Fig. 7, into two rectangular components, one of which is to be in the direction AX, and the other at right angles to AX. From B draw BC perpendicular to AX, and cutting it at C, and complete the rectangle ACBD. The lines AC and AD now represent the required components ; the component represented c Fig. 6. Fig. by AC has the given direction AX, and the other, represented by AD, has a direction at right angles to AX. If the angle BAX, the angle between the direction of the given quantity represented by AB and the given direction AX, be denoted by a it will be seen that AC AB = cos a, or AC = AB . cos a, and AD CB . ATJ . -_-- = - = sin a, or AD = AB . sin a. AJ5 A-t> That is, if R denote the magnitude of the given quantity represented by AB, and P and Q the magnitudes of the components represented by AC and AD respectively, we have 14 GENERAL PHYSICS. P = R . cos a and Q = R . sin a, where a is the angle between the direction of the given quantity R and the given direction of the component P. It is also evident, by application of Euc. i. 47 to the figure, that AB 2 = AC 2 + CB 2 = AC 2 + AD 2 , or R 2 = P 2 + Q 2 . Fig. 8. This relation may evidently be applied to find the resultant R of two vector quantities, P and Q, whose directions are at right angles to each other. If the directions of the quantities P and Q are not at right angles but make an angle, a, with each other as in Fig. 8, it can easily be proved with the aid of Euc. ii. 1 2 and 1 3 that R 2 - P 2 + Q 2 + 2PQ cos a. 15 CHAPTER III. MEASUREMENT OP LENGTH, AREA, AND VOLUME. 10. Units Of Length. The legal units of length in England are the yard and the metre. The yard is defined by Act of Parliament as the distance at 62 F. between the centres of the transverse lines on the two gold plugs in a bronze bar, originally deposited in the office of the Exchequer, but now kept at the Standard Office of the Board of Trade. The length thus defined and preserved is known as the standard yard, and copies of the standard are kept at the Houses of Parliament, the Royal Mint, the Royal Observatory at Greenwich, and the Royal Society of London. The multiples and sub-multiples of the yard are given in the familiar table of English long measure. We need only notice here the foot and the inch as sub-multiples and the mile as a multiple. Thus we have : 12 inches = 1 foot. 3 feet = 1 yard. 1760 yards = 1 mile. It will be understood that, although the yard is the legal unit or standard of length, any one of its sub-multiples or multiples, or, in fact, any specified part of it, may be taken as the unit of length in any set of length measurements. The metre is really the legal unit of length in France, but it was legalised in England in 1897. It is defined as the distance at C. between the ends of a platinum rod constructed by Borda. 16 . GENERAL PHYSICS. The metre was intended originally to be the ten-millionth part of the distance from the North Pole to the Equator measured along the meridian passing through Paris. Very careful mea- surements were made* for the purpose of determining this distance, and Borda constructed the platinum bar now used as the standard metre to be the ten -millionth part of the result then obtained. Later measurements, however, show that the distance from the North Pole to the Equator along the meridian of Paris is rather more than ten million times the length of Borda's bar r so that the metre is no longer defined as the ten-millionth part of this quadrant on the earth's surface, but simply as the distance at C. between the ends of Borda's platinum bar. Although the metre may be legally used in England as a standard of length it is not yet in common use in this country. It is, however, the standard most generally adopted in other European countries. The multiples and sub-multiples of the metre are given below Kilometre (km.) = 1000 metres. Hectometre = 100 Decametre = 10 ,, Metre Decimetre (dm.) = *1 metre. Centimetre (cm.) = *01 ,, Millimetre (mm.) = -001 ,, Of the multiples only the kilometre is in general use ; it is used on the Continent for specifying distance from place to place in the same way as we use the mile in England. The sub-multiples are all in use, but the centimetre and millimetre are most generally used. The unit of length generally adopted for scientific measure- ments is the centimetre. The foot and the inch are occasionally used in England for certain measurements, but it is now practi- cally the universal custom to employ the centimetre as the unit of length in all physical measurements. MEASUREMENT OF LENGTH. 17 The relative magnitude of the yard and the metre is given with sufficient accuracy by the following equivalents: 1 metre = 39 -.37 inches = 1-0936 yards. 1 yard = 0-9144 metre = 91 "44 cms. These equivalents may be reduced to the following approxi- mate values which are convenient for general use. 1 metre = 3 '28 feet. 1 foot = 30-48 cms. 1 inch = 2-54 cms. For rough calculations it is convenient to remember that 10 cms. = 4 inches (nearly), or, more exactly, 33 cms. = 13 inches; also that a millimetre is slightly less than the twenty-fifth of an inch. The relation between the mile and the kilometre is given by the equivalent : 1 kilometre = '62137 mile. 1 mile 1 -60935 kilometres. That is, a kilometre is nearly five-eighths of a mile, so that five miles is, roughly, equal to eight kilometres. 11. Measuring" Scales. The measuring scale in common use for physical measurements is the metre scale one metre in length. It is usually made of box-wood or metal, and is generally graduated to show decimetres, centimetres, and milli- metres along one measuring edge, and for convenience inches and tenths of an inch along another edge. Small steel scales showing a variety of small divisions of the inch and centimetre are also in use. The most accurate metre scales are made in metal. Steel, brass, and gun-metal have been used for this purpose, but it is probable that the new metal " invar," an alloy of nickel and steel, will be generally used in future. This metal has a very small coefficient of expansion with change of temperature, so that a scale 18 GENERAL PHYSICS. made of it would not be subject to any appreciable error due to change of length with change of temperature. Metre scales are generally made as line scales that is, the graduated metre extends from a line near one end to a line near the other end. Some scales are, however, made as end scales, the graduated metre extending from one end of the scale to the other. Borda's standard metre, for example, is. an end measure, while the standard yard is a line measure. 12. The Vernier. In certain cases a length may be measured with sufficient accuracy by simply applying the measuring edge of a suitable scale directly to it, and then reading off the required length from the graduations of the scale. With a scale graduated in tenths of an inch it is possible in this way, by estimating tenths of a scale division, to measure a length with fair accuracy to one-hundredth of an inch. Similarly, with a scale graduated in millimetres it is possible with care and practice to read to a tenth of a millimetre or one-hundredth of a centimetre. It is not possible to attain to greater accuracy than this by any further subdivision of the scale, so that when greater accuracy is required other methods have to be adopted. Of these methods, the vernier method is the simplest and most commonly used. A vernier is a short auxiliary scale used with the measuring scale for the purpose of reading the scale to some particular fraction of a scale division. The general principle on which a vernier is constructed may be stated concisely in the following way. If it is desired to construct a vernier to read to of a scale division, a length equal to (n 1) or (n + 1) scale divisions is takfcn and divided into n equal parts to give the vernier divisions. Thus, if we wish to make a vernier to read to -^ of a scale division we mark off a length equal to 19 (or 21) scale divisions and divide it into twenty equal parts. The small scale thus obtained would be a vernier scale which MEASUREMENT OF LENGTH. 19 would enable readings to be taken on the measuring scale to one-twentieth of a scale division. It will be seen that by this method of construction a division on a vernier reading to of a scale division differs from a scale division by -1 of a scale division ; it is either the -i- part of a scale division less, or the part greater than a scale division, according as it is made on the (n 1) or (n + 1) plan explained above. It is generally most convenient to make a vernier on (n 1 ) plan so that its divisions are less than the scale divisions by -i- of a scale division. This difference between the length of a vernier division and the length of a scale division given as a fraction of a scale division is known as the least count of a vernier. MEASURING SCALE The method of using a vernier can now be explained with the help of the following example. Suppose it is required to measure the length of the rod AB by means of the measuring scale and vernier shown in Fig. 9. It will be seen from the figure that the length of the vernier is equal to 9 scale divisions, and that it is divided into 10 equal parts ; it therefore reads to T V of a scale division, or its least count is -j 3 ^ of a scale division. When the measuring scale is applied directly to AB, as shown in the figure, the length of AB is found to be greater than three and less than four scale divisions. The vernier is then applied, as shown at BC, so that the zero division is placed exactly at the point whose position on the scale is to be determined. The length of AB is thus seen to be equal to three scale divisions and the 20 GENERAL PHYSICS. portion of the fourth division which lies to the left of the zero of the vernier. If we now pass along 4he vernier from the zero to the sixth division we see that the distance between each successive division and the scale division immediately to the left of it gets less and less, until, at the sixth division of the vernier, the two divisions are coincident. This evidently indicates that the distance from the zero of the vernier to the first scale division to the left of it (3) is 6 tenths of a scale division. For, from the construction of the vernier the corresponding distance at the fifth vernier division is 1 tenth of a scale division, at the fourth division it is 2 tenths, at the third 3 tenths, at the second 4 tenths, at the first 5 tenths, and at the zero 6 tenths. That is, the length of AB is 3*6 scale divisions. MEASURING SCALE VERNIER B Fig. 10. In the same way it can he made out that the length of the rod AB in Fig. 10 is 18-^-J- scale divisions, or 18*55 scale divisions. The manner in which the vernier divisions are marked and numbered should be noted. It will be seen that the numbering is in the same direction on the vernier as on the scale. The principle of the vernier is applied in a number of measuring instruments. The most important of these is the vernier callipers shown in Fig. 11. The purpose of the instrument is sufficiently indicated for general purposes in the figure. The details of its construction and the method of using it can be learnt satisfactorily only by practice in the laboratory. MEASUREMENT OF LENGTH. 21 13. The Micrometer Screw. Another important method of attaining great accuracy in the measurement of length depends upon the use of the micrometer screw. This is merely a very accurately cut screw of small pitch, pro- vided with a large head, which is divided round its circumference into a convenient number of equal parts arranged so that any fraction of a complete turn of the screw can be measured with an accuracy which depends upon the size of the head and the number of divisions into which its circumference is divided. It is called a micrometer screw because it is capable of measuring very small differences in length. The theory of the micrometer screw is simple. It is evident that for one com- plete turn of the screw the point moves through a distance equal to the pitch of the screw. Hence, if by dividing the head of the screw into 10, 100, or 1,000 parts, we can measure the tenth, hundredth, or thousandth part of a com- plete turn, we can measure by means of the point of the screw to one-tenth, one- hundredth, or one - thousandth of the pitch of the screw. For example, if the pitch of the screw is J mm., and the head is divided into 500 parts, the turning of the screw through one of these parts causes a displacement of the point of the screw through ^-^ of J mm., or '001 mm. A screw constructed in this would, therefore, be able to indicate a difference in way the 22 GENERAL PHYSICS. position of its point equal to the thousandth part of a millimetre. Fig. 1 2 indicates diagrammatically the manner in which the principle of the micrometer screw can be adapted for measuring short lengths or small differences in length. The screw gauge shown in Fig. 13 is constructed on this principle. The divisions"of the head are marked, as seen in the Fig. 12. figure, round the edge of a sleeve carried by the head, and fitting over the collar in which the screw works. The instru- ment is used for measuring the diameters of wires and for other similar measurements. The object to bejneasured is placed between the jaws of the gauge, and the difference; between the reading of the screw head Fig. 13. when the object is in position, and the zero reading when the jaws are in contact, gives the thickness of the object between the jaws. 14. The Cathetometer. The cathetometer is an instru- ment used for measuring differences in vertical height. A simple form of the instrument is shown in Fig. 14. It consists essentially of (a) a vertical scale engraved on, or attached MEASUREMENT OF LENGTH. 23 to a vertical metal rod, and (b) a telescope carried by a slide, which can be moved up and down the rod. This slide can be clamped at any point on the rod, and its position on the scale can then be read accurately by means of a vernier which moves with it over the scale. The telescope is generally mounted on the slide in such a way that it can be rotated round the rod as axis, and also round a horizontal axis, so that its line of sight can be elevated or depressed as required. It carries a spirit level in order that it may, when required, be set in a horizontal position. The metal rod or pillar which carries the telescope is usually mounted on a heavy base pro- vided with levelling screws, in order that it may be possible to adjust the rod and scale in a vertical position. In order to obtain accurate readings with the instrument, it is essential (1) that the scale rod should be vertical; and (2) that in any given measurement, or set of measurements, the axis of the tele- scope should make the same angle with the horizontal at all points on the scale at which readings are taken. The first adjustment is readily made by adjusting the levelling screws of the base until the tele- scope, after being set horizontal in any position, remains horizontal throughout a complete revolution round the rod as Fig. 14. 24 GENERAL PHYSICS. axis. The second condition is ensured if the telescope is horizontal in all positions, but it is evidently fulfilled (when the scale rod is vertical) for any position of the telescope relative to the horizontal, so long as the adjustments of the telescope remain unchanged throughout the readings. When the cathetometer is properly adjusted the vertical distance between any two points is readily measured by adjust- ing the telescope until one of the points is seen at the inter- section of the cross wire in the field of view, and then raising or lowering the telescope on the scale (without altering its adjust- ment on the slide) until the second point is seen in the same way. The difference of the readings of the vernier on the vertical scale for these two positions, then gives the required vertical distance between the points. It must be remembered that a good cathetometer is a very complicated instrument, and the full details of its construction and the methods of adjusting it and using it can only be learnt by practice in a laboratory. 15. Measurement Of Area. The unit of area in all physical measurements is the square on the unit of length as side. Thus, if the foot, inch, or centimetre is the unit of length, then the square foot, the square inch, or the square centimetre is the corresponding unit of area. As the centimetre is the unit of length generally adopted in all physical measurements, the square centimetre is the unit of area in most general use. When the unit of area is derived in this way from the unit of length, it is evident that the multiples and sub-multiples of the unit of area can be derived from the corresponding multiples and sub-multiples of the unit of length. Thus, since the square foot is the square on a side 1 foot or 1 2 inches long, it must contain 12 2 or 144 square inches. Similarly, since 1 centi- metre = 10 millimetres, we must have 1 sq. cm. = 10 2 sq. mms. = 100 sq. mms.; MEASUREMENT OF AREA. 25 and, in the same way, 1 square decimetre = 10 2 sq. cms. = 100 sq. cms. and, 1 square metre = 100 2 sq. cms. = 10,000 sq. cms. The numerical relations between the square foot or the square inch, and the square cm. can be derived from the linear relations already given. The following equivalents may, however, be given : 1 sq. foot = 929-04 sq. cms. 1 sq. in. = 6-4517 1 sq. cm. = '155 sq. inch. It is convenient to remember that 31 square inches is almost exactly equal to 200 square centimetres. The measurement of the area of any regular figure resolves itself into measurement of length. The necessary dimensions of the figure are measured by some suitable method of length measurement, and the area is calculated by the appropriate rule in mensuration. The area of any irregular figure can be found approximately by transferring the figure to squared paper, and counting the number of small squares of known area which are enclosed by it. It can also be found with fair accuracy by cutting out the figure in thin foil or cardboard of uniform thickness, and then corn- paring the weight of this piece of foil or cardboard with the weight of a known area of the same material. The area of an irregular plane figure can be measured accurately by means of a mathematical instrument known as the Planimeter. The theory and construction of this instrument are, however, beyond the scope of this work, and cannot here be considered. 16. Measurement of Volume. The unit of volume adopted in physical measurement is the cube on the unit of length as edge. Thus if the foot, inch, or centimetre is taken as the unit of length, the cubic foot, the cubic inch, or the cubic centimetre is the corresponding unit of volume. 26 GENERAL PHYSICS. The centimetre being the general unit of length the cubic centimetre is the unit of volume in general use in all physical measurements. When the unit of volume is derived in this way from the unit of length the multiples and sub-multiples of the unit of volume can be derived, as in the case of the units of area, from the cor- responding multiples and sub-multiples of the unit of length. Thus, since the cubic foot is a cube of 1 foot or 1 2 inches edge, it must contain 12 3 or 1,728 cubic inches. Similarly, since a cubic centimetre is a cube of 1 centimetre or 10 millimetres edge it must contain 10 3 or 1,000 cubic millimetres. In the same way we have 1 cubic decimetre = 10 3 cub. cms. = 1,000 cub. cms., or, 1 cubic metre = 100 3 cub. cms. = 1,000,000 cub. cms. The relative magnitude of the cubic inch or the cubic foot and the cubic centimetre is conveniently expressed by the following equivalents 1 cub. inch = 16 '388 cub. cms. 1 cub. era. = '06102 cub. in. From these values it will be seen that 1,000 cub. cms. is only very slightly greater than 61 cubic inches. The cubic centimetre is also the unit of capacity generally used in measuring the volume of a liquid or a gas in all scientific measurements. The litre, the unit of capacity adopted in the metric system of units, is a cubic decimetre or 1,000 cubic centimetres. It is the unit in which large volumes of a liquid or a gas are generally expressed. The English units of capacity are seldom used in scientific measurements. The gallon is the volume occupied by 1 pounds of pure water at 62 F., and is equal to 277*274 cubic inches. MEASUREMENT OF VOLUME. 27 The sub-multiples of the gallon, the pint, and the quart are defined by the relation 1 gallon = 4 quarts = 8 pints. The fluid ounce is the volume occupied by one ounce of pure water at 62 F., and is, therefore, the T ^ part of a gallon. It follows from this that a pint is equal to 20 fluid ounces. The measurement of the volume of any regular solid resolves itself, as in the measurement of area, into measurement of length. The necessary dimensions of the solid are measured by some suitable method of length measurement, and the volume of the solid is then calculated by the appropriate rule of mensuration. The volume of an irregularly shaped body can be measured by measuring the volume of water which it displaces, or, much more accurately, by determining its apparent loss of weight when weighed in water, or in some liquid in which it is insoluble, as explained later. . The volume of a liquid is generally measured by means of a graduated measuring vessel. Some of the vessels in common use for this purpose are shown in Figs. 15 and 16. The capacity, or internal volume, of any vessel or tube is generally found by finding the weight of the quantity of water or mercury which exactly fills it and then deducing the required volume from this weight, as explained in a later article. 17. Measurement of an Angle. The unit of angular measurement is derived from the right angle. The degree, which is the TTO P ai> t of a right angle, is the unit generally adopted in the measurement of angles. The subdivisions of the degree are the minute, and the second, a degree being divided into 60 minutes and a minute into 60 seconds. That is 60 seconds = 1 minute. 60 minutes = 1 degree. 28 GENERAL PHYSICS. CCm I Figs. 15 and 16. MEASUREMENT OF VOLUME. 29 or, in the usual notation, 60" = 1' 60' = 1 The usual method of measuring an angle in practice is by means of a divided circular scale. The arc of a circle is propor- tional to the angle which it subtends at the centre of the circle. so that by subdividing the circumference of a circle into 360 equal parts we subdivide the four right angles at the centre of the circle into 360 degrees. A circular scale for the accurate measurement of angles is generally divided to show divisions less than a degree, and is also provided with a vernier reading to some convenient fraction of a scale division. A common form of scale, for example, is divided into 20' divisions, and carries a vernier reading to V of a division. With this scale an angle at the centre of the scale could be measured to the nearest minute. For theoretical purposes an angle is frequently measured by the ratio of the arc which it subtends at the centre of a circle to the radius of the circle. This ratio gives what is called the circular measure of the angle. Since the ratio, ^ , which gives the circular measure of ' radius' an angle is of unit value when the arc is equal to the radius, it follows that the unit of circular measure is the angle which is subtended at the centre of the circle by an arc equal in length to the radius. This unit is called a radian. A right angle at the centre of a circle is subtended by an arc equal to one-fourth of the circumference. If r denote the radius of the circle, then 2irr, where TT = 3*1416, is its circumference, and is one-fourth of the circumference. The circular measure a of a right angle is, therefore, given by the ratio r or , 2 / 2 30 That is, or or GENERAL PHYSICS. - radians = 90, - - 90 x f> - 1 radian = - - = 57'2958 ( 7T 1 radian = 57 17' 44-9". It should be noted that an angle is not a physical quantity. It is merely a number, the ratio of two lengths, and its measure- ment is, in practice, essentially a measurement of length. 31 CHAPTER IV. MEASUREMENT OP TIME. 18. Units Of Time. The standard unit of time is derived from the period of rotation of the earth on its axis. The interval of time that elapses between two successive transits of a fixed star across the meridian of any place is almost exactly equal to the period of time in which the earth makes one complete revolution. The star is so distant that the direction in which it is seen from the earth is practically the same from all points on the earth's orbit round the sun, and the period between two successive transits of the star is, therefore, practically equal to the period of the earth's rotation on its axis. This period of time is known as a sidereal day and is the astronomical unit of time. The interval of time that elapses between two successive transits of the sun across the meridian of a place is not, however, equal to the time of one complete revolution of the earth on its axis, and is found also to vary from day to day. One reason for this is indicated in Fig. 1 7. Let A, B, and C represent the positions of the earth in its orbit round the sun, S, at noon on three consecutive days, at the place marked by the small arrow in the figure. If we consider the direction of the earth's rotation on its axis, as shown in the figure, we can see that the interval of time that elapses between the instant the earth is at A, with the small arrow pointing towards S, and the instant it is at B, with the small arrow again pointing to S, is the time 32 GENERAL PHYSICS. taken by the earth in rotating through one complete revolution and the angle ASB. Similarly, the interval between the positions B and C is the time in which the earth rotates through a complete revolution and the angle BSC. Now the angles ASB and BSC are not equal, for the earth moves with variable velocity in an elliptical orbit round the sun, and the line joining it to the sun does not sweep out equal angles in consecutive days. It follows from this that the interval of time which elapses between two successive transits of the sun across the meridian of a place, or^ in other words, the interval between noon by the sun on two successive days at any place is greater than the time in which the earth makes one complete revolution on its axis and varies from day to day throughout the year. Another cause of variation which pro- duces a similar effect to a smaller degree is the inclination of the earth's axis to the plane of its orbit round the sun. If, however, we take the average or mean value of this interval for a complete year, we get a definite interval of time known as the mean solar day. B The mean solar day is divided into 24 hours, the hour into 60 minutes, and the minute into 60 seconds. A mean solar second is, therefore, the g-g-.Vw P art of a mean solar day. This unit, the mean solar second, is the unit of time generally adopted in all physical measurements. The sidereal clay is obviously less than the mean solar day, and is calculated to be equal to 23 hours 56 minutes 4'09 seconds of mean solar time. 19. Instruments used for the Measurement of Time. The instrument generally used for the measurement of time is a clock or watch. MEASUREMENT OF TIME. 33 The mechanism and construction of a clock or watch cannot be considered here in any detail.* The following general points may, however, be noted. The mechanism of any instrument of this kind consists of four essential parts (a) the mainspring or weights, from which the motive power is derived; (ft) the pendulum or balance wheel, which determines by its motion the rate at which the mechanism moves; (c) the escapement, by which the pendulum or balance wheel is maintained in motion and is able, at the same time, to con- trol and regulate the action of the motive power ; and (d) the train of wheels by which the indicating hands are rotated. A simple form of escapement, known as the dead beat escapement, is shown in Fig. 18. In this figure the essential parts of the mechanism are easily distinguished. It is so constructed that the pendulum receives the successive impulses which maintain it in motion once during each swing, at the instant when it is at the middle point of its swing. These impulses are communi- cated by the teeth of the escapement wheel through the crutch and pendulum fork directly to the pendulum, and as they are communicated only when the pendulum is at the middle point of its swing, they have no disturbing effect on the time of swing. The teeth of the escapement wheel are thus allowed to " escape " from the pallets of the crutch at regular equal intervals, which are determined by the time of swing of the pendulum. An accurately made clock controlled by a pendulum, and made to go with practically perfect regularity, is called a standard clock. It may be regulated to indicate astronomical time with * See Ball's Experimental Mechanics. Lecture xx. Fig. 18. 34 GENERAL PHYSICS. 24 hours to the sidereal day, or mean solar time, in the usual way. An accurately made clock or watch controlled by a balance wheel, and constructed to go with the most perfect regularity possible, is called a chronometer. It is the instrument generally used in navigation and is usually regulated to indicate mean solar time. The principle involved in the regulation of the motion of a clock or watch by means of a pendulum or balance wheel is described in Art. 43. It is there explained that any body in vibratory motion executes each complete vibration, or each complete to-and-from movement, in the same time. This characteristic of vibratory motion, known as isochronism, is the fundamental principle of clock and watch construction. The period of vibration of the pendulum or balance wheel determines and regulates the rate at which the clock goes. The same principle is applied in the use of a tuning-fork for recording and measuring very short intervals of time. If the prong of a tuning-fork in vibration makes n complete vibrations per second it makes every complete vibration in ^ of a second, and if the vibrations are recorded in any suitable way they can be used to measure very short intervals of time. The chronograph, shown in Fig 1 9, is a simple form of apparatus in which a tuning-fork is used in this way to record and measure short intervals of time. A light metal style is attached to the end of one of the prongs of the fork, and the fork is mounted so that the tip of the style rests lightly on the surface of a sheet of smoked paper rolled round the large drum shown in the figure. This drum is rotated by hand or by clockwork, and the point of the style traces a line on the smoked surface on which it rests as the drum rotates. If the fork is not in vibration this line is a straight line, but if it is in vibration the line has a characteristic wavy form, in which the length of each wave corresponds to the period of a complete vibration. MEASUREMENT OF TIME. 35 The line thus traced on the smoked surface may evidently be used for measuring the interval of time between any two instants if the points on the trace corresponding to these instants can be marked. This might be done by arranging for the drum to receive a sudden small displacement parallel to its axis at the instants to be marked. While the fork is in vibration the displacements thus produced in the wavy trace on the smoked paper could readily be detected, and the interval of time between the instants at which the displacements were made would be Fig. 1 9. Chronograph. given by the number of complete waves in the portion of the trace between the displacements. The method in general use for marking the trace on a chrono- graph is, however, an electrical one which is simpler and more satisfactory in practice than any mechanical method. The drum and the tuning fork are connected to the secondary terminals of an induction coil, so that a spark passes between the point of the style and the drum whenever the primary circuit of the coil is 36 GENERAL PHYSICS. made or broken. This spark produces a small white mark on the smoked paper, and in this way the trace made by the style can be marked at any instant by merely making or breaking the primary circuit. When the frequency of the fork is high, very short intervals of time can in this way be recorded and measured. Thus, if the fork has a frequency of 500 vibrations per second, each complete wave of the trace on the smoked paper corresponds to 0*002 of a second, and a portion of the trace, estimated to contain 8 '3 complete wave lengths, would correspond to an interval of - 0166 of a second. The frequency of the tuning-forks used can readily be standardised by arranging for the pendulum of a standardised clock beating seconds to close the primary circuit of the induction coil at a certain instant in every beat, while the fork records its trace on the smoked paper. The interval between the marks on the trace will now correspond exactly to one second, and the average number of waves in this interval on the trace can readily be determined by counting them for several successive seconds.* This number is evidently the number of complete vibrations made by the fork in one second. * The drum is mounted on a screw as axis, so that it moves parallel to its length as it rotates. By this arrangement a very long spiral trace may be taken without any overlapping. 37 CHAPTER V. MEASUREMENT OF MASS. 20. Mass.* The mass of a body may be defined as the quantity of matter it contains. This definition is somewhat unsatisfactory from some points of view. The limited meaning which must be attached to the words, " quantity of matter," will be better understood at a later stage. It must be noted, however, that the mass of a body is a quantity which can be changed only by changing the quantity of matter in the body. Thus, if we add matter of any kind to a body we increase its mass, and if we take away any portion of the body we decrease its mass. Hence, if the mass of a body is found to change, it must be inferred that the body has gained or lost matter, and the quantity of matter gained or lost is measured by the change of mass. 21. Weight.* The fact that all bodies with which we have to deal possess weight is familiar to us from everyday experience. A body is said to be heavy or light according as its weight is great or small. It will be understood later that the weight of a body is due to the attraction exerted by the earth on the body. It is a general property of matter that any two pieces of matter mutually attract each other. The very large piece of matter which makes up the earth, and the small piece of matter which makes up the body, therefore attract each other mutually, and the force with which the earth pulls the body towards it is the weight of the body. * See also Arts. 38 and 39. Recent speculations as to the nature of mass need not here be considered. 38 GENERAL PHYSICS. The weight of any body may, therefore, be defined as the downward pull which the earth exerts on that body. The direction of the pull is vertically downwards, towards the centre of the earth, so that when a body falls freely the line along which it falls is a vertical line. The magnitude of the pull depends upon the mass of the body and upon its distance from the centre of the earth. In comparing mass and weight, it will be seen that the mass of a body is essentially constant, and cannot be changed without adding to or taking from the matter in the body. The weight of the body, on the other hand, depends upon its position relative to the earth, and changes as the distance of the body from the centre of the earth changes. The weight of a body, for example, is less at the equator than at the poles, and is found to increase slightly as the latitude of the place at which it is measured increases. This shows that the weight decreases as the distance from the centre of the earth increases. For the same reason the weight of a body decreases as its height above the sea level increases. 22. Units Of Mass. The standard unit of mass in England is the pound. The pound (avoirdupois) is denned as the mass of a piece of platinum, which is preserved with other standards at the Standards Office of the Board of Trade. Of the numerous multiples and sub-multiples of the pound we need only notice here the ounce and the grain. The ounce is the sixteenth part of a pound, and the grain is the seven- thousandth part of a pound, so that we have 1 pound =16 ounces = 7,000 grains, or, in the usual abbreviated notation, 1 Ib. = 16 ozs. = 7,000 grs. The standard unit from which the gramme of the metric system is derived is the standard kilogramme. This is the mass of a piece of platinum kept at the Bureau des poids et des MEASUREMENT OF MASS. 39 inesures in Paris, and known as the kilogramme des archives. It was originally constructed by Borda to be equal to the mass of a cubic decimetre of pure water at 4 C., but later measure- ments show that it is slightly greater than this mass. The derivatives of the standard kilogramme in general use are the gramme and its sub-multiples the decigramme, centigramme, and milligramme. Thus, we have 1 kilogramme (kg.) = 1,000 grammes. 1 gramme (grm. ) = 10 decigrammes = 100 centigrammes (cgs.). = 1,000 milligrammes (mgs. ). In physical measurements it is the general practice to express all masses in grammes, or, if they are very large, in kilo- grammes. Thus, we may have 896*423 grammes, or 126'432 kilogrammes. The mass of a cubic decimetre, or 1,000 cubic centimetres of pure water at 4 C., although not exactly equal to a kilogramme or 1000 grammes, is very nearly equal to this mass. A gramme is therefore almost exactly the mass of one cubic centimetre of pure water at 4 C. The mean result of recent measurements gives the exact mass of a cubic centimetre of water at 4 C. to be 0-999955 gramme. The relative magnitude of the pound and the gramme is given by the following equivalents. 1 kilogramme = 2 '2046 pounds. 1 gramme = '0022 pound = 15 '432 grains. 1 pound = 453'59 grammes. For ordinary purposes it is convenient to take 1 kilogramme = 2'2 pounds, and 1 pound = 453*6 grammes. The equivalent, 1 gramme = 15'432 grains, is easily remembered. 23. Comparison of Mass by the Process of Weighing-. If we are provided with a measuring scale derived from the standard of length and showing all necessary multiples and 40 GENERAL PHYSICS. sub-multiples of the unit, we have no difficulty in measuring any unknown length by means of the scale. The process by which the measurement can be made is obvious, and easily understood. If, however, we are provided with a copy of the unit of mass, and with all necessary multiples and sub-multiples of the unit, and are required to measure the mass of a given body in terms of the unit, we cannot proceed with the measurement without some understanding as to the process to be adopted in comparing the masses. The process generally adopted in comparing any two masses is that of weighing. This process is based on the understanding that the weight of a body is directly proportional to its mass, and that bodies which are equal in weight are therefore equal also in mass. The adoption of the process of weighing for the comparison of masses may be looked upon as an extension of the definition of mass. It implies that the mass of a body, or " the quantity of matter" in a body, is a quantity which is directly proportional to the force of attraction which the earth exerts on the body. The instrument by which masses are compared by the process of weighing is called a balance. A simple form of balance is shown in Fig. 20. It consists essentially of a horizontal beam balanced centrally on a knife-edge, and carrying a scale-pan suspended from each end of the beam at points equidistant from the central knife- edge. The beam is thus balanced as a horizontal lever on the central knife-edge as a fulcrum, and the scale-pans are suspended one on each side of the fulcrum, at the ends of equal arms. From this construction it is obvious that if the body whose mass is to be determined is placed in one pan, and multiples and sub-multiples of the unit of mass are placed in the other pan until the beam is exactly balanced in equilibrium on its knife- edge, the weight of the mass in one pan must be exactly equal MEASUREMENT OF MASS. 41 to the weight of the mass in the other pan; for these two weights act against each other on the lever, at points equidistant from the fulcrum, and can, therefore, balance each other exactly only when they are equal. The balance thus indicates when the weights of the masses in the scale pans are equal, and it is understood as the basis of the process of weighing, that the masses are equal when their weights are equal. If, therefore, we sum up the multiples and sub- multiples of the unit of mass which must be placed in one scale Fig. 20. Balance. pan to balance any given body in the other pan, we get the measure of the mass of the body, in terms of the unit of mass. The theory of the balance, and the details of its construction, will be dealt with more fully at a later stage. It may, however, be noted here, in anticipation, that the central knife-edge, made of steel or agate, rests upon a small plate or plane of steel or agate, and that the pans are suspended from the beam by means of an inverted stirrup arranged as shown in Fig. 20, so that a 42 GENERAL PHYSICS. small steel or agate plane in the stirrup rests on a knife-edge let into the upper edge of the beam. In this way, friction at the fulcrum is reduced to a minimum, and the length of an arm is determined definitely a% the distance between two knife-edges. In all well constructed balances the arms are exactly equal, and the three knife-edges on the beam are parallel and in the same plane. The plane should be horizontal when the beam is in such a position that its centre of gravity is vertically below the central knife-edge. This position is evidently the position in which the beam comes to rest of its own accord, if it is allowed to swing freely on the central knife-edge after removing the pans. It is sometimes called the zero position of the beam. The weights of the scale pans must evidently be exactly equal that is, the beam must balance in its zero position, with its centre of gravity vertically below the central knife-edge, when the pans are empty. This condition must be fulfilled before it can be said that masses in the scale pans are equal when the beam is balanced in its zero position. In order to indicate the position of the beam when swinging on its central knife-edge, a long thin pointer is attached to it as shown in Fig. 20. This pointer moves over a scale fixed at the base of the pillar supporting the beam, and is so adjusted that it points to the zero of the scale when the beam is in its zero position. The central pillar which supports the beam can, in most balances, be raised and lowered by means of a small eccentric cam fixed at the base of the pillar, and worked by the small handle shown at the front of the base board in Fig 20. When the pillar is raised the beam swings freely on the central knife- edge, and the balance is ready for use, but when the pillar is lowered the beam rests on supports provided for the purpose. In this way, the central knife-edge and the plane on which it rests are protected from wear when the balance is not in use. The masses used as standards in the process of weighing are MEASUREMENT OF MASS. 43 multiples and sub-multiples of the standard unit, or some con- venient derivative of it. They are usually called " weights," because they are used in " weighing," and are generally arranged in sets on a definite plan. The arrangement usually adopted is indicated in Fig. 21, which shows a box containing a set of weights for weighing masses up to 100 grammes. In this set the gramme is the unit of mass, and the multiples and sub-multiples provided include : Fig. 21. 100-gramme box weights. 50, 20, 20, 10 \ 5 2 2 1 '5 2 2 1 > Grammes. 05 02 02 01 ( 005 002 002 00 1) 50, 20, 20, ! Grammes. 5, Z 2, 1 J 5, 2 2, 1, Decigrammes. 5, 2,' "I 1, Centigrammes. 5, 2, 2, 1, Milligrammes. It will be seen that with this set of "weights," it is easy to- 44 GENERAL PHYSICS. make up to the nearest milligramme any desired mass up to 1 11*111 grammes. The gramme "weight" and all "weights" of higher denomina- tion are generally m^e of brass, and are often gilt or nickel plated to preserve them. The sub-multiples of the gramme are usually made of thick sheet platinum or aluminium. It will be seen later that the milligramme " weights " are not much used. Milligrammes, and tenths of a milligramme, are more conveniently determined by using a rider on the beam of the balance as explained in Chapter xv., where the construction and use of the balance are more fully dealt with than in this article. 45 CHAPTER VI. VELOCITY. 24. Motion of a Material Particle. Motion may be defined as change of position; when a body is changing its position it is said to move or to be in motion. The motion of a body is more conveniently studied if the motion of a material particle is first considered. A material particle is a particle of matter reduced to such small dimensions that the space which it occupies is reduced practically to a point. The position of a particle may, therefore, be indicated by a point, and as the particle moves from point to point it traces out a line. When a particle moves from one point to another the line which it follows is called a path of motion between the two points, and the straight line joining the two points is the displace- ment of the particle. 25. Motion of an Extended Body. An extended body that is, a body whose dimensions extend beyond those of a particle may be supposed to be made up of an infinite number of material particles. In what follows the relative positions of the particles of the body are supposed to be fixed, so that the size and shape of the body remain unchanged. That is, the body is supposed to be a rigid body, and to be, as such, incapable of .any change of configuration. An extended body is capable of two distinct kinds of motion. When a body moves in such a way that any straight line in it 46 GENERAL PHYSICS. remains parallel to its initial position throughout the motion, the motion of the body is said to be motion of translation. When a body moves in such a way that it rotates round a fixed axis, and the pa^hs of motion of all particles in it are circles round points on the axis as centres, the motion is known as motion of rotation. When a body moves in any way it can be shown that its motion at any instant is compounded of a simple motion of translation and a simple motion of rotation, as defined above. When a body is displaced from one position to another by simple translation the displacement is measured by the displace- ment of any particle in the body. In the case of simple motion of rotation the displacement of a body is measured as angular displacement. It is given by the angle through which the line joining any point in the body to the centre of its circular path of motion, rotates. It may be seen that the displacement of a particle or a body is a quantity which possesses direction as well as magnitude ; that is, it cannot be specified completely without giving its direction as well as its magnitude. 26. Velocity. When a particle is in motion it takes time in passing from point to point in its path, and a consideration of the time taken in passing over the distance between any two points introduces the idea of time rate of motion, or the distance passed over per unit of time. The rate of motion of a particle is generally called the velocity of the particle. The general unit adopted for the measurement of velocity is derived from the units of length and time, and the unit of length per unit of time is taken as the unit of velocity. That is, if the foot and second be taken as the units of length and time respec- tively, the foot per second is the corresponding unit of velocity ; or, if the centimetre and second are the units of length and time, the centimetre per second is the corresponding unit of velocity. VELOCITY. t 47 Hence, if a particle moves over a distance s units in a time t units, the average magnitude of its velocity for the time con- sidered is units of velocity. In the same way if a particle, in a very short time T, taken so as to include a particular instant in its time of motion, passes over a very short distance , the average magnitude of the velocity for g this short interval of time is given by -, and may be taken (if T is small enough) as the velocity of the particle at the particular instant considered. This idea of the velocity of a particle at a particular instant in its time of motion may also be presented as the velocity of a particle at a particular point in its path of motion. Thus, if a particle passes over a very short distance , taken so as to include a, particular point in its path of motion, in a very short time T, the average value of the velocity of the particle for this very short g time is given by -, and may be taken (if is small enough) as the velocity of the particle at a particular point in its path. For example, if a particle at a particular point in its path passes over a hundredth of a centimetre (taken so as to include the point) in a thousandth of a second, its velocity at that point cannot differ much from 10 cms. per second, the average magnitude of the velocity for the interval taken. It will be understood from what has been said above that if S and r are both infinitely small, s the limiting value of the ratio when 8 and T are both infinitely small gives the velocity of the particle at the instant at which T vanishes or at the point at which S vanishes. The direction of the velocity of a particle at any instant is the direction in which the particle is moving at that instant. The velocity of a particle is said to be of uniform magnitude when the particle passes over equal distances in equal times, no 48 GENERAL PHYSICS. matter whether the times be long or short. That is, the velocity is of uniform magnitude if the distance passed over by the particle in any given time is directly proportional to the time. Hence, if a particle moves witlyi velocity of uniform magnitude v for a time t. and the space passed over by the particle be denoted by 5, we have s = vt. When the velocity of a particle is not of uniform magnitude it is said to be of variable magnitude. From what has been said it will be understood that velocity is a vector quantity possessing direction as well as magnitude. It is necessary, therefore, in order to specify a velocity com- pletely to give both its magnitude and its direction. It follows, too, that the velocity of a particle is constant and invariable only if its direction as well as its magnitude remain unchanged. That is, if the velocity of a particle remains unchanged it must be of uniform magnitude, and the particle must move along a straight line. When this is the case the velocity of the particle is pro- perly called uniform velocity. In any other case the velocity is variable velocity. It has been proposed to use the term speed when velocity is considered as a magnitude only, without reference to direction. A velocity of uniform magnitude could then be spoken of as a uniform speed. It seems, however, unnecessary and undesirable to add to the number of terms already in use in this subject. When the velocity of a particle is variable the space covered in a given time cannot be determined by the simple relation given above. If the velocity of the particle at any instant be denoted by v the space passed over in a very short time, T, taken so as to include the given instant, differs very little from VT, and the shorter the time r the more nearly does VT give the space covered in that time. Hence, if a particle moves with variable velocity for a time, t, and we suppose this time to be divided into a very large number, n, of very small equal intervals, each VELOCITY. 49 denoted by r, then when n is large enough, the space covered by the particle in the time t is given by s = VIT + V 2 r + v 3 r . . + v n r, where v v 2 , v 3 v n denote the velocities of the particles at the middle instants of the short intervals taken in order from the first to the %th, or last. That is, s (# t -f i\ 2 -j- v 3 . . . v n )r. But T -, n (V, + V + V ' V n) and, therefore, s . t. n ft) I _, ft] i- -I- ffi Q) Now, if n is infinitely great, - - is the average velocity of the particle during the time t. If this average velocity is denoted by v, we have s = vL The investigation given above may be put in a graphical form which is more easily followed and leads to a more definite result. If a curve be plotted so that the ordinate at any point represents the velocity of the particle at the instant indicated by that point, and the abscissa represents the time of motion measured from a particular instant as starting point, the curve will show how the velocity of the particle varies from instant to instant during the time of motion. Let CD, Fig. 22, be a velocity curve plotted in this way for a particular case, and consider the motion of the particle for the very short interval of time represented by ab. The velocity at the beginning of this interval is represented by ac and at the end of the interval by bd. If the velocity remained constant throughout the interval at the value it has at the beginning of the interval, the space described during the interval would be represented by the area of the rectangle abce. The number that 4 50 GENERAL PHYSICS. measures the length of etc is the same as the number which measures the velocity of the particle at the instant indicated by the point a, and the number that measures the length of ab is the same as the number that measures the duration of the interval of time it represents ; the product of these two numbers is, therefore, the number that measures the area of the rectangle abce contained by ac and ab, and also the space passed over by a particle moving with a uniform velocity represented by ac for a time represented by ab. This space is, therefore, represented by the area of the rectangle in the usual way; that is, the number that measures the one is the same as the number that measures the other. o A a b B x Time Fig. 22. Similarly, if the velocity of the particle were the same throughout the interval as it is at the end of the interval, the space passed over during the interval would be represented by the rectangle abdf. Now the space actually passed over during the interval must be less than that represented by abdf and greater than that represented by abec, and the difference between each of these extreme values and the space represented by the strip abdc can be made as small as we please by making ab small enough. Hence, when ab is small enough the space actually passed over by the particle in the short time represented by ab is represented VELOCITY. 51 by the area of the strip abed which stands on ab as its base, and is bounded along cd by the portion of the curve intercepted between the ordinates ac and bd which form its sides. It follows from this that if we consider the motion of the particle for any finite time represented by AB, the space passed over in that time must be represented by the area ABDC which lies between the ordinates AC and BD, and is bounded by the curve along CD. The time represented by AB may be divided into a very large number of small equal intervals similar to that represented by ab, and if AB be divided into a corresponding number of short equal lengths, the space passed over in each interval is represented by the area of the strip similar to abec which stands on the length which represents it. The total space passed over must, therefore, be represented by the area ABDC, which is made up of all the strips which stand on the short lengths into which AB is divided. It will be seen that if AE, the height of the rectangle ABFE, shown 'in Fig. 22, is such that the area of the rectangle is equal to the area ABDC, then AE is the mean or average of the ordinates between A and B and represents the average velocity denoted by v above. When a body moves with motion of translation, the velocity of the body at any instant is the velocity of any particle in it at that instant. Since velocity is a vector quantity, velocities may be com- pounded or resolved by any of the rules for compounding or resolving vector quantities. The application of the parallelo- gram and triangle rules to the composition and resolution of velocities, gives rise to the theorems known as the parallelogram of velocities and the triangle of velocities. These theorems are merely the statement of the general rules for vectors with specific relation to velocities. Numerical Examples. 1. A bullet is projected horizontally from the top of a tower, and as it falls its velocity at a certain instant is 52 GENERAL PHYSICS. known to be made up of a horizontal component 80 feet per second in magnitude, and a vertical component 64 ft. per second in magnitude. Find its velocity. Here, if we take AB, 80 units long, to represent the horizontal component, and AC, 64 units long, to represent the vertical com- ponent, and then complete the rectangle ABDC, as in Fig. 23, the diagonal AD, drawn from A to D, represents the resultant of the two components, and gives the magnitude and direction of the velocity of the bullet at the instant considered. If the figure is drawn carefully to scale, and the length of AD measured, it will be found to be about 102-4 units long. The magni- tude of the velocity is, therefore, 102-4 feet per second, and its direction is such that it makes an angle BAD with the horizontal. This angle can be measured with a protractor, and the velocity of the bullet at the instant considered, can then be fully specified. B B Fig. 23. Instead of adopting a graphical method the magnitude of the- velocity can be calculated by the relation, Rr P- given in Art. 9. We get and, therefore, R2 = 80 2 + 64-, R - 102-45. The angle BAD can also be specified mathematically. From the figure it will be seen that t = tan BAD ; that is, tan BAD = -t>A 80 = = ; or, BAD is an angle whose tangent is - The magnitude and direction of the velocity of the bullet at the given instant are thus completely determined. 2. A bullet is fired from a rifle in a direction making an angle of 30 above the horizontal. At the instant the bullet leaves the muzzle of the rifle its velocity is 1,000 feet per second, find the magnitude of the horizontal and vertical components of this velocity. VELOCITY. 53 Let AB, Fig. 24, represent the velocity of the bullet in magnitude and direction. Construct on AB as diagonal the rectangle ACBD with the side AC horizontal and the side AD vertical. The sides AC and AD now represent, respectively, the horizontal and vertical components of the velocity represented by AB. The lengths of AC and AB can now be found in several ways. If the figure is drawn accurately to scale the lengths can be measured directly. If this is done it will be found that AC is about 866 units long, and AD 500 units long, indicating that the horizontal component is about 866 ft. per sec., and the vertical component 500 ft. per sec. From the geometry of the figure it is easily seen that AB = ! =, or AC = AB . .N/^ = 500 V3 = 866'025. AC v/3 2 and, |g~p or AD = AB. \= 500. That is, the horizontal component is 500 V3 ft. per sec., or 866 '025 ft. per sec., and the vertical component is 500 ft. per sec. The same result is obtained more concisely and expeditiously if we apply the relation given in Art. 9. If P be taken to denote the horizontal component, and Q the vertical component in feet per second, we have at once P = R cos = 1,000 . cos 30 = 1,000. = 500 \/3, and Q = R sin a = 1,000 . sin 30 = 1,000 x i = 500. That is, the horizontal component is 500 vl ft. per sec., and the vertical component is 500 ft. per sec. 27. Relative Velocity. The velocity of a point B relative to a point A is the rate at which the point B changes its position relative to A. If the two points are in motion with the same velocity there can be no change in their relative position, and the velocity of one relative to the other is zero. If, however, the points are in motion with different velocities, the velocity of one relative to the other depends upon the magnitude and direction of their individual velocities. 54 GENERAL PHYSICS. Let the velocity of the point A in Fig. 25 be represented in magnitude and direction by AP, and the velocity of the point B by BQ. Now, the velocity of B relative to A will not be affected if we impress the same velocity on the two points. Imagine, therefore, a velocity equal and opposite to the velocity of A to be impressed on each point, and let this velocity be represented in magnitude and direction by AE for the point A, and by BE, for the point B. It will be seen from the figure that the result of this is to reduce the point A to rest, and to give the point B a velocity compounded of the velocities repre- Fig. 25. sented by BE and BQ. This velocity of B is represented in the figure by BS, and is evidently the velocity of B relative to A, for the point A is now at rest. That is, the velocity of a point B relative to another point A is a velocity compounded of B's actual velocity and a velocity equal and opposite in direction to A's velocity. It will be seen at once by drawing a figure that the velocity of A relative to B is equal and opposite in direction to the velocity of B relative to A. If the points A and B move along the same line, and if velocity is considered to be positive for one direction along the line, and VELOCITY. 55 negative for the opposite direction, the velocity of B relative to A is evidently obtained by adding A's velocity with its sign changed to B's velocity that is, by subtracting A's velocity fron^ B's velocity. What has been said above with reference to the relative velocity of two points applies also to the relative velocity of any two bodies moving with motion of translation. Examples. 1. Two bodies A and B move along the same straight line with uniform velocities ; the velocity of A is 20 ft. per sec., and the velocity of B is 15ft. per sec., find the velocity of B relative to A when the bodies move (1) in the same direction, (2) in opposite directions. Here if we take velocity in the direction in which A moves to be positive we get the following results : (1) The velocity of B relative to A, when A and B are moving in the same direction is given by (15 - 20) ft. per sec. or - 5ft. per sec. That is, B's velocity relative to A is 5ft. per sec. in a direction opposite to that in which A (and in this case B also) is moving. This means that if B is in front of A, B is getting nearer to A at the rate of 5ft. per sec., or if A is in front of B, B is getting further away from A at the rate of 5 ft. per sec. (2) The velocity of B relative to A when A and B are moving in opposite directions is given by ( - 15 - 20) ft. per sec. or - 35ft. per sec. That is B's velocity relative to A is 35ft. per sec. in a direction opposite to that in which A is moving. If we suppose the two bodies A and B to be two trains, the velocity of B relative to A obtained as above is the velocity which the train B appears to have to a passenger on A who looks only at the train B. 2. A steamer, A, travelling due north at a speed of 15 knots passes another steamer, B, travelling due east at a speed of 20 knots. Find the velocity of the steamer B relative to the steamer A. The velocity of the steamer B relative to the steamer A is the velocity obtained by compounding B's velocity with a velocity equal and opposite to that of A. Hence, if BE in Fig. 26 represents B's velocity in magnitude and direction, and BS similarly represents a velocity equal and opposite to A's velocity. BR will, by the parallelogram of velocities, represent the velocity of B relative to A. Since BE and BS are respectively 20 units and 15 units in length, and the angle EBS is a right angle, it follows that BR 2 = BE 2 + BS 2 . 56 That is, or GENERAL PHYSICS. BR 2 = 20 2 + 15 2 = 625 BR = 25. The velocity of B relative to A is therefore 25 knots in the direction represented by BR. This direction lies between S.E. and E.S.E., making an angle of nearly 37 with BE. That is, to a passenger on board the steamer A, the steamer B appears to move away from him in a direction between SE and ESE with a speed of 25 knots. It will be understood that all the velocities with which we Fig. 26. have to deal are relative velocities. When we speak of the velocity of a body we mean its velocity relative to some point at rest on the earth's surface. Motion and rest are in fact relative terms, and the use of either term implies the existence of some point of reference expressed or understood. We cannot specify the absolute position of a point in space, and cannot therefore attach any real meaning to the terms " absolute velocity " and " absolute rest " which are sometimes used. 57 CHAPTER VII. ACCELERATION. 28. Acceleration. When the velocity of a particle changes, the time in which any given change takes place depends upon the time rate at which the velocity changes from instant to instant. The rate of change of velocity, or the change of velocity per unit of time is called acceleration. The unit of acceleration is unit change of velocity per unit of time. Thus, if a foot-per-second is the unit of velocity, a change of velocity at the rate of a foot- per-second per second is the corresponding unit of acceleration. Similarly, if a centimetre-per- second is the unit of velocity a change of velocity at the rate of a centimetre-per-second per second is the corresponding unit of acceleration. Acceleration as here denned evidently applies to rate of decrease of velocity, as well as to rate of increase of velocity. Sometimes the term acceleration is limited to the rate of increase of velocity, and the term retardation is adopted for rate of decrease of velocity. It is, however, more convenient to use only the one term acceleration, and to consider retardation as negative acceleration. In determining change of velocity it must be remembered that velocity is a vector quantity, and that a change of velocity may involve a change of direction as well as a change in magnitude. Change of velocity must, therefore, be found by applying the rule for finding the difference of two vector quantities. 58 GENERAL PHYSICS. It is convenient, however, to consider separately the two cases in which change of velocity involves (1) a change in magnitude without change of direction, and (2) a change in direction with or without a change of magnitude. The first case, in which the velocity of a particle changes in magnitude without changing in direction, is evidently a special case in which the particle moves without change of direction along a straight line. In this case the velocity at any instant is always in the same direction, and the difference of the velocities at the beginning and end of any interval of time gives the change of velocity in the time. Hence if u denote the velocity of the particle at any instant, and v the velocity at an instant t units of time later, the change of velocity in the time t is given by (v u), and the average rate of change of velocity, or average change of velocity per unit of time during this time is given by -. That is, - is the average acceleration during the t t time t. In the same way, it follows that if the velocity of the particle at any instant is , and it changes in a very short time T to u , then - - is the average acceleration for the very short interval of time r, and if T is small enough the value of - - may be taken as the acceleration of the particle at the particular instant considered. For example, if the velocity of a particle at any instant is 10 cms. per second, and a thousandth of a second later it is lO'Ol cms. per second, the change of velocity in '001 second is '01 cm. per second, and the average acceleration during this short interval is-, or 10 cms.-per-second per second. Now, the acceleration at the instant first considered cannot differ much from this value, for the interval of time taken is too short to allow of much change, and it is clear that the shorter the ACCELERATION. 59 interval is, the more exactly will the average acceleration for the interval give the acceleration at the particular instant considered. The acceleration of a particle, in this case, is said to be of uniform magnitude when equal changes of velocity take place in equal times, however long or short the times may be. That is, the acceleration is of uniform magnitude when the change of velocity in any time is directly proportional to the time. Hence, if a particle moving in a straight line is subject to an acceleration of uniform magnitude a, the change of velocity which takes place in a time t is given by at. That is, if the velocities at the beginning and end of a time t are denoted by u and v respectively, we have v u at. or, v = u + at. In this case an acceleration of uniform magnitude may be said to be a uniform acceleration, for, since there is no change of direction to be considered, the acceleration is constant and invariable in magnitude and direction. The direction of the acceleration in this case must evidently be along the line of motion, and will be in the same direction as the motion of the particle, or in the opposite direction, according as the motion is accelerated or retarded. If the motion is accelerated the velocity is increased, and the acceleration is positive in sign, but if the motion is retarded the velocity is decreased, and the acceleration is negative in sign. The second case, in which the velocity of a particle changes in direction with or without a change of magnitude, is evidently the general case in which the particle moves along any line. In this case the direction of the velocity changes from instant to instant, and the difference of the velocities at any two given instants must be determined by the rule for vectors. It will be seen, too, that the acceleration may also change in direction from instant to instant, so that we cannot find the average 60 GENERAL PHYSICS. acceleration for any interval of time by dividing the change of velocity in that time by the time. As a general rule, we can deal only, in this case, with acceleration at an instant. Thus, in Fig. 27, let AB represent the velocity of the particle, in magnitude and direction, at a particular instant, and let AC represent its velocity, similarly, at an instant a very short time later, then by the triangle rule given in Art. 8, BC represents the change in velocity during this very short interval of time. If, now, the magnitude of this change of velocity be denoted by S, and the very short interval of time in which it takes place by T, then S/r denotes the average acceleration for this short interval of time, and may be taken, if T is small enough, to give the acceleration of the particle at the instant when the velocity of the particle is represented by AB. The direction of this Fig. 27. acceleration is then represented by the direction of BC. The acceleration at any instant, obtained in this way, may evidently vary in magnitude and in direction from instant to instant; that is, the acceleration is. in general, variable. If, however, the magnitude remains constant, the acceleration is said to be of uniform magnitude, and if both magnitude and direction are constant and invariable from instant to instant, the acceleration is called uniform acceleration, as in the case considered above. A particle which is not moving in a straight line may evidently be subject to uniform acceleration, for this merely means that the velocity of the particle is changing at a uniform rate in a constant direction which is not at any instant the same as that in which the particle is moving. It should be noted that acceleration possesses direction as well as magnitude, and is, therefore, a vector quantity. Accelerations ACCELERATION. 61 may therefore be compounded and resolved by the usual rules for vector quantities. The theorems known as the parallelogram of accelerations, the triangle of accelerations, and the polygon of accelerations deal merely with the application of these general rules to the special case of accelerations. 29. Uniformly Accelerated Motion in a Straight Line. The motion of a particle along a straight line with uniform acceleration is a case of special importance. If a particle starts from rest and moves for a time t subject to a uniform acceleration a, it evidently gains a units of velocity every unit of time during the motion, and, therefore, acquires a velocity at at the end of the time. That is, if v denote its final velocity, or its velocity at the end of the time considered,. we have v = at. Similarly, when a particle moves with uniform acceleration a along a straight line, and we consider its motion for any interval of time t during the motion, it will be seen that if u denote its initial velocity, or its velocity at the beginning of the time, and * r its final velocity, or its velocity at the end of the time, we have for the velocity is increased by a units of velocity every unit of time, and, therefore, gains at units in the time t. It will be seen, too, that since the velocity of a particle moving with uniform acceleration in a straight line changes uniformly with time, its mean or average value for any interval of time is the arithmetic mean of its initial and final values for the interval considered, and is equal, also, to the actual velocity of the particle at the middle of the interval. Thus, let u denote the initial velocity of the particle at the beginning of the time t, and imagine this time to be divided into n equal intervals, each equal to - : then, if a denote the uniform acceleration to which the n 62 GENERAL PHYSICS. particle is subject, the successive velocities of the particle at successive instants, taken from the beginning of the time at equal intervals, each equal to -, are given by t 2t 3 u. u + a - , u -\- a , u -\- a . . . u + at. n n 11 These velocities form an arithmetical progression of (n + 1) terms, however large n may be, and their sum by the usual algebraic rule for the summation of a series in arithmetical progression is {u + (M + at)} ^ The mean or average of ithese velocities is, therefore, - , and if n is supposed to 2i be infinitely great, this is evidently the mean or average velocity jof the particle for the time t in the sense explained in Art. 26. The mean or average velocity of the particle for the time t is thus the arithmetic mean of u, and (u -f- at) the initial and final velocities of the particle for the time, and its value is, therefore, .equal to u + 9 , which is the actual velocity of the particle at the middle of the time. The space passed over by the particle in any given interval of time can now be readily found. For if v denote the average velocity of the particle for any time t f then the space or distance passed over in this time is given by s = t't as explained in Art. 26. If the particle starts from rest and moves with uniform .acceleration a in a straight line for a time t, the average velocity of the particle for the given time is J at, and the space ,or distance passed over in this time is given by s = (i at)t, 4 or s = |- at 2 . Similarly, if the particle at any instant has an initial velocity ACCELERATION. 63 u, and moves in a straight line with uniform acceleration a for a time /, its average velocity for the time is (u + J #0> an( ^ ^he space passed over in the time is given by or s = ut 4- 2 rt ^ 2> The graphical method explained in Art. 26 can easily be applied in this case. The velocity increases uniformly with the time of motion so that the curve showing how the velocity varies with the time is a straight line. If the time of motion is measured from rest, the straight line passes through the origin as shown in Fig. 28, where 0V represents this velocity curve. From this figure it will be seen that if OA represents a time t A Time B X Fig 28. from rest, the space passed over in this time is represented by the area of the triangle OAC that is, by |- OA . AC. Now OA represents the time t and AC represents the velocity, at, acquired by the particle in the time t, so that the space passed over in the time t from rest is given by s i . t . at, or s = -i at 2 . Similarly, it can be seen that the space passed over in any interval of time t, represented by AB, is represented by the area ABDC. Now, if CE be drawn parallel to AB and cutting BD in E, it can be seen from the figure that ABDC = ABEC + CED, and the area of ABDC = AC . AB + -J AB . DE. 64 GENERAL PHYSICS. In this relation the area of ABDC represents the space passed over in the time t, AC represents the initial velocity of the particle at the beginning of the time t, and DE represents the additional velocity, at^ acquired by the particle in the time t. Hence, it follows that s = ut + -i- . t. at, or s ut + -J af. The same result might have been obtained by writing The area ABDC = - - . AB, which indicates that _ u + (it + at) ~Y~ ~' ' or s = ut + J at'\ as before. In applying these formulae it must be remembered that the quantities denoted by s, u, v, a. and t, must be expressed in con- sistent units. That is, for example, if the centimetre and second are taken as the units of length and time respectively, the unit of velocity must be the centimetre-per-second and the unit of acceleration the centimetre-per-second per second. It must be remembered, too, that the sign of a must be taken positive or nega- tive according as the motion is accelerated or retarded. Or, more generally, if the direction in which the particle is moving along the line at any instant be taken as the positive direction, the opposite direction along the line must be taken as the negative direction, and this sign convention must be taken to apply to all the quantities involved in any formula. Numerical Example. For example, if a particle moving in a straight line subject to a uniform acceleration of 20 cms. per sec. per sec. in a direction opposed to that in which it is moving, has, at a particular instant, a velocity of 45 cms. per sec., find (a) the velocity of the particle 5 seconds later, and (ft) the distance passed over iu this interval of 5 seconds. ACCELERATION. 65 (a) Here, by applying the relation, v = u + at, we get v = 45 -j- ( - 20) 5 ; or v = 45 - 100 = - 55. That is, the particle at the end of the given interval of 5 seconds is moving with a velocity of 55 cms. per sec. in a direction opposite to that in which it was moving at the instant first considered. (b) Here, by applying the relation, a = ut + & at-, we get a - 45 x 5 + & ( - 20) 25 ; or s = 225 - 250 - -25. This result means that at the end of the 5 seconds the particle is 25 cms. from its starting point (at the beginning of the 5 seconds) measured in a direction opposite to that in which it was then moving. In the formulae given above s gives the space or distance passed over so long as the particle moves in the same direction throughout the time considered. In this case, however, the particle evidently moves for 2J seconds in the initial direction until its initial velocity is reduced to zero ; it then turns back and moves for 2| seconds in the opposite direction. It will be found, by working out the results, that during the first interval of 2| seconds the particle moves over 50 "625 cms. in the positive direction, and during the remaining 2| seconds it moves over 75*625 cms. in the negative direction, so that at the end of the 5 seconds it is (75 '625 - 50'625) cms., or 25 cms. from the starting point in the negative direction. It is sometimes convenient to eliminate t from the formulae a = \ af 2 and v = at and also from s = ut x at 2 and v u + at. In the first case we get >- 2 = a?t 2 = 2 a . | af = 2 as, or v- = 2 as ; 66 GENERAL PHYSICS. and in the second case, /;- = u* + 2 uat + a-t' 2 ; or f> 2 = if- + 2 a (ut + ^ at-) ; or v 2 = u 2 + 2 as. These results are useful when the relation between v, r, a, and s is required. When a particle moves from rest with uniform acceleration in a straight line the characteristics of the motion are concisely expressed by the two formulae v = at, . . . (1) s = J at\ . . . (2) which are given above. Fig. 29. In these formula? a is a constant, and they, therefore, indicate (1) That the velocity acquired by the particle during any time measured from the instant of starting is directly pro- portional to the time. (2) That the space passed over by the particle in any time measured from the instant of starting is directly propor- tional to the square of the time. The motion of a particle moving with uniform acceleration in a straight line may be studied experimently by the method of the following experiment : Experiment 1. Set up a long inclined plane (with a V-groove cut along its length) at a small angle with the horizontal, so that a large steel bearing ball will roll slowly down the plane along the groove. ACCELERATION. 67 A scale should be fixed on the plane, parallel to the groove, as shown in Fig. 29, so that the position of the ball can be read at any instant during its motion down the plane. The motion of the ball down the plane is a rolling motion compounded of motion of rotation and motion of translation, but the particle at the centre of the ball evidently moves in a straight line down the plane. Now adjust a metronome to tick half seconds or use a clock which ticks half seconds distinctly, and follow the motion of the ball down the plane in the following manner. Let the ball rest in the groove, supported by a small block of wood, with its centre opposite the zero of the scale. At a particular tick of the clock or metronome remove the block suddenly away from the ball so as to let the ball begin at this instant to roll down the plane. Then follow the motion of the ball down the plane, and at every 750 75" (Seconds) 20 Fig. 30. -successive tick read off its position on the scale. If this operation is repeated several times it will be found that, with practice, perfectly consistent readings may be obtained. The inclination of the plane to ,the horizontal should be small, in order that the motion of the ball may be slow. Record the readings obtained in this way and then plot from the readings a smooth curve, showing how the distance travelled by the ball from rest varies with the time of motion. A curve similiar to that shown in Fig. 30 will be obtained. If this .curve is examined it will be found that the ordinate at any point representing the distance travelled from rest, is proportional to the square of the abscissa representing the time of motion. That is, the distance travelled from rest is directly proportional to the square of the time of motion, and the centre of the ball moves down the plane 68 GENERAL PHYSICS. with uniform acceleration in a straight line. If we now apply the relation, s = 4 ^> the value of a can be determined by substituting in the relation values of s and t obtained from the co-ordinates of any suitable point on the curve. The velocity of th centre of the ball at any instant, can also be deduced from this curve. Let ab, in Fig. 31, represent any very short interval of time during the motion of the ball down the plane. Then de, the difference of the ordinates ac and bd, evidently repre- sents the distance passed over in the short time represented by ab. Hence if we divide the distance represented by de by the time repre- sented by ab, we get the average velocity of the ball during the short interval of time represented by ab, and this average velocity may be taken as the actual velocity at the middle of the interval. Time Fig. 31. (Seconds) If therefore we divide the whole time of motion into a number of consecutive short intervals similar to that represented by ab, and plot at the middle point of the short length representing each interval, an ordinate representing the velocity at the middle of the interval, as obtained above, we obtain a curve which shows how the velocity of the ball varies with the time of motion. The curve obtained in this way is known as the velocity curve. In this case it is a straight line, as shown in Fig. 31. This means that the ordinate at any point on the line is directly proportional to the abscissa, and indicates, therefore, that the velocity of the ball at any instant is directly proportional to the time of motion from rest. If we apply the relation v = at to this line the value of a is readily found by substituting in the relation the values of v and t given by the co-ordinates of any point on the line, and then calculating out the ACCELERATION. 6 9 value of a. This value should agree with the value obtained from the space curve by the help of the relation s = J a 2 . In plotting the velocity curve it will generally be found sufficient to divide the time of motion into half second intervals, and it will be necessary to plot the ordinates on a larger scale than those of the space curve. 30. Acceleration due to Gravity. When a body falls freely from rest it falls vertically in a straight line, and it is found that the distance travelled from rest in any time is directly proportional to the square of the time. This shows, that a body in falling is subject to uniform acceleration directed vertically downwards along the line of fall. This acceleration is known as the acceleration due to gravity. Its value differs slightly at different points on the earth's surface, but for places in Great Britain it may be taken for ordinary purposes, as 32 - 2_ feet-per-sec. per sec., or 981*2 cms.-per-sec. per sec. That is, a body falling freely in vacuo (so as to be free from air resistance) is subject to a uniform increase of velocity at the rate of 3 2 -2 feet-per-sec. per sec., Fig 32 or 981 centimetres - per - sec. per sec. Similarly, if a body is projected vertically upwards it is subject to a decrease of velocity at the same rate. The direction of the acceleration due to gravity is vertically down- ward, and so causes acceleration of downward motion and the retardation of upward motion. When a body is projected horizontally or in any direction it is still subject to the acceleration due to gravity that is, the change in its velocity from instant to instant, as explained in Art. 28, always takes place in the same direction and at the same rate. Hence, if the velocity of the body can be represented at a given instant by AB (Fig. 32), and by AC at an instant t units of time later, then BC, which represents the change of velocity in 70 GENERAL PHYSICS. the time t, will be directed vertically downwards, and if this change of velocity be denoted by v, then - will give the acceler- ation* due to gravity. - In the same way, if the velocity of the body at any instant is represented by AB, the velocity at any instant t units of time later is given by compounding with the velocity represented by AB, a velocity represented by BC, taken in a vertical direction and equal to gt, where g denotes the acceleration due to gravity. The acceleration due to gravity is usually denoted by g, and the formulae of the foregoing article, when applied to the motion of bodies subject to this acceleration, are usually written with g instead of a, so that we have 1} = gt, or v = u + gt, and, s = 4 gfi, or s = ut -\- J gt' 2 . It will be understood in applying these formulae that g must be taken as positive or negative according as the positive direction of motion is taken vertically downwards or vertically upwards. The value of the acceleration due to gravity at any place is determined best by the pendulum method explained in Art. 34. It can, however, be determined roughly by direct methods. The chronograph described in Art. 19 may, for example, be adjusted to record electrically the time taken by a suitable body, such as a steel ball or a bullet, in falling from rest through a known distance. Then if 5 denote the distance, and t the time recorded by the chronograph, we have s = J gt 2 , and from this relation g can be determined. A common form of this method in which the time record is traced by the tuning fork on the falling body itself is described below. * It should be noted that in this case gives the acceleration whether t t be large or small, because the acceleration is uniform in magnitude and direction. See Art. 28. ACCELERATION. 1 Experiment 2. Arrange, as shown in Fig. 33, that a long strip of plate glass, G, falls vertically between guides in such a way that as it falls past the end of the tuning fork T, a short bristle on the end of one of the prongs traces, when the fork is in vibration, the usual wavy trace on the smoked surface of the glass. The trace need not be taken from the starting point of the strip's fall. It is better to let the strip fall from a point as far as possible above the fork, and to let the trace be taken at the end of the fall where the strip is moving very rapidly. An indiarubber pad should be arranged to receive the strip at the end of its fall. The acceleration due to gravity can now be calculated from data given by the trace on the strip of glass in the following way. Measure off on the trace the lengths of two consecutive portions each containing the same number of complete waves. These lengths evidently give the distances through which the plate falls in two conse- cutive equal intervals of time. Let dj and d 2 denote these distances, and let t denote the duration of the equal intervals of time. Then, d^/t is the average velocity of the plate during the first interval, and gives, therefore, the actual velocity of the plate at the middle of this interval. Similarly, d 2 /t gives the actual velocity of the plate at the middle of the second interval. A change of velocity, ~ l , therefore, takes place in the interval between the middle of the first interval and the middle of the second interval ; this interval is itself equal to t, and the rate of change of velocity, or the acceleration of the plate is, therefore, given by Now, t is equal to p(n seconds, if p denote the number of 6 complete waves in each of the measured portions of the trace, and n the number of vibrations per second made by the fork. Wo, therefore, have (do - di) n- Fig. 33. 72 GENERAL PHYSICS. The best method of getting an accurate result from a good trace by this method of calculation is as follows. Number the wave lengths on the trace from crest to crest, from 1 to 50, using a fine needle point to write the numbers on the smoked surface. Then find a "number of values of d and d. 2 for intervals corresponding to the time of twenty vibrations of the fork, by making the measurements indicated in the table given below. 4. d+ d 2 - di. Portion of Trace Measured. cms. Portion of Trace Measured. cms. . cms. 1 -21 21 - 41 2-22 22 - 42 . 3 - 23 23 - 43 . . 4 - 24 24 - 44 5-25 25 - 45 . . 6-26 26 - 46 . . 7-27 27 - 47 8-28 28 - 48 9 - 29 29 - 49 10 - 30 ... 30 - 50 ... Mean. The values obtained in this way for d and d. 2 change progressively as the portions of the trace from which they are taken advance along the trace, but the value of (d 2 - d ) should be constant, and the mean of the 10 values given by this table should be fairly free from error. The tabular scheme given above is arranged to give the mean of 10 values of (cZ 2 - d^ when the measured portions of the trace contain 20 complete wave lengths; it may obviously be modified in these particulars as may be required. The value of the acceleration due to gravity may be deter- mined with fair accuracy by methods of this kind, but the only accurate method is the pendulum method referred to above. 73 CHAPTER VIII. CIRCULAR MOTION AND SIMPLE HARMONIC MOTION. 31. Angular Velocity. In considering the motion of rotation of a body, the body is supposed to be rigid that is, the particles which make up the body are supposed to be fixed in position relative to each other, and not subject to any relative displace- ment such as might be produced by any deformation of the body. When a rigid body rotates round any straight line as axis, every particle in the body moves in a circle round a point in the axis as centre, and the angular velocity of the rotation is measured by the angle which the radius joining any particle to the centre of its path of motion, describes per unit of time. This angle is evidently the same for every particle in the body, and is always expressed in circular measure. Hence if the angular yelnfiit.y of a rotating body is denoted by jjjj^jthe Bangle which the radius joining any particle at a distance r from the axis, describes in a very short time r, is wr, and since this angle is expressed in circular measure, the arc over which the particle moves is given by rtor or (rw)r. That is, the particle moves over a distance (rw)r in a very short time r, and its linear velocity is therefore given by rw, where r denotes the distance of the particle from the axis of rotation, and (u denotes the angular velocity of the body. The linear velocity of any particle in the body is thus directly proportional to its distance from the axis of rotation. If a rotating body rotates through an angle (circular 74 GENERAL PHYSICS. measure) in a time t, then 0/t gives the average angular velocity for the time t. If at any instant the body rotates through a very small angle , in a very short time T (taken to include the instant considered), then <5/r gives the angular velocity of the body at the instant considered. If the body rotates through equal angles in equal times, however long or short the times may be, the angular velocity is uniform, and the angle described by the body in any time t is given by = wt, where w denotes the uniform angular velocity of the body. Angular acceleration bears the same relation to angular velocity as linear acceleration bears to linear velocity. It may be defined as the rate of change of angular velocity, or the change of angular velocity per unit of time. 32. Motion in a Circle. When a particle moves round a circle with a velocity of uniform magnitude it is subject to acceleration, for, although the magnitude of the velocity is constant, its direction changes continuously from point to point on the circle. Imagine a particle to move round the circle PQ, Fig. 34, with a velocity of uniform magnitude v t and suppose it to move over any very short arc in a very short time r. The direction of the velocity of the particle at the point is along the tangent to the circle at that point. Hence, if PR represent the direction of the tangent at P, and QS the direction of the tangent at Q, the velocity of the particle is, at P, along PR, and at Q, along QS, so that the direction of its velocity changes through the very small angle POQ in the very small time r, in which it passes from P to Q. In Fig. 35, let the velocity of the particle be represented in magnitude and direction by AB at the point P, and by AC at the point Q. The change of velocity which takes place in the very short time r, as the particle moves from P to Q, will then CIRCULAR AND HARMONIC MOTION. 75 be represented in magnitude and direction by BC as explained in Art. 28. Since AB and AC are equal, each being v units in length, and the angle BAG, being equal to the angle POQ, is very small, the line BC is practically coincident with the arc of a circle described with centre A and radius AB or AC. Hence, if the angle BAG be denoted by a, in circular measure, the length of BC is measured by va, and the change of velocity represented by BC is also measured by va. That is, the change of velocity which takes place as the particle moves from P to Q, in the time T, is measured by va, and the average Fig. 34. magnitude of the acceleration for this short interval of time is measured by . Now the angle POQ, in Fig. 34, is equal to the angle BAC, a . which is denoted by a. and it will be seen that is the angular T * V velocity of the particle round O, and is therefore equal to - where r denotes the radius of the circle. The average acceleration v 2 of the particle for the time T is therefore given by, for a v v v . - - = v . - . r r r 76 GENERAL PHYSICS. This is the average value of the acceleration of the particle for any very small interval of time, so that if the interval of time be assumed to be infinitely small, it gives the value of the acceleration at any instant during the motion of the particle. The acceleration of the particle is thus of constant magnitude for since v and r are both constant must be constant. r The direction of this acceleration can also be determined from Fig. 35. It will be seen that when the interval of time r is infinitely small, the angle BAG is infinitely small, and the direction of BC is at right angles to AB and AC. This indicates that when the particle is at P, and the direction of its velocity is along the tangent PR, the direction of its acceleration is along the radius PO towards the centre of the circle. That is, the acceleration of the particle, at any point in its path, is directed towards the centre of the circle along the radius at that point. Hence, when a particle moves in a circle of radius r, with a velocity of constant magnitude v, it is subject to an acceleration 02 of constant magnitude directed towards the centre of the circle, at all points in its path. 33. Simple Harmonic Motion. Let P, Fig. 36, be any point on the circumference of the circle APB, and AB any diameter of the circle. From P draw Pp perpendicular to the diameter AB and meeting it at the point p. This point p is the projection of the point P on the diameter AB. For different positions of the point P on the circumference of the circle, the point p will have different positions on the diameter AB, for the point p will in all positions be the foot of the per- pendicular from P on to the diameter. Now, imagine the point P to move round the circumference of the circle with uniform speed, and consider the corresponding motion of the point p along the diameter AB. It will be seen that as P moves round and round the circle the point p moves CIRCULAR AND HARMONIC MOTION. 77 backwards and forwards along the diameter AB, making a com- plete backward and forward movement for each complete revolu- tion made by P. The point P makes a complete revolution from any starting point on the circumference of the circle every time it passes through the starting point ; the point p, therefore, raake& a complete backward and forward movement from any starting point on the diameter every time it passes through the starting point in the same direction as it had at the instant of starting. Thus when P moves round the circle from B through Q, A, and Q' back to B, the pointy moves along the diameter AB from B r through to A, and back through to B. Or, as P moves from Q. through A, Q.' and B, back to Q, the point p moves from to A, back through to B, and then back to again. The point j?, moving in this way, is said to move with simple harmonic motion along the line AB. That is, if a point moves round the circumference of a circle with uniform speed, the pro- jection of this point on any diameter of the circle moves back- ward and forward along the diameter in simple harmonic motion. The point P moves round the circle with uniform speed, and,, therefore, describes each complete revolution in the same time. The point p makes a complete backward and forward movement for each complete revolution made by P, and must, therefore,, describe each complete movement in a definite constant period of time equal to the time occupied by P in making one complete revolution. This period of time is known as the period of the motion. In the case of a point moving in simple harmonic motion the period of the motion may, therefore, be defined as the time occupied by the point in making one complete back- ward and forward movement. The line AB along which the pointy moves in simple harmonic motion is the path of the motion, and O is the middle point or centre of this path. The distance OA or OB is, therefore, the greatest distance the point travels from during its motion, This distance is called the amplitude of the motion. 78 GENERAL PHYSICS. The distance of a point in simple harmonic motion from the centre of its path is sometimes called the displacement of the point. If this term is used, the amplitude of the motion may be defined as the maximum^ displacement of the point during the motion. The displacement at any instant of a point in simple harmonic motion may be expressed in terms of the period and amplitude of the motion. Thus, in Fig. 36, if w denote angular velocity of -the point P round 0, and if we suppose the point P to start from Q, and to take a time t to travel from Q to P, the angle POQ >will be denoted by wt, and the displacement, Op, of the point p by OA sin ut. For in the figure we have ~ = sin OPp or Op = OP sin OP/;. But OP = OA, and OPp = POQ = M t, and, therefore, Op == OA sin tot. Now, if r denote the amplitude and t the period of the motion, 2?r 27r we have OA = r and T = or co = fr . It follows, there- to fore, that . 2irt Op r sin . This result shows that during a complete period that is, as t .changes from to T the displacement OP varies in the same CIRCULAR AND HARMONIC MOTION. way as the sine of an angle varies as the magnitude of the angle changes from to 2?r. This law of the variation of the displace- ment with time during each complete period is the characteristic of simple harmonic motion. If we plot a curve showing how the value of r sin -=- varies with ^, the ordinate of the curve gives the displacement, Op, of the point p at any instant during the motion, and the curve, shown in Fig. 37, is called the displacement curve. It is readily plotted in any particular case by finding graphically, as in Fig. 36, the values of Op for a number of successive positions of the point P, and then plotting these values as ordinates and the corresponding Time Fig. 37. % values of t as abscissae. The curve is similar in form to the curve showing how the sine of an angle varies with the angle, and is sometimes called the sine curve or the curve of sines. The velocity of the point p at any instant is evidently the component of the velocity of P in a direction parallel to AB. Thus, in Fig. 38, if the velocity of P is represented by PT the velocity of p will be represented by PR. That is, if the velocity of P is denoted by v the velocity of p is denoted by v cos TPK or v cos wi, for TPR = SPO = POQ = wt. This velocity may, like the displacement, be expressed in terms of r, the amplitude, and T, the period of the motion, for we have v = ^=-, and w = -=-, *aS ey , ^irt so that v cos tot may be written as -7=-cos^=-. It will be seen 80 GENERAL PHYSICS. from this result that the velocity varies as the cosine of the angle "'j while the displacement varies as the sine of the same angle. f) That is, the velocity has its maximum value ^=~ when the dis- placement is zero, and the displacement has its maximum value r when the velocity is zero. It should be noticed that the maximum velocity of p when at 0, the middle point of its path, is the same as the velocity of the point P round the circle, for when P is at Q or Q', the direction of its velocity is parallel to AB. In the same way the acceleration of the point p is the com- ponent of the acceleration of P in a direction parallel to AB. Since P moves in a circle with uniform speed v it is subject to o;2 an acceleration of constant magnitude directed towards the centre of the circle. Thus, in Fig 39, if the acceleration of P is represented by PT, the acceleration of p is represented by PR. That is, the acceleration of p is always directed towards 0, the # 2 centre of its path of motion, and is equal to sin wt or 4?T 2 r . 27T* m-2 S111 m The acceleration of the point p thus varies in the same way as the displacement, and is, in fact, directly proportional to the CIRCULAR AND HARMONIC MOTION. 81 4wV . 2irt 47T 2 / 2irt\ ^t displacement, for -- sin = -^ (r sin -^- j, and ? sm -^r denotes the displacement of the point. That is, if x denote the displacement at any instant of a point moving in simple harmonic motion, then the acceleration of the point at that /27T\ 2 instant is ( -=- J x, where T denotes the period of the motion. Hence, if a point move in simple harmonic motion of period T and amplitude r, the displacement, velocity, and acceleration of the point at the end of any time t t reckoned from an instant when the point passes through the centre of its path in the positive direction, have the values given below. Let the displacement be denoted by .r, the velocity by u, and the acceleration by a, then, from the results obtained above we have x = r sm -^ Bin-Tjr = \^j x. The relation a (-? J & is an important one. It indicates that the acceleration a is directly proportional to the displace- ment x, and it can be seen from Fig. 39 that it is always directed towards the centre of the path of motion. 34. The Simple Pendulum. A simple pendulum consists of a particle suspended by a thread so fine that its mass and weight may be neglected. The particle forms the bob of the pendulum, and the length of the suspension thread from the point of suspension to the particle is called the length of the pendulum. When the pendulum is set in vibration the particle which 6 82 GENERAL PHYSICS. forms the bob of the pendulum swings backwards and forwards through an arc of a circle whose centre is at the point of suspension. If this arc is small compared with the length of the pendulum, the motion of the pendulum is practically iso- chronous, and approximates very closely to simple harmonic motion. The period of vibration of a simple pendulum may be determined theoretically by the following method. Let OP, Fig. 40, represent a simple pendulum suspended from the point 0, and let the particle P oscillate backwards and P Fig. 40. forwards through the small arc QPR. If this arc is very small the motion of the particle may be considered without sensible error as simple harmonic motion along QSE, the chord of the arc. The velocity of the particle at P, the middle point of its path, is, therefore (Art. 33), the same as the velocity of a particle which moves round a circle described on QR as diameter with uniform speed, and makes a complete revolution in the time of one complete vibration of the particle. Hence, if SR, the radius of this circle, be denoted by r, the velocity of the particle at the CIRCULAR AND HARMONIC MOTION. S3 middle point of its path by v, and the period of vibration of the pendulum by t, we have v = - . But the velocity of the particle at P, the middle point of its path, is the velocity acquired in falling from R to P through the vertical distance SP. Hence, if SP be denoted by h, and the acceleration due to gravity by g, we have, by Art. 29, the relation, v 2 - 2gh. That is, , l^-) = 20*, Now, in Fig. 40 we have TS . SP = (SR) 2 by Euc. iii. 35. Hence, if OP, the length of the pendulum, be denoted by /, we have (21 - h)h = f 2 , or 2lh >- /i 2 r 2 . But when the arc QPR is very small, k is very small, and k' 2 may be neglected. That is, when the amplitude of vibration of the pendulum is very small, we have r = -2lh', and if we substitute this value of r 2 in the relation obtained above, we have 47T 2 / or r- = g. That is, f 2 = ^ J or t = l 84 GENERAL PHYSICS. This result shows that for vibrations of very small amplitude the period of vibration of a simple pendulum is constant for a given length, and varies directly as the square root of the length for different lengths. For ^example, if the length of one simple pendulum is four times the length of another, the period of vibration of the longer pendulum will be twice that of the shorter. Experiment 3. Set up a simple pendulum and find its period of vibration when the length of the pendulum has, in turn, the following values : 100 cms., 81 cms., 64 cms., 49 cms., 36 cms., and 2-5 cms. The period of vibration of the pendulum is most conveniently observed by noting the average time occupied by 10, 20, 50, or 100 complete vibrations. The vibrations should be counted, beginning at 0, and proceeding 0, 1, 2, 3, &c., as the bob of the pendulum passes in a given direction (say to the right) through the middle point of its swing. It will be found that the periods of vibration thus found are in the ratio ^/IOO : x/Sl : Joi : C/49 : -N /36 : x /25, or 10 : 9 : 8 : 7 : 6 : 5. That is, the period of vibration is found to be directly proportional to the square root of the length when the length varies. It will be seen that if t, the period of vibration of a simple pendulum of known length, /, is determined accurately by experiment, the value of g, the acceleration due to gravity, can be at once calculated from the relation = V- It must be remembered, however., that a simple pendulum is a theoretical conception and cannot be realised in practice. The nearest approach to it for practical purposes is a small heavy sphere, such as a single shot, or small bullet, suspended by a very fine thread or fibre. A rough determination of the acceleration due to gravity may be made by a simple pendulum of this kind. The length of the pendulum, from the point of suspension to the centre of the bob, is carefully measured as the pendulum hangs ready for use, and the period of vibration is- CIRCULAR AND HARMONIC MOTION. 85 determined as accurately as possible by the method explained above. The value of g can then be calculated from the relation given above. The most accurate experimental determinations of the ac- celeration due to gravity are made by means of the compound pendulum. Any rigid body mounted so as to be capable of vibration round a fixed axis under the action of its weight is called a compound pendulum. If the body is of regular form, such that its moment of inertia (Art. 45), can be calculated from its dimensions, the length of the equivalent simple pendulum, which has the same period of vibration as the body, can be calculated from the radius of gyration of the body about an axis through the centre of gravity and the distance of the centre of gravity of the body from the axis of rotation. Hence, if the period of vibration of a compound pendulum be determined with great accuracy by special methods, and the length of the equivalent simple pendulum is calculated from data found by exact measurement of the necessary dimensions of the pendulum, the acceleration due to gravity can be found by the relation given above. The theory of the compound pendulum and the details of the methods of determining the acceleration due to gravity by its use are, however, beyond the scope of this book, and cannot be further considered. Example. Find the length of the seconds pendulum at a place where the acceleration due to gravity is 32 '18 ft. -per- sec. per second. The "seconds" pendulum is the simple pendulum which would 3>eat seconds or make half a complete vibration in one second. The period of vibration of the seconds pendulum is, therefore, 2 seconds, :so that by applying the relation 86 GENERAL PHYSICS. 32-18 x 4 we have l = 4 x 9-8696' or / = 3-2604. That is, the required Tength of the seconds pendulum at a place when the accumulation due to gravity is 32' 18 ft. -per-se it must be in one of these states, and must continue, without change, in that state. If the law is considered in relation to the body in motion it may be taken as defining a general property of matter. It states that any body or piece of matter is unable by its own action to change its state of rest or motion ; unless acted on by an external " force " it remains at rest or continues to move with uniform motion in a straight line. This property of matter is called inertia. If the law is considered in relation to the motion of the body it serves to define "force." The law states that the action of force compels a body to change its state of rest or motion. Force may, therefore, be defined as that which causes change in 88 GENERAL PHYSICS. the motion of a body. It is important to understand that the change of motion produced by the action of a force is not a sudden change produced when the force first acts on the body, but a continuous progressive change which goes on during the whole time the force acts. This law is sometimes used as a means of defining what is meant by equal intervals of time. It will be seen that equal times may be defined as the times in which a body free from the action of force moves over equal distances. The law cannot be verified directly by experiment, for it is impossible in practice to free a body entirely from the action of force. General observation shows, however, that the more com- pletely a body is freed from the action of force the less appreci- able becomes the change in its state of motion. For example, in the case of the motion of a small truck on straight horizontal rails, it is found that the more the opposing forces of friction are reduced the more nearly does the motion of the truck approximate to uniform motion in a straight line. We are familiar from everyday experience with the fact that a body in motion tends to continue in motion till stopped by the application of force. If a train or carriage of any kind in rapid motion is stopped suddenly,* the passengers and luggage in the carriage tend to continue their onward motion. If they are securely fixed in position they are brought to rest with the train by the resistance of their supports ; if they are not securely fixed they may actually continue their motion after the train has stopped until they are brought to rest by the resistance of some fixed support. That is, they are apparently thrown forward against whatever may be in front of them. *It may be noticed that when a train is stopped gradually in the ordinary way, the passengers frequently experience a jerk backwards instead of forwards. This is due to the fact that as the train slows up the muscles of the body are braced to reduce the forward motion of the parts which are not directly supported, and the action of these muscles produces the jerk backwards if the train stops before they can be relaxed. FORCE. 89 When a ball is thrown vertically upwards inside a railway carriage in motion it is, while in the air, practically free from the action of any force affecting its motion in a horizontal direction. Its horizontal motion, which is the same as that of the train at the instant of leaving the hand, must, therefore, continue without change while the ball is in the air. That is, if the motion of the train does not change while the ball is in the air, the two move forward together with the same horizontal motion. We know from experience that this is the case ; if the ball is thrown vertically upwards from the hand, it keeps verti- cally over the hand while in the air, and ultimately returns to the hand. It is also a matter of common observation that in order to set a body at rest in motion, the force applied must be greater than that required to overcome the frictional and other forces opposing the motion. The excess of force is required to produce the change from rest to motion, and if the excess is maintained the change of motion continues, and the velocity of the body steadily increases. If the excess is not maintained, and the applied force is reduced to equality with the opposing forces when a certain velocity has been attained, the motion continues without further change, for the resultant of the forces acting on the body is zero. If the applied force is removed, or reduced so as to be less than the opposing forces,* the body is gradually brought to rest again. 36. Newton's Second Law Of Motion. Newton's second law of motion states that change of motion is proportional to the impressed force, and takes place in the direction in which that force acts. It is evident that the term " motion," as used in this law, applies to a measurable quantity, for it is said to be proport-ional to the impressed force. Newton explains that the term " motion," * These forces are here supposed to act only while the body is in motion. 90 GENERAL PHYSICS. as here used, involves the mass as well as the velocity of the body in motion, and must be taken, when applied to any body, as the product of the mass of the body into its velocity at the instant considered.* That is, if in denote the mass of any body, and v its velocity at any instant, then the quantity mv is the motion of the body at that instant in the sense in which Newton uses the term in his statement of this second law. This quantity is now generally called momentum. It is a vector quantity, and its direction at any instant is the same as the direction of the velocity at that instant. It will be seen, too, that the element of time must also be involved in the law, for the " change of motion " produced by the action of the impressed force, must depend upon the time in which the change is supposed to take place. In accordance with Newton's explanations on this point we may interpret u change of motion " to mean change of motion per unit time, or the time-rate of change of motion. We may now re-state the law in the following terms. The time-rate of change of momentum of a body at any instant is directly proportional to the force acting on the body at that instant, and takes place in the direction in which the force acts. If the force acting on the body is constant in magnitude and direction, the rate of change of momentum is also constant in magnitude and direction, but if the force is variable the rate of change of momentum also varies from instant to instant. If at any instant the rate of change of velocity, or the acceleration of a body of mass m is denoted by a, the rate of change of momentum of the body must be ma ; that is, the force acting on the body at that instant is directly proportional to ma, and the direction of the force is the same as the direction of the acceleration. The simplest case illustrative of this general result is FORCE. 91 that in which a body moves along a straight line under the action of a constant force acting in the same direction as that in which the body is moving. In this case let u denote the velocity of the body at any instant, and v velocity at an instant t units of time later; then, if m denote the mass of the body the change of momentum during the time is given by (mv mu), and the rate of change of momentum by mv mu m (v ),.- . .1 i i ,1 - , or - - -- -. I he force acting on the body is, there- t t fore, proportional to --- , and since the force is constant, y; _ , it follows that - is constant. But - - is evidently the t v average acceleration of the body for the time t, and if this is constant for all values of t, it follows that the body moves with uniform acceleration along the straight line. Hence, if a body of mass in moves with uniform acceleration a along a straight line, the force acting on it is constant, and proportional to ma. The motion of a body falling vertically under the action of its own weight is an example of this case of motion. The weight of the body is practically constant for a short fall, and the body is known by experiment to fall vertically in a straight line with the uniform acceleration known as the acceleration due to gravity. In connection with the general result that the force acting on a body at any instant is directly proportional to the product of the mass of the body into the acceleration at that instant, it must be remembered that the direction of the force is, in all cases, the same as the direction of the acceleration. 37. Unit Force. In the foregoing article it has been shown that if a body of mass m is subject at any instant to an acclera- tion a, the force acting on the body at that instant is pro- portional to ma, in accordance with Newton's second law of 92 GENERAL PHYSICS. motion. We may, therefore, write F is proportional to ma, or F = kma, where k is a constant. If we now agree that when m and a are both of unit value, F shall also be of unit value, the value of k becomes equal to 1, and we may write F = ma. That is, if we define the unit of force as that force which pro- duces unit acceleration in unit mass, we may measure the force which produces an acceleration of a units in a mass of m units by the formula, F = ma as given above. The unit of force in the English F.P.S.' system of units will be that force which produces an acceleration of 1 ft.-per-sec. per sec. in a mass of 1 pound. This unit is called a poundal. The unit of force in the C.G.S. system will, similarly, be that force which produces an acceleration of 1 cm.-per-sec. per second in a mass of 1 gramme. This unit is called the dyne. It should be noticed that the relation F = kma, derived from Newton's second law, takes the from F = ma as the result of the definition adopted for the unit of force. It follows, therefore, in using the formula F ma, that F will always be expressed in terms of a unit of force consistent with the units in which m and a are expressed. Thus, if m is in pounds, and a in feet-per-sec. per sec., F will be in poundals. Similarly, if m is in grammes, and a in cms.-per-sec. per sec., F will be in dynes. 38. Mass. The meaning of the term mass can now be more fully explained and understood. It must be remembered that forces can be specified and compared without involving in any way the idea of mass. A force may, for example, be definitely FORCE. 93 specified as the force which will extend a certain standard spiral spring through a given distance. Similarly, equal forces may be defined as forces which extend the same spiral spring to the same extent, and unequal forces may be compared by comparing the extents to which they extend the same spring. The mass of a body is evidently the quantity which measures the inertia of the body. In explaining Newton's second law of motion the term mass has been used without any explanation. It will be seen, however, that the idea of mass as a measurable quantity is derived from this law, and that the interpretation of the law includes the explanation of what is really meant by mass. From the relation F = kma deduced from the second law in Art. 37 above, we get 1 F m = 7 - k a This result shows that the mass of a body, as the term i& used in this law. is a quantity which is directly proportional to the force required to produce a given acceleration of the body, or inversely proportional to the acceleration produced by a given force. The three statements given below follow directly from this result. 1. A mass may be definitely specified as the mass on which the action of a given force produces a given acceleration. 2. Equal masses may be defined as masses on which the action, of the same force produces the same acceleration. 3. Masses may be compared (a) by comparing the accelera- tions produced by the action of the same force on the masses to be compared ; or (b) by comparing the forces which, when acting on the masses, produce the same acceleration. In the one case (a) the masses would be inversely proportional to the accelera- tions, and in the other case (&), the masses would be directly proportional to the forces. 94 GENERAL PHYSICS. It will be seen at once from the last of these three statements I [3 (b)] that masses may be compared by comparing their weights, provided the acceleration due to gravity is the same for all j bodies whatever may bg the size or material of the bodies. If this is the case the weights of different masses are evidently forces which produce the same acceleration when acting on the masses, and are, therefore, directly proportional to the masses. Galileo and Newton both proved by direct experiment that the acceleration due to gravity is the same for all bodies, and is quite independent of the size and material of a body. It follows that the comparison of masses by the process of " weigh- ing," as described in Art. 23, is in strict accordance with the -definition of mass derived from Newton's second law. It should be noticed that we can use the relation, F kiim, which we derived from this second law to define a unit of force, .as above, or to define a unit of mass. If we selected a unit of length, a unit of time, and a unit of force (such as the force required to extend a standard spiral spring a specified distance) as fundamental units, we might define the unit of mass as the mass in which the unit of force produces unit acceleration. If, on the other hand, we follow the general practice and select a unit of length, a unit of time, and a unit of mass (Art. 4) as fundamental units, we define unit force as that force which produces unit acceleration by its action on unit mass, as explained above. In either case the relation F = kma would reduce to F ma, but the system of units would be essentially different in the two cases. 39. Weight. It has already been explained that the weight .of a body at any place on the earth's surface is the force with which the earth attracts the body towards it. This force acts towards the centre of the earth and its direction at any place .determines the vertical direction at that place. When a body falls at any place the force acting on it is its FORCE. 95 own weight and the acceleration with which it falls is the acceleration due to gravity at the place. If, therefore, we apply the formula F = ma to the case of a body falling freely at any place, and if we write W instead of F for the particular force called weight, we get the formula W = mg, where can be determined with fair accuracy if the instants referred to above are recorded on a chronograph trace as explained in Art. 19. They may also be determined roughly with the aid of a clock or metronome which ticks sharply and distinctly. The mass A is released at a certain tick, the position of the ring at C is adjusted by repeated trials until it removes the rider at the next tick, and the position of D is adjusted in the same way until it is so placed that the mass A strikes it at the instant the third tick is heard. That is, the starting instant, the instant of removing the rider, and the instant A touches D, are made to coincide with three successive ticks of the clock or metronome. The times t l and t. 2 are thus made equal to one another, and to the interval between two successive ticks. If this interval is exactly one second we have s a -, or a = s. i x r That is, the acceleration is numerically equal to the space passed over in the second interval of time. These intervals of time may also be measured by weighing the quantity of water or mercury which escapes in each interval through a small hole in a vessel containing the liquid. This is the water clock method adopted by early experimenters with Atwood's machine. The distance s is obviously the distance between the upper faces of the ring at C and the platform at D, diminished by the length of mass A. FORCE. TCI If the experiments indicated above are carried out in the manner here described, it will be found that the more com- pletely the sources of error in the experiments are removed the more nearly do the results obtained agree with those deduced from Newton's second law of motion. It will be seen that if we accept Newton's second law of motion as true, we can use Atwood's machine to determine the acceleration due to gravity. From the relation mg = ( 2 M + m) a, 2M -f- m we get (j = m 'a- so that if a is determined by experiment, as explained above, and M and m are known, the value of y can be calculated directly from this result. Atwood's machine is of great interest historically and theo- retically, but it is extremely difficult to obtain anything like accurate results by its use for experimental work. ' The main sources of error are those due to the neglect of (a) the mass of the string ; (6) the inertia of the wheel ; and (c) the friction and air resistance which oppose motion. The moving force (the weight of the rider) should really be split up into the following parts : j\ , which acts on the masses A and B and the mass of the rider, and which produces the acceleration of these masses. A'j, which acts on the mass of the string, and sets it in motion with the acceleration of the system. ,v. 2 , which acts on the wheel and sets it in rotation round its axis with an angular acceleration, such that the linear acceleration of a point on the rim where the string touches the wheel is the same as that of the string. A- 3 , which is neutralised by the opposing forces of friction and air resistance. Of these f l is the real moving force which produces the accelera- tion of the moving system, and it will be seen that / x = / - (x' -f a- + .<'. 102 Fig. 48. C4ENERAL PHYSICS. The error which results from neglect- ing x lt x, 2 , and :>'.., and taking /, the weight of the rider, as the moving force, will depend upon the magnitude of (/! + tf-2 + ^ 3 ), compared with /, and may be very great. If the string is very light, and the wheel is light, properly designed, and mounted on frictionless bearings, r^, ,/'.,, and r st may, for rough purposes, be neglected, pro- vided / is not small. For more accurate work, however, these errors must be cor- rected and allowed for by methods which we cannot here consider. Correction should also be made for the change in the moving force caused by the fact that as the system moves the weight of the string on one side of the wheel increases, while the weight on the other side de- creases. In all elementary problems and experi- ments relating to Atwood's machine, the mass of the string, the inertia of the wheel, and the opposing forces of fric- tion, are usually neglected. A form of Atwood's machine suitable for accurate work is shown in Fig. 43. The experimental results which can be obtained by this machine, even under the best conditions and with due regard to the sources of error, are not, however, of a very high degree of accuracy. 41. Newton's Third Law of Motion. Newton's third law of FORCE. 103 motion states that to every action there is an equal and opposite reaction ; or that the mutual actions of two bodies are equal in magnitude and opposite in direction. The terms "action" and "reaction" in this law apply to forces. The law implies that force can be exerted only by one piece of matter on another, and that the action between any two bodies is mutual, so that each body may be considered to exert force on the other. This mutual action between two bodies is generally called stress. Hence, if a stress exists between two bodies A and B, and the force exerted by A on B is taken as the action, then the equal and opposite force exerted by B on A is called the reaction. Action and reaction are thus merely opposite aspects of the stress between the bodies. When the stress between any two bodies is such that each presses against the other, the stress is of the particular kind known as pressure. Thus, if we press with the hand against a wall, the hand presses on the wall, and the wall resists or presses back against the hand. The stress between the hand and the wall is thus a pressure ; the hand exerts pressure on the wall, and the wall exerts an equal and opposite pressure on the hand. That is, the action and reaction are equal and opposite. Similarly, when a book rests on a table it exerts a pressure vertically downwards on the table ; at the same time the table resists arid exerts pressure vertically upwards on the book. This upward pressure exerted by the table on the book must be equal and opposite to the weight of the book, for the two forces acting on the book are its weight acting downwards, and the resistance or upward pressure of the table acting upwards, and since the book remains at rest, these forces must be exactly equal and opposite to each other. Hence, if the downward pressure of the book on the table be taken as the action, the resistance or upward pressure of the table on the book is the reaction, and these forces are equal and opposite in accordance 104 GENERAL PHYSICS. with Newton's third law, and each is equal to the weight of the book. If a box rests on the floor of a lift in motion the stress between the under surface ofthe box and the floor of the lift is a pressure, and the action and reaction between the surfaces are equal and opposite ; but they are not necessarily equal to the weight of the box. Let the pressure of the box on the floor of the lift be denoted by P, then the reaction of the floor acts vertically upwards and is also equal to P, as shown in Fig. 44. The forces acting on the box are, therefore, its weight, AY, acting vertically downwards, and P the resistance from the floor acting vertically upwards. Hence, if P = AA 7 , the resultant force acting on the box is of zero value, and the box (and lift) must either be at rest or moving with uniform motion in a straight line up or down. In this case the pressure of the box on the floor of the lift would be equal to its weight. If, however, AA r is greater than P, the box is under the action of a force (W -- P) acting p downwards, and will, therefore, be subject to Fig. 44. the downward acceleration caused by the action of this force on the mass of the box. In this case the pressure of the box on the floor of the lift is lets than its weight. The weight may, in fact, be divided into two parts a part, P, which exerts pressure on the floor, and the remainder (W P), which gives the box its down- ward acceleration. If, again, P is greater than W, the box is moving under the action of a force (P AV) acting upwards, and will be subject to the acceleration caused by the action of this force on the mass of the box. In this case the pressure of the box on the floor of the lift is greater than its weight, for the floor not only supports the weight of the box, but exerts the additional force (P AV) which gives the box its upward acceleration. FORCE. 105 When the stress between any two bodies is such that each body exerts a pull towards itself on the other, the stress is of the type called tension. Thus, if two persons pull against each other along a rope, as in a "tug of war," the stress between them is a tension. The stress is properly considered to act across any transverse section of the rope between the two portions of the system separated by this section. If the rope is at rest the tension is practically the same at all points in its length. For, if T l denotes the tension at a point, A (Fig. 45), and T 2 the tension at another point, B, it is evident from the figure that the force acting on the portion AB of the rope * is (T T - T 2 ), if we assume T t to be greater than T,, and if we neglect the weight of the rope. But if the rope is at rest, the force acting on any portion of it must be of zero value that is, T 1 and T 2 must be equal. Hence, when the rope is at rest, and Fig. 45. the tension the same at all points in it, we generally speak of the tension of the rope, and consider the stress to act by means of the rope between the two persons pulling at its ends. When a body hangs at rest from a nail by a thread, as in Fig. 46, the tension in the thread at any point, A, is equal to the weight of the body and the piece of thread below this point. For this weight, W, acting at A, is evidently balanced, as shown in the figure, by T, the tension in the string at this point. If we neglect the weight of the thread the tension is evidently the same at all points in the thread, and is equal to the weight of the body. The body thus exerts a downward pull, equal to its weight, on the nail, and the nail exerts an equal upward pull on the body. The thread in this case may be looked upon as the * The left-hand arrow at A and the right-hand arrow at B evidently indicate the forces acting on the portion AB. A 106 GENERAL PHYSICS. medium or connection between the body and the nail, by means of which the stress between these two bodies is maintained. If a mass hanging by a thread is in motion, as in the case of the masses of Atwood's machine, the tension of the thread is not in general equal to the weight of the mass. Let the mass, A, in Fig. 47, be supposed to be moving up or down in a vertical line. The forces acting on the mass are its weight, W, acting vertically downwards, and the tension in the thread, T, acting vertically upwards. If the mass is at rest, as above, or moving up or down with uniform velocity, the resultant force acting on it must be of zero value, and T must be equal to W. That is, in this case, the tension in the string would be equal to the weight of the mass. If the mass is moving upwards or downwards with uniform acceleration downwards, T must be less than W and the difference ( W T) is the force which gives the mass its downward acceleration. If the mass is moving upwards or downwards with uniform acceleration upwards. T must be 4 _ greater than W and the difference (T W) is the force which gives the mass its upward acceleration. It must be remembered that in each of these cases the stress at the point of attachment of the string to the mass is the tension in the string at that point ; the string exerts an upward pull on the mass equal to the tension T, and the mass exerts an equal downward pull on the string. The weight W is an external force exerted by the earth on the mass ; the reaction to this is the equal and opposite force exerted by the mass on the earth. In applying the third law of motion to any body or system in motion it must be remembered that the action and reaction must FORCE. 107 be taken at the same point, or across the same section. Thus, if AB (Fig. 48) represents a body in motion in the direction of the arrow at B, and we consider the stress across a transverse section at C, we can say that the action and reaction at this section are equal and opposite. Similarly, if we consider the stress across the transverse section at D, we can also say that the action and reaction at this section are equal and opposite. The stress at D will not, however, in general be the same as at C. Let P and P' denote the stresses at C and D respectively, then the resultant force cwting on the portion CD is evidently the difference between P and P'.* If the body is in motion with uniform velocity, P is equal to P'. If the body is subject to acceleration in the direction of its motion, then P must be greater than P', and the difference Fig. 48. (P P') is the force which gives the mass of the portion CD the acceleration with which it moves. If the body is subject to retardation then P' must be greater than P, and the difference (P' P) will be such as to produce in the mass of CD the retardation to which it is subject. Thus, in the case of the motion of a horse and cart along a level road, the horse and cart move as one system, and the difference between the stresses at any two vertical sections of the system gives the force which at any instant determines the motion of the portion of the system between the sections. The force which causes the motion of the system as a whole is derived from the stress between the horse's feet and the * External forces, such as the weight of the body and frictional resistance to motion, are here neglected. 108 GENERAL PHYSICS. ground. The component parallel to the road of the reaction of the ground on the horse's feet is the force which acts on the system at any instant, and the system moves forward with uniform velocity, or is, subject to acceleration or retardation according as this force is equal to, or greater or less than, the forces which oppose motion. The reaction of the ground on the horse's feet is equal and opposite to the " action " of the feet on the ground, and depends, therefore, for its value on the muscular effort exerted by the horse. It will be understood from the examples which have been given above that, in determining the forces which act on any portion of a system which moves as a whole, the stresses due to its connection with the rest of the system must be considered, as well as the external forces which may act on it. This may be more fully understood from a study of the fol- lowing numerical example. Numerical Example. Three masses, A, B, and C, of 10 grammes, 5 grammes, and 12 grammes mass respective!}', are Fig. 49. connected in line by a fine thread and arranged on the wheel of an Atwood's machine in the manner shown in Fig. 49 ; find the tension in the thread connecting A and B, and in the thread connecting B and C when the system is in motion under the action of the weights of the masses. The thread being fine the mass and weight of the connecting threads may be neglected, and the tension in either thread may be considered to be the same at all points in its length. Let T denote the tension in the thread connecting A and B, and T' the tension in the thread connecting B and C. Consider first the FORCE. 109 motion of the mass A. The forces acting on it are its weight W, acting vertically downwards, and the tension T, in the thread AB, acting vertically upwards. The mass is evidently subject to down- ward acceleration, so that the resultant force acting on it is downwards and equal to (W - T). The mass of A is 10 grammes, and its weight, W, is (10 x 981) dynes (Art. 39), so that by applying the relation F = ma, we get 9,810 - T = 10a, (1) where a is the acceleration of the mass A, and, therefore, of the whole system of masses. Consider next the motion of the mass B. The forces acting on this mass are its weight, W, acting vertically downwards, the tension, T, in the thread BA, also acting vertically downwards, and the tension, T', in the thread BC, acting vertically upwards. The mass moves with downward acceleration, a, so that the resultant force acting on it is downward, and is equal to (W + T - T'). The mass of B is 5 grammes, and its weight W, is (5 x 981) dynes, so that by again applying the relation F = ma, we get 4,905 + T - T = 5a (2) In the same way by considering the motion of the mass C, we get the result T' - 11,772 = 12a. ....... (3) From the three equations thus obtained we can find the values of T, T', and a in the usual way. It will be found that T = 8,720, T' = 13,080, and a = 109. T and T' being expressed in dynes, and a in cms.-per-sec. per sec. These results may be arrived at more expeditiously by considering first the system as a whole, and finding the acceleration of its motion. The mass of the system is 27 grammes, and the force to which its acceleration is due, is evidently the difference between 15 gramme- weights and 12 gramme- weights, or 3 gramme- weights, or (3 x 981) dynes. If, therefore, we apply the relation F = ma to the system, we get 3 x 981 = 27a ; or a = 109. That is, the acceleration of the system is 109 cms.-per-sec. per sec. 110 GENERAL PHYSICS. If we substitute this value of a in equation (1) obtained above, we get 9,810 - T = 1,090; or T = 8,720. That is, the tension iif the thread AB is 8,720 dynes, or 8f gramme- weights. Similarly, by substituting for a in equation (3), we get T - 11,772= 1,308; or T' = 13,080. That is, the tension in the thread BC is 13, 080 dynes, or 131 gramme- weights. 42. Motion in a Circle. It has been shown in Art. 32 that a particle moving in a circle of radius r, with a velocity of v 2 constant magnitude v, is subject to a constant acceleration directed towards the centre of the circle. Hence, if m denote the mass of the particle, the magnitude of the force which acts on the particle and keeps it moving in its circular path is a , and the direction of this force always passes through the centre of the circle. For example, the moon moves round the earth in an approxi- mately circular path, and the force constraining it to move in this path is the force of attraction exerted on it by the earth. Hence, if M denote the mass of the moon, V the magnitude of its velocity round the earth, and R its distance from the centre of the earth, the force of attraction exerted on it by the earth must be equal to 5. K, Example. A stone of 100 grammes mass is whirled round in a vertical circle, at the end of a string 100 cms. long, at a uniform rate of 10 complete revolutions per minute. Find the tension in the string when the stone is (a) at the top of its path, and (b) at the bottom of its path. (a) Let W denote the weight of the stone, and T the tension in the string ; then when the stone is at the top of its path the force act in y on the stone towards the centre of the circle is evidently W + T. Hence, FORCE. Ml if m denote the mass of the stone, r the radius of the circle in which it moves, and v its velocity in this circular path, we must have From the data of the question we know that in C.G.S. units we have m - 100 (grammes). r = 100 (cms.) 10 X 2007T 1007T . w = go" = -3- (cms- per sec.) and W = 100 x 981 = 98,100 (dynes). Hence, we get, 98.100 + T = - 100x 1 % xlOI = -'xlO.. That is, T = ir 3 x 10 4 - 98,100 = 98,696 - 98,100 = 596. Or the tension on the string when the stone is at the top of its path is 596 dynes, or "61 gramme-weights. (b) Similarly, when the stone is at the bottom of its path, the force acting on the stone towards the centre of the circle is T - W. Hence, as above, we get T - w = e* T or T - 98, 100 = ir 2 x 10 4 . That is * T - 98,100 + 98,696 = 196,796. That is, the tension on the string at the lowest point in its path is 196,796 dynes, or about 200*6 gramme-weights. It is important to realise, in connection with the result a obtained above, that -- is not the magnitude of a new force which acts on the body in virtue of its circular motion, and in addition to any other forces which may be acting on it ; it is the magnitude of the resultant of the forces actually acting on the body. That is, if a body of mass m moves in a circle of radius r, 112 GENERAL PHYSICS. with a velocity of constant magnitude v, the resultant force acting on the body at any point in its path is directed towards A>1 ,0*2 the centre of the circle, and its magnitude is . When a particle moves in a circle, it may be said, in accord- ance with Newton's third law of motion, that a stress exists between the particle and the centre of the circle. This stress acts on the particle towards the centre, and on the centre towards the particle, or away from the centre. These two aspects of this stress have been called the centripetal and centrifugal forces. The existence of this stress between a particle in circular motion and the centre of its path explains why a flywheel or emery-wheel in rapid rotation sometimes "bursts." Stress exists between every particle of the wheel and the axis of rotation, as a tension in the intervening material of the wheel, and if this stress becomes at any point too great for the strength of the material to withstand, the wheel " bursts " into fragments. At the instant of bursting each fragment is freed from the constraint which compels it to move in a circle, and will, therefore, continue its motion in the same direction and with the velocity which it has at that instant. 43. Simple Harmonic Motion. It has been shown in Art. 33, that when a particle moves in simple harmonic motion, the acceleration of the particle is directed towards the centre of ,f) \ 2 the path of motion, and is of magnitude I x, where T ) denotes the period of the motion and x the displacement of the particle. Hence, if m denote the mass of the particle, the force acting on the particle at any instant during its motion is 27T\ 2 47T 2 W T -\ mx, or -^g- x. That is, the force acting on the particle at any instant during its motion is directly proportional to its displacement at that instant, and is directed towards the centre of the path of motion. FORCE. 113 Hence, if a body is so constrained that the force which acts on it, as the result of any displacement from its position of rest, is proportional to the displacement and directed towards its position of rest, the body will move in simple harmonic motion along the line of displacement, and its position of rest will be the centre of its path of motion. Also, if the force acting on the particle is denoted by lex, where k is a constant, we have or k ~ For example, if a small bullet of mass m, hangs by a thin elastic cord which is stretched by the weight of the bullet through a distance, d, the force which will act on the bullet as the result of a small vertical displacement, ^, from its position of rest will be* ~ . x, and will be directed towards the position of Cu rest. The bullet will, therefore, move up and down in simple harmonic motion, and as mg 4?r 2 m ~d~ "f 2 "' the period of its motion is given by T= 27T ff 44. Moment of a Force. The moment of a force about any point is denned as the product of the magnitude of the force into the length of the perpendicular from the point on to the * The force causing unit elongation of the cord is denoted by ~ ; the force resulting from a change, x, in the elongation is, therefore, denoted by^.a. 114 GENERAL PHYSICS. line of action of the force. Thus, in Fig. 50, if a force of magnitude, F, act along the line AB the moment of the force round the point O is measured by the product of F into the length of the perpendicular OP. That is, if the moment of the force be denoted by M, and the length of OP by d, we have M = Fd. The moment of the force F about the point thus depends upon F, the magnitude of the force, and d, the length of the arm OP, and if either of these quantities is zero the moment is zero. When d is zero the point is on the line of action of the force, so that the moment of a force round any point on its line of action is zero. If the force F be supposed to act on a rigid body free to rotate round a fixed axis passing through O at right angles to the plane of the paper, the force tends to produce rotation of A P B Fig. 50. the body round this axis, and the moment of the force round may be taken as a measure of the effect of the force in tending to produce rotation round the axis passing through 0. In order to distinguish between the two possible directions of rotation a moment is considered to be of positive sign if it tends to produce rotation in a direction opposite to that of the hands of a clock, and of negative sign if it tends to produce rotation in the same direction as the hands of a clock. It can be shown by experiment that two forces acting on the same body and tending to produce rotation in opposite directions round the same axis, will balance each other exactly if their moments round the axis of rotation are of equal magnitude. This shows that the moment of a force round a FORCE. 115 point is a real measure of the effect of the force in tending to produce rotation round the point. It follows also from this that if a number of forces * act on a body tending to produce rotation round the same axis, the total equivalent moment of the system of forces round the axis is the algebraic sum of the moments of the individual forces round the axis. Experiment 4. Take a flat strip of wood, such as a half -metre scale, and balance it on a knife-edge placed horizontally at right angles to the length of the scale. Now take masses of 100 grammes and 200 grammes and suspend them from the scale, one on each side of the knife-edge, by means of threads. The threads should be looped at their upper ends, so that the masses can be suspended by passing the loop over the scale. Adjust the positions of the masses on the scale until an exact balance is obtained and the scale balances on the knife- edge in a horizontal position. This is most conveniently done by setting the loop of the thread carrying the smaller mass at any con- venient distance from the knife-edge and then sliding the loop of the thread carrying the other mass along the scale until a balance is obtained. It will then be found that the distances of the suspension loops from the knife-edge are inversely proportional to the masses carried by these loops. That is, if d^ and d. 2 denote the distances from the knife-edge of the points of suspension of the 100 grammes mass and the 200 grammes mass respectively, then d l : d z : : 2 : 1, and it will be found on trial that this relation is true for all corresponding values of t/j and d. 2 . Similarly, it may be found by trial with other masses that the distances d l and d% are always inversely proportional to the masses. That is, m^ and m. 2 denote the masses, we always find that d L : d. 2 : : m. 2 : m^ This experiment proves that moments tending to produce rotation round the same axis balance each other when they are of equal magnitude. The scale is a rigid body free to rotate about the knife-edge as axis, .and the weights of the suspended masses acting on it at the points of suspension tend to produce rotation round this axis in opposite directions. The axis is horizontal, and the weights act vertically, so that the weight of each mass acts in a plane at right angles to the axis. Hence, when the scale is horizontal, the moment of the weight * Each force is supposed to act in a plane at right angles to the axis, so that the forces considered are either all in the same plane or in parallel planes. 116 GENERAL PHYSICS. of each mass round the axis is the product of the magnitude of the weight into the distance of the point of suspension of the mass from the knife-edge. If, therefore, the masses are denoted by m and m*, and the distances of the points of suspension of these masses from the knife-edge by d l and^ 2 respectively, the moments of the weights of the masses round the knife-edge are given by m 1 gd l and mgd z , where g denotes the acceleration due to gravity. Hence, if these moments are equal when a balance is obtained, we should have This, however, is the result actually obtained by the experiment. It may, therefore, be considered as established that forces of equal moment round any axis have equal effects in tending to produce rotation round that axis. Fig. 51. This principle, established by this experiment, is sometimes called the principle of moments. If the force F in Fig. 50 is represented as in Fig. 51 by a length QR taken on its line of action, AB, the moment of the force round the point O will be represented by twice the area of the triangle QOR. For, by definition, the moment of the force round O is measured by the product of the measure of QR into the measure of OP, and this product is also the measure of twice the area of the triangle QOR. It will be seen that, in dealing with forces whose lines of action all lie in the same plane, the moments of the forces round an axis at right angles to the plane become the moments of the forces round the point at which the plane cuts the axis. FORCE. 117 45. Motion Of Rotation. If a body rotates round an axis with angular acceleration a, the linear acceleration of a particle of the body at a distance r from the axis is ra, and the force acting on the particle is mra, where m denotes the mass of the particle. The direction of the linear acceleration of the particle is tangen- tial to the circle in which it moves round the axis, so that the direction of the force acting on the particle is also along the tangent to this circle. The moment of the force acting on the particle round the axis of rotation is, therefore, measured by mra . r or mr 2 a. Now the moments of the forces acting on the particles of the body are all in the same direction, so that the total moment to which the body is subject can be obtained by simply adding together the moments for all the particles of the body. Hence, if m v m. 2 , m 3 , m . . . denote the masses of the particles which make up the body, and r v r z , r 3 , r 4 . . . denote respectively the distances of these particles from the axis of rotation, the total moment of the forces acting on the body round the axis of rotation is given by G = niji'i 2 a + m. 2 r.2 2 a + %?' 3 2 a -f . . . or G = a [m^i 2 -f m. 2 r.f -f m s r/ + . . . ]. The quantity [m^\ 2 -f w 2 r 2 ' 2 + W 3 r 3 2 + . . . ] is the sum of all the products obtained by multiplying the mass of every particle in the body into the square of its distance from the axis of rotation. It is called the moment of inertia of the body round the axis of rotation. If this quantity be here denoted by I, we have G = la. This is the relation which corresponds, in the case of motion of rotation, to the relation, F = ma, in the case of motion of translation. It should be noted that G denotes the moment of the force 118 C4ENERAL PHYSICS. acting on the body, taken round the axis of rotation, I denotes the moment of inertia of the body round the axis, and a denotes the angular acceleration of the body. 46. Impulse. It 4ias now been established that when a body of mass m moves with acceleration a, the force acting on the mass is given by the relation, F = ma, in units consistent with those in which m and a are expressed. It has also been explained in Art. 36 that this relation is equivalent to the statement that the force acting on a body is measured by the time-rate at which the momentum of the body changes. That is, the force acting on a body at any instant is measured by the rate of change of momentum of the body at that instant. In the case of some forces, however, it is practically impossible to apply this method of measurement. If a force acts on a body for a very short interval of time and, it may be, is not even constant during that interval, it is impossible to determine the rate of change of momentum which it produces in the body at any instant. For example, when a golf club strikes a ball the time during which the club acts on the ball is so very short that it is prac- tically impossible to determine the force exerted on the ball at any instant by the rate of change in its momentum at that instant. A force of this kind is called an impulsive force, and is usually measured, not by the rate of change of momentum which it produces, but by the total change of momentum which it produces in the body on which it acts. The total change of momentum produced by an impulsive force is called an impulse. Thus, if a golf ball of mass m is struck by a club, and leaves the club with a velocity v, the impulse given to the ball is measured by mVy and this is taken as the measure of the whole action of the club on the ball during the stroke. 47. Impact of Inelastic Bodies. If a body of some inelastic material, such as clay, is in motion, and strikes against FORCE. 119 another body of the same material at rest, or also in motion, the bodies do not rebound from each other after the impact, but adhere together, and, if free to move, move on as one mass. Suppose an inelastic body, A, of mass m v moving with uniform velocity v lt to overtake another inelastic body, B, of mass m 2 moving in the same direction with velocity v 2 ; and that after the impact the two bodies move on as one mass with velocity v, in the same direction as before. It will be clear that during the time of impact the one mass acts impulsively on the other in accordance with Newton's second law of motion ; the forward impulse communicated by A to B is (m, 2 v m 2 # 2 ), or m 2 (v v.-,), and the backward impulse communicated by B to A is (m^i m^), or m x (v x v), and these two impulses being related as " action " and " reaction " must be equal. That is, we have m 2 (v v. 2 ) = raj (i\ - v) ; Or m^} + ni"2 v 2 ( m i ~^~ m -2) v ' This result shows that the total momentum of the two bodies is unchanged by their impact. The total momentum of the bodies before impact is (m 1 v l + ?ft^ 2 ), and the total momentum after impact is (m-i + m 2 ) v, and the relation obtained above shows that these two quantities are equal. This result is an example of the general principle of conserva- tion of momentum. This principle states that the total momentum of an isolated system of bodies is constant, and cannot be changed by any mutual action between the bodies. The principle follows directly from the second law of motion, for if mutual force takes place between any two bodies of the system, the momentum which one gains will be exactly equal to the momentum which the other loses, and the total momentum of the system will remain unchanged by this transfer of momentum from one body to the other. In applying the principle to simple cases of direct impact 120 GENERAL PHYSICS. between inelastic bodies, it must be remembered that momentum is a vector quantity, and that it is necessary, therefore, to dis- tinguish between momenta in opposite directions by difference in sign. Examples. 1. An inelastic body of 20 grammes mass, moving with a velocity of 10 cms. per second, meets another inelastic body of 10 grammes mass moving in the opposite direction with a velocity of 35 cms. per second, find the velocity of the combined masses after the impact. Here, if we take the momentum of the first body of 20 grammes mass to be positive in sign, the momentum of the other body moving in the opposite direction will be negative, and the total momentum before impact is given by {(20 x 10) - (10 x 35)} units; or - 150 units (cm. gramme). That is, 150 units in the same direction as the momentum of the body of 10 grammes mass. Hence, if v denote the velocity of the com- bined masses after impact, we have 30 v = - 150, or v = - 5. That is, the combined masses move after impact in the same direction as the body of 10 grammes mass, with a velocity of 5 cms. per second. 2. A bullet of 20 grammes mass is fired from a rifle of 4,000 grammes mass, and leaves the barrel with a velocity of 30,000 cms. per second. If the rifle when fired is suspended freely, with its barrel in a horizontal position, find the initial velocity of its recoil. Here the system considered is made up of three bodies, if we neglect the suspension strings, cartridge case, wads, etc. These three are the rifle, the bullet, and the charge of powder, and of these we may neglect the charge of powder, since no data are given respecting it. Considering only the rifle and the bullet the momentum before firing is of zero value. After firing, if we take the momentum of the bullet to be positive, the total momentum of the two bodies is 20 x 30,000 - 4,000v, where v denotes the initial velocity of the recoil of the rifle. FORCE. 121 We, therefore, have 600,000- 4,000 1>=0, or, 4,000 v = 600,000. That is, v = 150. The initial velocity of the rifle in recoil is, therefore, 150 cms. per second. This question may also be solved by assuming that the charge of powder when exploded gives equal impulses in opposite directions to the bullet and the rifle. This assumption at once gives 4,000 v = 600,000, or v = 150, as before. 122 CHAPTER X. WORK AND ENERGY. 48. Work. When the point at which a force acts is dis- placed, work is said to be done, either by the force or against the force. If the displacement is along the line of action of the force, then work is done ly the force if the direction of the displacement is the same as that of the force, and work is done against the force if the direction of the displacement is opposite to that of the force. F A B~ ~X Fig. 52. Fig. 53. Thus, let a force F be supposed to act at a point A along AX, and let the point of application be displaced from A to B along the line of action of the force. Then if the direction of the displacement AB is the same as that of the force, as in Fig. 52,. work is done by the force. If, however, the direction of the displacement AB is opposite to that of the force, as in Fig. 53, work is done against the force. If the displacement is not along the line of action of the force, but inclined to it, then work is done by the force or against the force, according as the component of the displacement along the line of action of the force is in the same direction as the force, or in the opposite direction. WORK AND ENERGY. 123 Thus, let a force F be supposed to act at a point A in a direction AX, and let the point of application be displaced from A to B in a direction inclined to the line of action of the force. Then, if the direction of A&, the component of the displacement AB, along the line of action of the force is in the same direction as the force, as in Fig. 54, work is done by the force. If, how- ever, the direction of the component Ab is opposite to that of the force, as in Fig. 55, work is done against the force. The work done by or against a force is measured by the A lr X ^ Fig. 54. product of the magnitude of the force into the displacement or its component along the line of action of the force. Thus, if the point of application of a force F is displaced through a distance s along the line of action of the force in the lr A Fig. 55. same direction as the force, then W, the work done by the force, is given by W = Fs. That is, if a force F acts through a distance s, the work done is given by W = FA Similarly, if the point of application is displaced through a distance s in the direction opposite to that in which the force acts, the work done against the force is given by W = F*'. 124 GENERAL PHYSICS. If the displacement s be taken as positive when in the same direction as the force, and negative when in the opposite direction, work done by the force will be positive in sign, and work done against tha force will be negative in sign. When the point of application of a force is displaced through a distance s in a direction making an angle with the direction of the force, as in Fig. 56, the component of the displacement in the direction of the force is s . cos 0, and the work done by the force is given by W = F . s cos 0. The same result is obtained if we consider F cos 0, the com- ponent of the force in the direction of the displacement to act through the displacement s, for the work done by the force F cos acting through a distance s is given by W = F cos . s. It should be noticed that when the point of application of a force is displaced in a direction at right angles to the direction of the force, no work is done by or against the force. If two forces act on a body in opposite directions along the same line, and the body is displaced in the direction of one of the forces, work is done by one force against the other. Thus, if two forces, P and F, act on the body in opposite directions, and the body is displaced through a distance s in the direction of the force P, then the work done by the force P is Ps, and the work done against the force F is Fs. In this case, if we assume P and F to be the only forces acting WORK AND ENERGY. 125 on the body, the work done on the body is given by (Ps Fs). That is, if we consider work done to be positive when done by a force, and negative when done against a force the work done on a body during any translational displacement of the body is the algebraic sum of the work done by all the forces acting on the body. Work is a scalar quantity and not a vector quantity. That is, in the measurement of work we have to deal with magnitude only, without reference to direction. 49. Units Of Work. The unit of work is derived from the unit of force and the unit of length. It is the work done when unit force acts through unit distance. Hence, in the C.G.S. system the unit of work is the work done when a force of one dyne acts through a distance of one centimetre. This unit of work is called an erg. The unit of work in the English F.P.S. system is the work done when a force of one poundal acts through a distance of one foot. This unit of work is called a foot-poundal. Work is very commonly expressed in gravitational units, in which the unit of force taken is the weight of unit mass. The work done when the iveight of one gramme acts through a distance of one centimetre, is called a centimetre-gramme, and is equal at any place to g ergs, where g is the acceleration due to gravity at that place. That is, in London a centimetre-gramme is equal to 9 80 '6 ergs. Similarly, the work done when the weight of one pound acts through a distance of one foot is called a foot- pound, and is equal at any place to // poundals, where g is the acceleration due to gravity at that place. That is, in London a foot-pound is equal to about 32'IS foot-poundals. From these definitions of units of work, it will be seen that the work done against the weight of the mass in lifting a mass of M grammes through a vertical distance of h centimetres is Mh centimetre-grammes, or M.hg ergs. Similarly, the work done in lifting a mass of M pounds 126 GENERAL PHYSICS. vertically through a distance of h feet is M/a foot-pounds, or M/M/ foot-poundals. The dyne being a very small unit of force, the erg is a very small unit of work. For example, the work done in lifting this book from the floor on to a table would be something like twenty million ergs. A larger unit, called a joule, containing ten million, or 10 7 ergs is, therefore, sometimes used. It can be calculated from the relations already given that 1 foot-poundal = 4'214 X 10 5 ergs, and 1 foot-pound = 12,283 centimetre-grammes = 1-356 x 10 7 ergs= 1 -356 joules. For rough purposes it may be remembered that a joule, or 10 7 ergs, is nearly three-quarters of a foot-pound. 50. Energy. Energy is capacity for doing work as defined in the foregoing article. A body may possess energy in virtue of being in motion, for a body in motion is able to do work against an opposing force until it comes to rest. A body may also possess energy in virtue of the configuration of its parts. A compressed spiral spring, for example, possesses energy in virtue of the configuration which constitutes its com- pression, for it is able to do work against an opposing force in expanding. A system of bodies which exert force mutually on each other may, in the same way, possess energy in virtue of the configura- tion of the system. For, if the system resists any change of configuration impressed on it by the action of an external force, it will be able to do work against an external force in recovering its original configuration. It will be seen that when a body in motion does ivork against a force acting on the body, it loses energy, but if work is done by WORK AND ENERGY. 127 a force acting on the body in increasing its momentum, it gains energy. In either case the loss or gain of energy is measured l>y the work clone by or against the force. Similarly, when a body, or system of bodies, does ivork against an external force in undergoing a change of configuration, the body, or system of bodies, loses energy by the change, but if work is done by an external force in producing a change of con- figuration, then the body, or system of bodies, gains energy by the change. In either case the loss or gain of energy by the body, or system of bodies, is measured by the work done by or against the .external force. Energy is thus measured as work, and the units employed in its measurement are the same as those employed for the measurement of work. The energy which a body possesses in virtue of its motion is called kinetic energy, and the energy which a body, or system of bodies, possesses in virtue of configuration, is called potential energy. It will be found that in the measurement of energy we are generally called upon to measure the energy which a body, -or system of bodies, gains or loses, and not the whole energy which the body, or system of bodies, may possess. 51. Kinetic Energy. As stated above, the energy which a body possesses in virtue of its motion, is called kinetic energy. If a body of mass m moves from rest, with motion of transla- tion under the action of a constant force F, the body moves with uniform acceleration along a straight line in the direction in which the force acts. If the force be allowed to act on the body through a distance s, the work done by the force is Fs, and this is also, as explained above, the measure of the kinetic energy gained by the body. But if a denote the acceleration of the body, and v the velocity it acquires in moving over the distance s, we know that 128 GENERAL PHYSICS. F = ma, and also, by the relation given in Art. 29, that v 2 = 2as. That is, & .b = ma, and s = ^-. iy2 mv 2 It follows, therefore, that Fs = ma . = . Jft 2 That is, the kinetic energy of a body of mass m moving with- out rotation with a velocity v, is given by -| mv 2 . This may be written in the form K.E. = J mv 2 . If m is expressed in grammes, and v in cms. per second, the kinetic energy is expressed in ergs. Similarly, if m is expressed in pounds, and v in feet per second, the kinetic energy is expressed in foot-poundals. In the same way if a body of mass m, moving without rotation under the action of a constant force, F, acting in the direction of motion, changes its velocity from u to v in moving over a distance, s, the work done by the force is Fs, and this is also the measure of the kinetic energy gained by the body as it moves over the distance s, and its velocity changes from u to v. But, as above, if a denote the acceleration of the body we know that F = ma, and also that v 2 = u 2 -f- 2as, or v 2 -- u 2 = 2as+ That is, TJ, ^ - " 2 = ma, and s - ( Za It follows, therefore, that v* - u z m (v 2 - u 2 ) -"* ir nr - That is, the kinetic energy gained by the body is - / or | mv 2 mu' 2 ; This result is in accordance with that obtained above, for if we take the kinetic energy of the mass to be -^ mv 2 when moving WORK AND ENERGY. 129 with a velocity v, and J mu 2 when moving with a velocity u, the gain of kinetic energy when the velocity increases from u to v is evidently J mv 2 mu 2 , as obtained above. Similarly, when the velocity of the mass decreases from u to v, the loss of kinetic energy is given by ^ mu 2 Examples. 1. A body of 10 Ibs. mass is moving without rotation, with a velocity of 64 ft. per second, find its kinetic energy in foot- pounds. The kinetic energy of the mass is given by the relation K. E. = mv*. That is, K.E. = & . 10 x 64 2 = 20,480. Since m is expressed in pounds, and v in feet per second, the kinetic energy will be in foot-poundals. The kinetic energy of the mass is, therefore, 20,480 foot-poundals, or, if we take the acceleration due to gravity as 32 feet-per-sec. . 20,480 per sec., the kinetic energy is ^ , or 640 foot-pounds. 2. A mass of 1 kilogramme moving without rotation does work against a constant opposing force through a distance of 1 metre, and in doing this work its velocity is reduced from 500 cms. per sec. to 400 cms. per second ; find the amount of work done against the opposing force, and also the magnitude of this force. From the data of the question, the mass of the body is 1,000 grammes, and its velocity is reduced from 500 cms. per sec. to 400 cms. per sec., in doing work against the opposing force. The kinetic energy lost by the body in doing this work is given, therefore, in ergs by the relation Loss of K.E. = m (a 2 - v 2 ) ~2~ 1,000 (500 2 - 400 2 ) L ~2~ = 45,000,000. That is, the loss of kinetic energy is 45,000,000 ergs, or 4 '5 joules. If the opposing force against which work is done, be denoted by F in dynes, the work done in overcoming this force through a distance 9 130 GENERAL PHYSICS. of 1 metre, or 100 cms., is (F x 100) ergs, and we have, therefore, 1 OOF -45,000,000, or F = 450,000. 450 000 That is, the force is et^ual to 450,000 dynes, or gramme-weights, "81 if we take the value of , 2,000. That is, the average force is 2,000 gramme-weights, or (2,000 x 981) dynes. The work done in producing the elongation is given by W = Fs, where s is the elongation or the distance through which the force acts. W is therefore given in ergs by W = 2,000 x 981 x 20, or W = 39,240,000. That is, the work done in producing an elongation of 20 cms. is 39,240,000 ergs, or 3 '924 joules. (6) Similarly, the force applied in increasing the elongation of the spring from 20 cms. to 30 cms. increases uniformly with the elonga- tion from an initial value of 4,000 gramme-weights to a final value of (30 x 200), or 6,000 gramme-weights, and the average value of the force exerted is given in gramme-weights by g = 4.000 + 6.000 = 6>500 That is, the average value of the force is 5,000 gramme-weights, or (5.000 x 981) dynes. The work done in increasing the elongation is given by W is, therefore, given in ergs by W = 5,000 x 981 x 10, or W = 49,050,000. That is, the work done in increasing the elongation from 20 cms. to 30 cms. is 49,050,000 ergs, or 4-905 joules. This example may also be worked by the graphical method explained above. The reader will find it a useful exercise to work it out for himself. 59. Kinetic Energy Of Rotation. The kinetic energy of a particle of mass m moving with a velocity v is J mv 2 . The kinetic energy of a body of mass m moving with motion of 152 GENERAL PHYSICS. translation with velocity v is also measured by J mv 2 , for every particle in it is moving with the same velocity. The kinetic energy of a body in rotation is not, however, given by this relation, for the linear "velocity of motion is not the same for every particle in it. When a rigid body rotates round an axis with angular velocity w, the velocity of any particle in the body at a distance r from the axis is TLJ. If the mass of the particle is denoted by m, the kinetic energy of the particle is \ m(rw) z , or J mr z tD 2 9 and the total kinetic energy of the rotating body is the sum of the kinetic energy of all its particles. Hence, if m 1} m. 2 , m s , w 4 , . . . denote the masses of the particles which make up the body, and r b r 2 , r s , r 4 , . . . denote respectively the distances of these particles from the axis of rotation, the total kinetic energy of the body is given by "o i 22 i 1 2 2 i 1 ft. o 2 i The expression here given in square brackets is seen to be the moment of inertia of the body round the axis of rotation, as explained in Art. 45, so that if I denote this moment of inertia, we have El T O = \ l or 2*63. The magnitude of this angle is about 67 40'. The resultant of the given system of forces is thus completely determined in magnitude and direction. 172 CHAPTER XII. CENTRE OF GRAVITY. 67. Centre Of Gravity. If a force is applied to a block of stone by means of a rope attached to the block at a particular point, the force can be said to be applied to the block at the point of attachment of the rope. The force of attraction exerted by the earth on any body near it cannot, however, be said to be applied to the body at any particular point. The weight of each particle of the body acts on the particle at the point where it is situated in the body, and the weight of the body as a whole is really a system of parallel forces, infinite in number, made up of the weights of all the particles which make up the body, acting at the points where these particles are situated. If a body held in any position is rotated through any angle into another position the direction of the system of parallel forces which constitutes the weight of the body does not change relative to the earth, for it is always vertical, but it does change relative to a fired line in the body. If the body, for example, is rotated through any angle in a vertical plane, the direction of the system in the body relative to a fixed line in the body changes through the same angle. That is, as the position of the body is changed, the forces of the system remain of the same magnitude and act at the same points in the body, but the direction of the system relative to any fixed line in the body, in general, changes. It follows, therefore, as explained in Art. 63, that the resultant of the system passes, for all positions CENTRE OF GRAVITY. 173 of the body, through a point which is fixed in position relative to the body. If the system is considered relative to the body only, this point is the centre of the system, as defined in Art. 63, and is called the centre of gravity of the body. The centre of gravity of a body may, therefore, be defined as the point through which the resultant of the weights of the particles which make up the body passes for all positions of the body. The weight of the body may thus be considered as a single resultant force, acting at the centre of gravity of the body. It will be understood that the centre of gravity of a body is not necessarily in the body itself; it may be at a point outside the material of the body, but the position of the point is fixed relative to the body and must be specified in this relation. For example, the centre of gravity of a length of wire bent in the form of a circular ring is at the centre of the ring. When we speak of the centre of gravity of a body as a definite ^ )V -- fixed point in the configuration of the body it is understood that the body is rigid. If the body is made up of movable parts the centre of gravity is fixed for any given configuration of the body, but changes its position with change of configuration. For example, the position of the centre of gravity of a bicycle depends upon the arrangement of its parts, but for any given arrangement it is a definite fixed point. 68. Method of Finding- the Position of the Centre Of Gravity Of a Body. The method of finding theoretically the centre of gravity of a body is based on the result given in Art. 63. Thus, if two particles of weights w 1 and w 2 are placed at points A and B, Fig. 75, their centre of gravity is on the line AB at the point D which divides the line into two segments AD and DB is the ratio o> 9 : w r Similarly, 174 GENERAL PHYSICS. the centre of gravity of three particles of weights w 1 , w. 2 , and oj 3 at the points A, B, and C is on the line CD at the point E which divides the line into two segments DE and EC in the ratio (a) L + tu 2 ) : &>... Any body may be supposed to be made up of its constituent particles, and as we can, by the method here indicated, find the centre of gravity of any system of particles, we can, in theory, find the centre of gravity of any body. In simple cases where the body is of regular form, and its particles are arranged symmetrically about any point or line in the body the application of the method is comparatively easy. Thus, the centre of gravity of a straight uniform filament, such as a straight piece of very fine uniform wire, is evidently at B D Fig. 77. its middle point. The filament may be considered as a row of particles of equal weight distributed uniformly along a straight line, and by pairing particles equidistant from the centre it can be seen that the centre of gravity of every pair, and, therefore, the centre of gravity of the filament, as a whole, is at the middle point. From this result the centre of gravity of any regular plane lamina, or thin sheet of matter uniformly distributed over any regular plane area, can readily be found. Thus, the rectangular lamina ABCD in Fig. 76 may be supposed to be made up, as indicated in the figure, of an infinite number of straight filaments or linear elements arranged parallel to the sides AB and CD. The centre of gravity of each of these elements is CENTRE OF GRAVITY. 175 at its middle point, so that the centres of gravity of all the elements which make up the lamina lie along the median line EF which joins the mid points, E and F, of the lines AB and CD. It follows from this that the centre of gravity of the lamina must be somewhere on this median line. In the same way, by dividing the lamina into linear elements parallel to the sides AD and BC, it can be shown that the centre of gravity must lie on the median line HK. Hence, if the centre of gravity of the lamina lies on the line EF and also on the line HK, it must be at the point G where these two lines intersect. In this case it might have been stated at once, after showing that the centre of gravity of the lamina lies on the median line EF, that it lies at the middle point of the line. The linear elements into which the lamina is divided parallel to AB and CD are equal, and their weights acting at their centres of gravity on the median line EF are uniformly distributed along this line. It follows, therefore, that the centre of gravity of the lamina made up of these elements must be at the middle of the line. It will be seen from what has been said above that the centre of gravity of a lamina in the form of a square, a rectangle, or a parallelogram, is at the intersection of the median lines of the. figure. It is easily shown geometrically that this point is also the intersection of the diagonals of the figure. In the same way it can be shown that the centre of gravity of a lamina in the form of any regular plane figure, is at the intersection of any two median lines of the figure. Thus, the centre of gravity of a circular lamina is at its centre, an the centre of gravity of a lamina in the form of any regular polygon is at the centre of the circle circumscribing the polygon. In the case of a triangular lamina ABC, Fig. 77, it can be shown by dividing the lamina into linear elements parallel to any one of the sides, that the centre of gravity must lie on the 176 CENTRE OF GRAVITY. median line joining the middle point of that side to the opposite angular point, and must, therefore, be at G, the point of inter- section of any two of the three median lines AD, BE, and CF. It is easily proved geometrically that this point is so placed on each of the median lines that its distance from the mid-point of the side to which the line is drawn is one-third the length of the line. That is, DG = -J DA, EG = J EB, and FG = -J- FC. Just as a lamina may be divided for the purpose of finding its centre of gravity into linear elements, and a linear element into particles, so a solid body may be divided into infinitely thin laminar elements or slices. If the laminar elements into which a solid can be divided are of any regular form for which the centre of gravity is known, and if they are arranged in a regular and definite manner, the centre of gravity of the solid can, in general, be found by an extension of the method indicated above. A cylinder, for example, may be divided into an infinite number of circular laminae or slices at right angles to the axis ; the centre of gravity of each lamina is at its centre, on the axis of the cylinder, and as the lamina are distributed uniformly along the axis, the centre of gravity of the cylinder must be at the middle point of its axis. Each lamina may be supposed to be replaced by a particle of the same weight placed at the centre of gravity of the lamina on the axis of the cylinder. The cylinder as a whole thus becomes equivalent to a line of equiil particles uniformly distributed along the axis, and its centre of gravity must, therefore, be at the middle point of the axis. Similarly, the centre of gravity of any prism whose cross section is a regular polygon, is at the middle point of its axis, and the centre of gravity of a triangular prism is at the middle point of the line parallel to its edges, which passes through the centres of gravity of the transverse triangular laminae, into which the prism may be divided. The centre of gravity of a sphere is obviously, from con- CENTRE OF GRAVITY. 177 siderations of symmetry, at the centre of the sphere. It can he seen, also, that if the sphere is divided into laminar elements at right angles to any diameter, the centre of gravity must lie on that diameter, and must, therefore, be at the point where any two diameters intersect. This point is the centre of the sphere. The centre of gravity of a cone must evidently lie on the line joining the apex of the cone to the centre of the base, for if the cone be supposed divided into laminar elements parallel to its base, the centre of gravity of each element lies on this line, and the centre of gravity of the cone made up of these elements must, therefore, be at a point on the line. The elements are not, however, of equal weight, so that the weight is not distributed uniformly along this line, and the centre of gravity is, consequently, not at the middle point of the line, but at a point nearer the base of the cone. The exact position of the centre of gravity of the cone on this line evidently depends upon the manner in which the weight is distributed along this line. It can be seen without much difficulty that the weight at any point on the line is pro- portional to the square of the distance of the point from the apex of the cone, and it can be shown from this that the centre of gravity of the cone is at a point on the line whose distance from the base of the cone is one-fourth the length of the line. In the same way it is found that the centre of gravity of a pyramid is on the line joining the apex of the pyramid to the centre of gravity of the base at a point whose distance from the base is one-fourth the length of the line. It will be seen from the examples given above that the usual method of finding the centre of gravity of a body is to divide the body into suitable elements for each of which the position of the centre of gravity is known. If the centres of gravity of these elements all lie on a straight line, the centre of gravity of the body must lie on that line, and if another similar straight 12 178 GENERAL PHYSICS. line can be found, the centre of gravity must be at the point of intersection of the two straight lines. The exact position of this point can then be found by geometry. If there is only orte way of dividing the body into elements whose centres of gravity all lie on a straight line, the elements may be supposed to be replaced by particles of the same weight placed at their centres of gravity. The problem of finding the centre of gravity of the body is thus reduced to finding the centre of gravity of a line of particles of known weight arranged along a straight line. The most general method of finding the centre of gravity of u number of particles whose weights and positions are known is indicated below for particles in one plane. oc 2 B 1 X* C x, A * r Let o> Fig. 78. n denote the weights of a number of particles at the point A, B, C, . . . N in the plane of the paper, Fig. 78, and let (x v ft), (x. 2 , y,), (^, &), . . . ( x >i> y n ) ^ e the co-ordinates of these points with reference to the axes OX and OY. Then, if x and y denote the co-ordinates of the centre of gravity of the particles, we have Ct> 2 and CENTRE OF GRAVITY. 179 That is. () and This result is readily obtained geometrically by working out the co-ordinates of the centre of gravity of the particles by applying the method of Art. 63. Thus, the co-ordi- nate of the centre of gravity of the particles at A and B obtained by this method will be found to be f^ 1 ' 1 ' 1 w * ' /2 land ix \ Wi 4- W 2 / / w i !/i __ty*\ an( j con tinued application of this result leads b)! + 0> 2 / for any number of particles to the general result given above. The result is, however, more easily obtained by supposing the particles to be in a horizontal plane, so that their weights act at right angles to the plane of the axes of co-ordinates, and then taking moments about OX to find ~, and about OY to find y. The moment of the resultant of the weights of the particles, acting at their centre of gravity, about either axis, must be equal to the algebraic sum of the moments of the weights of the individual particles about the same axis. Hence, by taking moments about OX we have 2 (tax) or x ' . as above. 2,(o>) Similarly, by taking moments round OY we get _ 2 (*) 2() In these results the expressions 2 (wx) and 2 (wy) must be taken to mean the algebraic sum of the quantities of the form we 180 GENERAL PHYSICS. and wy for, unless the particles are all in one quadrant, as in Fig. 78, the co-ordinates of the points at which the particles are placed may be positive or negative. If the particles are distributed along a straight line their centre < of gravity is also on this line, and if the line be taken as the axis of .> then we have -f (t).> ,''., + Wo /., + . . . + w 10 \_ + W.> 4- U>. ; . . . + W n where .' r /.>, ' ;> . . . . / are the distances of the particles from any point on the line, taken as origin, and x is the distance of the centre of gravity from the same point. The determination of the centre of gravity of a body by dividing the body into infinitesimal elements, and then applying the results given above, usually requires advanced mathematical methods, and cannot be further considered. If. however, any body can be divided into a few finite parts for which the positions of the centres of gravity are known, the parts may be supposed to be replaced by particles of the same weight at their centres of gravity; and the centre of gravity of these particles that is r the centre of gravity of the body can then be found by the method of Art. 63, or by applying the results given above. Examples. 1. Find the centre of gravity of a uniform lamina in the form of an irregular quadrilateral. The lamina ABCD, shown in Fig. 79, may be divided into two triangular lumimv, ABC and ADC. Draw the diagonal AC and let O be its middle point. The centre of gravity of the triangular lamina ABC will then be on the median line BO at a point E, such that OE = J OB. Similarly, the centre of gravity of the triangular lamina ADC is on the median line DO at a point F, such that OF = J OD. The triangular lamina? ABC and ADC may therefore be supposed to be replaced by particles of the same weights, placed at the points E and F respectively. The centre of gravity of these two particles will be on the line EF at a point G which divides the line into two segments, which are inversely proportional to the weights of the adjacent particles. The weights of the particles at E and F are CENTRE OF GRAVITY. 181 evidently proportional to the areas of the triangles ABC and ADC, and the line EF must therefore be divided at G, so that GE : GF :: Area of triangle : Area of triangle ADC ABC. This relation determines the position of the point G on the line EF, and this point is the centre of gravity of the lamina. Instead of finding the position of G on the line EF in the manner given above, it would be possible, by dividing the lamina ABCD into two triangular lamina- BAD and BCD, to find another line, E'F', on which the centre of gravity must lie. The point G would then be determined by the intersection of the two lines EF and E'F'. 2. Find the centre of gravity of a uniform wire bent into the form of a triangle. A D a Fig. 80. The bent wire ABC, Fig. 80, may be divided into three straight lengths AB, BC, and C-A, and the centres of gravity of these lengths are at their mid-points D, E, and F. The three lengths AB, BC, and CA may, therefore, be replaced by particles of the same weights at the points D, E, and F. The weights of the particles at D, E. and F are thus proportional to the lengths of the sides AB, BC, and CA of the triangle ABC. The centre of gravity of the two particles at D and E will, there- fore, be on the line DE at a point H, which divides the line, so that we have HD : HE :: BC : AB. The two particles at D and E anay, therefore, be replaced by a single particle of their combined weight at the point H, and it follows that the centre of gravity of the three particles at D, E, and F, or the centre of gravity of the bent wire, must lie on the line FH. Similarly, it can be seen that the centre of gravity of the two particles at E and F is on 182 GENERAL PHYSICS. the line EF at a point K, such that EK : KF :: CA : BC, and it follows that the centre of gravity of the bent wire must lie on the line DK. Since the centre of gravity of the wire is thus found to lie on the line FH, and also on the line DK, it must be at the point G where these two lines intersect. By following up the geometry of the figure with the help of Euc. vi. 3, it can be seen that the lines FH and DK bisect respectively the angles EFD and FDE ; and that the point G, the centre of gravity of the bent wire, is the centre of the circle inscribed in the triangle DEF. 3. A body of given material is made in the form of three cylinders arranged end to end with their axes in the same line. The cylinders are of the same length, but their diameters in order are in the ratio 1:2:3. Find the position of the centre of the gravity of the body. Since the cylinders are of the same length and material, and their | ; c X / 4 9 Fig. 81. diameters are in the ratio 1:2:3, their weights must be in the ratio 1:4:9. Hence, if 21 denote the length of each cylinder, and distances be measured along the common axis of the cylinders from the narrow end of the body as origin, we get, by dividing the body into the three cylindrical parts of which it consists, and then applying the relation, _ 2 (coo?) That is, 58^ 29J --- That is, the centre of gravity of the body is at a point G, Fig. 81, such that OG = 4f . I, where I is half the length of each cylinder. 4. A circular lamina has a circular piece cut out ; the radius of the piece cut out is one-half the radius of the lamina, and its centre is at a point whose distance from the centre of the lamina is one-fourth of CENTRE OF GRAVITY. 183 the radius of the lamina. Find the centre of gravity of the portion of the circular lamina which remains after the circular piece is cut out. The centre of gravity of the lamina as a whole is at its centre G, Fig. 82. The centre of gravity of the piece cut out is at a point A whose distance from G is one-fourth the radius of the lamina. The centre of gravity of the remaining portion of the lamina must, therefore, be at some point X on the line AG produced. This is evident, for the centre of gravity of the whole lamina must lie on the line joining the centres of gravity of the two parts here con- sidered. That is, the points AG and X must lie in a straight line. If the weight of the piece cut out be taken as 1 unit, the weight of the whole lamina will be 4 units, and the weight of the portion remaining will be 3 units. Fig. 82. Hence, if the distance GX be denoted by x and the radius df the lamina by r, and we apply the relation, or r 4' and r X = 1-2 * It should be noticed that, in this relation, when distances are measured from the centre of gravity of the body as origin, we must have x = 0. In this case with G as origin the distance GA is positive and GX negative according to the usual convention. 184 GENERAL PHYSICS. That is, the centre of gravity of the portion of the lamina which remains after the piece is cut out, is on the line AG produced, at a point X whose distance from G, the centre of the lamina, is one-twelfth of the radius of the lamina. Instead of applying the general relation, as above, it would be simpler in this case simply to take moments about G. If the line AGX s supposed to be horizontal the weights of the lamina and the two parts considered act vertically at right angles to AGX, and by taking moments round G, we get 5. Three bodies A, B, and C, whose weights are respectively 2 units, 3 units, and 4 units, are suspended by threads from the ceiling of a room. The distances of the centres of gravity of these bodies are respectively 2 feet, 4 feet, and 6 feet from one wall of the room, 3 feet, 5 feet, and 7 feet" from an adjacent wall, and 3 feet, 6 feet, and 9 feet from the floor. Find the position of the centre of gravity of the three bodies. The bodies may here be considered as particles placed at their respective centres of gravity. It will be seen by an extension of the result given above for particles in one plane that if x lt x. 2 , x s denote the distances of the bodies from the first wall, y lt y. 2 , y 3 their distances from the adjacent wall, and z lt z. 2 , z% their distances from the floor, we have where x,l/, and z denote the distances of the centre of gravity of the three bodies from the reference planes specified above. Hence, by applying these results we get - (2 x 2) + (3 x 4) + (4 x 6) 40 2 + 3-4 = -9 =: 4 *' - _ (2 x 3) + (3 x 5) + (4 x 7) _ 49 _ ^ 4 2+3 + 4 -"9" r _ (2 x 3) + (3 x 6) + (4 x 9) _ 60 _ 2+3+4 ~9~ That is, the centre of gravity of the three bodies is at a point distant 4* feet from the first wall, 5* feet from the adjacent wall, and 6 feet from the floor. CENTRE OF GRAVITY. 185 69. Centre Of Mass. It has been explained in Art. 68 that if particles of weights w v w. 2 , w ;3 . . . w,, lie in the same plane at points whose co-ordinates with reference to known axes are (#$,), (x 2 . y. 2 ), (x s . y s ) . . . (.<' H . y n ), the position of the centre of gravity of the particles is given by the relations _ <"I A 'I + fty'g + fry's . . . + fay'tt and If the masses of these particles be denoted by m v in.,, . . . w n their weights will be given by mtf, m. 2 y, g . . . m n g, and the relations given above reduce to _ _ m^ + mflz + It will be seen that in making the necessary substitutions the quantity g appears in both numerator and denominator of these relations, and, therefore, cancels out in the final form given above. It follows from this result that the position of the point whose co-ordinates are (,c, ~y) t really depends upon the distribu- tion of mass in the plane. That is, the position of the point called the centre of gravity of the body really depends upon the distribution of mass in the body, and is really the centre of mass of the body. The centre of mass of a body is most conveniently denned by the method indicated in Example 5 in Art. 68. If the distance of any particle in the body from three reference planes at right angles to each other and having a common point at the origin 186 GENERAL PHYSICS. be denoted by .r, ?/, and :, and if the mass of the particles be denoted by m, then the point whose co-ordinates are _ _ = is the centre of mass of the body. It will be seen that if a body is subject to the action of any force, such that the forces to which the individual particles of the body are subject are all parallel in direction, and proportional to the masses of the particles in magnitude, the centre of this system of parallel forces, and the point at which the force acting on the body may be supposed to act, is the centre of mass of the body. The attraction of the earth on any body is a force of this kind, as explained in Art. 67, and it follows that the centre of gravity of a body is at its centre of mass. The centre of mass of a body has some important dynamical properties. When a body is moving with pure motion of translation its velocity, as explained in Art. 26, is the velocity of any point in it ; but, if the body is moving in any way, its velocity, in accordance with Newton's laws of motion, is the velocity of its centre of mass. Similarly, the momentum of a body moving in any way is given by the product of the mass of the body into the velocity of its centre of mass. It can be shown, too, that the kinetic energy of a body in motion is the kinetic energy of the whole mass moving with the velocity of the centre of mass, together with the sum of the kinetic energies of its component particles relative to the centre of mass. 70. Experimental Method of Finding the Centre of Gravity Of a Body. If a body is suspended by a thread attached to the body at any point, the body will hang when at rest with its centre of gravity vertically below the point of suspension. The forces acting on the suspended body are its CENTRE OF GRAVITY. 187 weight acting vertically downwards at its centre of gravity G, as shown in Fig. 83, and the tension in the suspension thread acting vertically upwards at the point of suspension A. When the body hangs at rest these two forces balance each other ; they must therefore be equal in magnitude, and must act in opposite directions along the same straight line. That is, the vertical through G and the vertical through A are in the same straight line, and the point G is therefore vertically below the point A. Hence, if a body is suspended by a thread attached at any point A, the centre of gravity of the body lies on the prolongation of the direction of the thread through A. Similiarly, if the body is suspended by a thread attached at another point B, the centre of gravity lies on the prolongation of the direction of the thread through B. The centre of gravity will therefore be found at the inter- section of the two directions thus determined in the body. In many cases it is practically impossible to apply this method experimentally for the direc- tions of the lines on which the centre of gravity is known to lie cannot in general be marked, and the point at which they intersect cannot be determined. In the case of some bodies, how- ever, of open structure, such as a bird cage or any open frame work, it is generally possible to mark the prolongations of the suspension thread by means of threads, and so to fix the position of the centre of gravity at the point where these threads inter- sect. In the case of thin plane sheets or lamina of any rigid material, the centre of gravity is easily found by this method as- explained below. Experiment 6. Take a plane sheet of cardboard or tin-plate of any irregular shape, and find the position of its centre of gravity by Fig. 83. 188 GENERAL PHYSICS. suspending it by a thread attached successively at two different points, A and B, taken anywhere at the edge of the sheet. When the sheet hangs at rest by a thread attached at the point A, mark the prolongation of the direction of the thread on the face of the suspended sheet by means of a plumb line through A. Similarly, when the sheet hangs at rest by a thread attached at the point B, mark the prolongation of thread in the same way on the face of the sheet. The point at which the two lines thus drawn on the face of the sheet intersect is the centre of gravity of the sheet. Suspend the sheet by attaching the thread at other points, and note that the prolongation of the direction of the thread in every cape passes through the same point. From what has been said above, it will be understood that if a body is balanced at rest on a point or pivot, the centre of gravity lies on the vertical line passing through the point. Similarly, if a body is balanced at rest on a knife-edge the centre of gravity lies in the vertical plane through the knife-edge. Experiment 7. Find the centre of gravity of a rectangular block of non-uniform material by balancing it on a knife-edge taken parallel successively to the length, breadth, and thickness of the block. Mark on the block in each case the position of the plane in which the centre of gravity lies, and specify from the data thus obtained the position of the centre of gravity as determined by the point at which these three planes intersect. 189 CHAPTER XIII. i EQUILIBRIUM OP FORCES. 71. Equilibrium. When the forces acting on a body balance each other so that they cannot produce any change in the body's state of rest or motion the forces are in equilibrium. It follows from this that the magnitude of the resultant of a system of forces in equilibrium must be zero, and that the algebraic sum of the moments of the forces about any and every point must be zero. If the resultant is zero there can be no change in the motion of translation, and if the algebraic sum of the moments is zero, there can be no change in the motion of rotation. It is evident from what has been said that a body acted on by a system of forces in equilibrium must either be at rest or in uniform motion in a straight line. A body acted on by a system of forces in equilibrium is sometimes said to be in equilibrium. 72. Stable, Unstable, and Neutral Equilibrium. A body at rest* under the action of several forces which balance each other is said to be in equilibrium. If a body in equilibrium receives a small displacement, and the forces acting on it tend to restore it to its original position, it is said to be in stable equilibrium. If, however, when the body receives a small displacement, the forces acting on the body tend to increase the displacement, and * Or in uniform motion in a straight line. 190 GENERAL PHYSICS. so to displace the body further from its original position, the body is said to be in unstable equilibrium. In some cases when the body is displaced the forces acting on it tend neither to restore the body to its original position nor to displace it further from this position ; that is, the equilibrium of the body is not disturbed by the displacement. In any such case the body is said to be in neutral equilibrium. Thus, a cone resting on a plane horizontal surface, under the action of its weight and the resistance of the plane, is in stable equilibrium if it rests on its base, in unstable equilibrium if it is balanced on its apex, and in neutral equilibrium if it rests on its side. The equilibrium of heavy bodies is more fully considered in Arts. 79 and 80. 73. Equilibrium of Two Forces acting- at a Point. If two forces act at a point the obvious condition of equili- brium is that the forces must be equal in magnitude, and must act in opposite directions. It follows from this that two forces, acting at different points, will be in equilibrium if they are equal in magnitude and act in opposite directions along the same straight line. It will be seen, too, that if a number of forces act along the same straight line, and the magnitudes offerees acting in opposite directions be distinguished by difference of sign, the forces will be in equilibrium if the algebraic sum of their magnitudes is zero. 74. Equilibrium of Three Forces acting- at a Point. Three forces acting at a point will be in equilibrium if any one of them is equal in magnitude and opposite in direction to the resultant of the other two. It follows from this that if three forces acting at a point are in equilibrium their lines of action must all be in the same plane. The resultant of any two of the forces acts at the same point as the forces, and its line of action is in the same plane as EQUILIBRIUM OF FORCES. 191 the lines of action of the two forces of which it is the resultant. Hence, if the third force is opposite in direction to this resultant its line of action must also lie in the same plane as the two other forces. It will be seen, too, that if three forces acting at three different points are in equilibrium the lines of action of the forces must all meet at the same point, and must all lie in the same plane. For, if the three forces are in equilibrium any one of them and the resultant of the other two must be equal in magnitude, and must act in the opposite directions along the same straight line. But the line of action of the resultant of any two of the forces lies in the same plane as the lines of action of the forces^ and passes through the point at which these lines intersect. The line of action of the third force must, therefore, also lie in the same plane as the lines of action of the two other forces, and must pass through the point at which these lines intersect. That is, the lines of action of the three forces meet at one point, and lie in one plane. Hence, if any three forces are in equilibrium they may be considered as acting at a point, for their lines of action meet at a point at which the forces may be supposed to act. If three forces acting at a point are represented in magnitude and direction by lines drawn in order from any starting point, the three lines so drawn must form a closed figure. For, if the forces are in equilibrium the resultant is of zero magnitude, and the closing line, which represents the resultant in the representa- tion diagram, must therefore be of zero length. That is, the end-point of the diagram must coincide with the starting point, and the lines of the diagram must, therefore, form a dosed figure. In this case the closed figure will be a triangle, and the result here considered leads to the theorem, known as the triangle of forces, and its converse. The triangle of forces theorem states that if three forces 192 GENERAL PHYSICS. acting at a point can be represented in magnitude and direction by the sides of a triangle taken in order they must be in equilibrium. The converse of this theorem states that if three forces acting at a point are in equilibrium they can be represented in magnitude and direction by the sides of a triangle taken in order. Since all triangles, whose sides are parallel to three given directions, must be similar triangles, it follows from the converse theorem that if three forces acting at a point are in equilibrium, and a triangle can be found whose sides represent the forces in direction, they must also represent them in magnitude. (P) Fig 84. That is, if a triangle can be found whose sides are parallel to the lines of action of the forces, the lengths of the sides must be in the same ratio as the magnitudes of the forces. Thus, if three forces, P, Q, and R, acting at a point (Fig. 84), are in equilibrium, and a triangle ABC is drawn with its sides AB, BC, and CA parallel respectively to the lines of action of the forces P, Q, and E, then the lengths of the sides AB, BC, and CA must be in the same ratio as the magnitudes of the forces P, Q, and K. That is, we must have AB : BC : CA :: P : Q : R. This result is of great service in solving simple statical problems . relating to the equilibrium of three forces whose lines of action are not parallel. EQUILIBRIUM OF FORCES. 193 Examples. 1. A heavy particle C is suspended by two threads, AC and BC, from points A and B, 5 feet apart on a horizontal line, f the threads AC and BC are respectively 3 feet and 4 feet in length, d the particle weighs 10 grammes, find the tension in each thread. Let A and B (in Fig. 85) represent the two points from which the rticle at C is suspended by the threads AC and BC. The three forces acting on the particle at the point C are its weight, W, acting vertically downwards, the tension T x in the thread AC, and the tension T 2 in thread BC. Through the point C draw CD vertically upwards, and through A draw AD parallel to CB. It will now be seen that the sides DC, CA, and AD of the triangle DCA are parallel respectively to the lines of action of the forces W, T lt and T,,. It follows, therefore, that the magnitudes of these forces are proportional to the lengths of the corresponding sides. Since the sides AB, BC, and CA of the triangle ABC are re- spectively 5 feet, 4 feet, and 3 feet in length, it follows that the angle BCA is a right angle. It can now be easily proved that the triangle ADC is similar to the triangle ABC, and that its sides DC, CA, and AD are in the ratio 5:4:3. Since the three forces acting at C are in equilibrium, we get at once by the triangle of forces that :: 5:4:3. Fig. 85. The value of W is given as 10 gramme-weights, so that we have 10:T i: :5:4 and 10:T, ::5:3 That is, the tension in the thread AC is equal to the weight of 8 grammes, and the tension in BC to the weight of 6 grammes. This problem might have been worked by noticing that the sides AB, BC, and CA of the triangle ABC, are respectively perpendicular to the lines of action of the forces W, Tj, and T. 2 . Hence, if we suppose the triangle rotated through 90* in its own plane its sides 13 194 GENERAL PHYSICS. would be parallel to the lines of action of the forces, and the lengths of the sides would be proportional to the magnitudes of the forces. That is, we have ^ AB:BC:CA::W:T i: T 2 , or W: T I: T 2 :: 5: 4: 3, as obtained above. The construction given above is, however, more general, and can be applied to any similar problem. 2. A heavy particle rests on a plane inclined at an angle of 30 to the horizontal, and is kept in position by a thread parallel to the plane. If the reaction of the plane be supposed to be at right angles to the plane, find its magnitude : find, also, the tension in the thread. Let AB (in Fig. 86) represent a vertical section of the inclined plane on which the particle rests at the point P. The three forces acting on the particle at P, as shown in the figure, are its weight, W, acting vertically downwards, the reaction R acting at right angles to the plane, and the tension T in the thread acting upwards in a vertical plane (the plane of the figure), and parallel to the inclined plane. Through P draw a line vertically downwards and through any point, M, on this line ; draw MN at right angles to AB. It will be seen that the sides PM, MN, and NP of the triangle PMN are parallel respectively to the lines of action of the forces W, R, and T, and that the lengths of these lines are, therefore, propor- tional to the magnitudes of the forces they represent. Since the line AB makes an angle of 30 with the horizontal, the line MN must make an angle of 30 with the vertical. That is, the EQUILIBRIUM OF FORCES. 195 angle PMN in the triangle PMN is 30, and the sides PM, MN, and NP of the triangle must, therefore, be in the ratio 2 : *J'3 : 1. We, therefore, have W :T::2:1, T-f, and W:R::2:V3, WV3. or E= 2 That is, if W denote the weight of the particle, the tension of the w w Vs~ thread is , and the reaction of the plane is - ' or '866 W. [Construct the triangle ABC by drawing a horizontal line through A and a vertical line through B to cut the horizontal through A at the point C. It will be seen at once that this triangle is similar to the triangle MPN, and that the forces W, T, and R are proportional, therefore, to the lengths of the sides AB, BC, and AC. The side AB is usually called the length, the side BC the height, and the side AC the base of the inclined plane AB. We, therefore, have T _ Height ' W . Length' R Base W = Lenyth' Since these ratios are readily determined from the triangle ABC when the inclination of the plane to the horizontal is known, the triangle MPN need not be used in solving problems of this type.] 3. A ladder, weighing 24 pounds, rests on a horizontal floor against a vertical wall. The foot of the ladder is 6 feet from the wall, and the top rests on the wall at a height of 12 feet above the floor. If the centre of gravity of the ladder is at a distance of one-third the length of the ladder from its lower end, and if the reaction of the wall is assumed to act at right angles to the face of the wall, find the reactions at the wall and at the floor. In Fig. 87 let AB represent the ladder resting on the floor at A and against the wall at B, and let G denote the position of its centre of gravity. The three forces acting on the ladder are its weight, W, acting vertically downwards at G, the reaction of the wall, R 19 acting at B at right angles to the face of the wall, and the reaction of the floor, R 2 , acting at A. 196 GENERAL PHYSICS. The three forces are in equilibrium, so that their lines of action must meet at one point. The directions of the forces W and R x are known, and the point at which their lines of action intersect can be determined. The direction of the force R. 2 is thus found to be along AO, for its line of action must also pass through 0. The three forces W, R 1? and R 2 may thus be supposed to act at the point 0. From produce the line of action of the force W to cut the horizontal line through A at the point C. Then, since the lines OC, CA, and AO of the triangle AOC are parallel respectively to tile lines of action of the forces W, R 15 and R 2 , the length of these lines must be proportional to the magnitude of the forces. From the data of the question the length of OC is 12 feet, and the length of AC must be 2 feet, for, since AG = ^ AB, we must have AC = i AD, and AD is known to be 6 feet in length. The side AO is, therefore, Vl48 feet, or 12-165 feet in length. We, therefore, have W:R i:: 12:2, and W:R 2 ::12: Vl48 WV148 R '= T'2 ' Hence, since the ladder weighs 24 pounds, we have W = 24, and R 2 = 2\/l48 = 24-33. t of That is, the reaction at the wall, R ls is equal to the weight 4 pounds, and the reaction at the floor, R 2 , is equal to the weight of 2 Vl48, or 24 "33 pounds. The direction of R. 2 is seen from the figure to be such that it makes an angle equal to the angle AOC with the vertical on the same side of the vertical as the ladder. That is, the direction of R., makes with the vertical an angle whose tangent is . 75. Equilibrium of a Number of Forces acting* at a Point. If a number of forces acting at a point are in equili- brium their resultant must be of zero magnitude. Hence, if the forces be represented in magnitude and direction by lines EQUILIBRIUM OF FORCES. 197 drawn in order from any starting point, the lines so drawn must, as explained above, form a closed figure. In the case of forces whose lines of action all lie in one plane (Art. 61), this closed figure will be a polygon, so that the condition for the equilibrium of a number of co-planar forces acting at a point is usually stated in a theorem known as the polygon of forces. This theorem states that if a number of co-planar forces acting at a point can be represented in magnitude and direction by the sides of a closed polygon taken in order the forces must be in equilibrium. The converse of the theorem states that if a number of co- planar forces acting at a point are in equilibrium, they can be represented in magnitude and direction by the sides of a polygon taken in order. Fig. 88. It is important to notice, however, that it cannot be said, as in the case of the triangle of forces, that if the sides of a polygon are parallel respectively to the lines of action of the forces, the lengths of the sides are proportional to the magnitudes of the forces. This evidently cannot be true, for polygons whose sides are parallel are not necessarily similar figures. For example, the two polygons shown in Fig. 88 have their sides parallel, but they are evidently not similar figures. The converse of the polygon of forces is thus much less useful than the converse of the triangle of forces in the solution of statical problems. 76. General Condition for the Equilibrium of Forces acting 1 at a Point. If a number of forces acting at a point 0, 198 GENERAL PHYSICS. as in Fig. 89, are resolved into rectangular components along two straight lines, X'OX and YOY'. drawn at right angles to each other through the point O, and the algebraic sum of the components along each of the straight lines is zero, the forces will be in equilibrium. For, if X denote the algebraic sum of the components along X'OX and Y denote the algebraic sum of the components along Y'OY, then the resultant of the system of forces is given by R 2 = X 2 + Y 2 , and, in order that R may be zero, it is evidently necessary that X and Y should each be of zero value. The quantities X 2 and Y 2 being squares are necessarily of positive sign, so that their sum can be zero only when each quantity is of zero value. Y \ Y] Fig. 89. 77. Equilibrium of a System of Co-planar Forces acting* at Different Points in the Plane. A system of co-planar forces acting at different points in the plane will be in equilibrium if the algebraic sum of the moments of the forces is zero about every point in the plane. It has been explained in Art. 65 that a system of co-planar forces must either be in equilibrium or must reduce to a single resultant force or a couple. If the system reduces to a single resultant force, the algebraic sum of the moments will have different values for different points in the plane, and will be of zero value only for points on the line of action of the resultant. EQUILIBRIUM OF FORCES. 199 If the system reduces to a couple, the algebraic sum of the forces will have the same value for all points in the plane and will not be of zero value for any point. It follows, therefore, that,. if the algebraic sum of the moments of the forces is zero about every point in the plane, the system must be in equilibrium. In order to prove, in any given case, that a system of co- planar forces is in equilibrium, it is evidently sufficient to show that the algebraic sum of the moments is zero about any three points not in a straight line. For, if the system has a resultant, these three points must lie on its line of action. This, however, is impossible, for the three points are not in a straight line. The system cannot, therefore, have a resultant. It should be noted, as the converse of what is stated above, that, if a system of co-planar forces is in equilibrium, the algebraic sum of the moments of the forces is zero about any and every point in the plane. The conditions for the equilibrium of a system of co-planar forces may be stated in another way. Let each force of the system be resolved into two components at right angles along two given directions in the plane, then the system will be in equilibrium if (1) The algebraic sum of the components along each of these two directions is zero, and (2) The algebraic sum of the moments of the forces about any one point in the plane is zero. If the first of these two conditions is fulfilled, the system cannot, as explained in Art. 7 6, reduce to a single resultant ; and if the second condition is fulfilled, the system evidently cannot reduce to a couple. The system must, therefore, be in equilibrium. 78. Equilibrium of Parallel Forces. A co-planar system of parallel forces is merely a special case of the general co -planar system considered in Art. 77. The system will evidently be in 200 GENERAL PHYSICS. equilibrium if the algebraic sum of the forces is zero, and if the algebraic sum of the moments of the forces about any one point in the plane is zero. The special case of three parallel forces in equilibrium should, however, be noticed. Let the forces P, Q, and R, shown in Fig. 90, be in equilibrium, and let any transverse line ABC cut their lines of action at the points A, B, and C respectively. Since the three forces are in equilibrium, any one of them may be considered as equal and opposite to the resultant of the other two. Hence, if the force R, reversed and acting at C, be taken as the resultant of the forces P and Q acting at A and B respectively, we have, by Art. 63, P _ CB Q~CA Similarly, if the force P, reversed and acting at A, be taken Fig. 90. as the resultant of the forces Q and R acting at B and C respectively, we have Q _ CA R ~ BA' It follows at once from these two results that P : Q : R : : BC : AC : AB. That is, when three parallel forces are in equilibrium the magnitude of each force is proportional to the segment inter- cepted between the lines of action of the other two forces on any transverse line drawn across the lines of action of the forces. EQUILIBRIUM OF FORCES. 201 79. Equilibrium of a Heavy Body Supported at a Point. If a body is suspended by a thread attached at a point, or if it is balanced on a pivot point, or if it rests on a plane which it touches at only one point, it is in each case supported by a single force acting at the point of support. Hence, when a body is supported in this way it is acted on only by its weight and the force supporting it, and if these two forces are in equilibrium they must be equal in magnitude, and must act in opposite directions along the same straight line. The weight of the body acts vertically downwards through the centre of gravity of the body. The supporting force acting at the point of support, is, therefore, equal to the weight of the body, and must act vertically upwards through the centre of gravity of the body. That is, when the body is in equilibrium the centre of gravity of the body lies on the vertical line through the point of support. When the body is suspended by a thread the centre of gravity is vertically below the point of support, but when the body is balanced on a point, or when it rests on a plane which it touches at only one point, the centre of gravity is vertically above the point of support. These three cases are shown at A, B, and C (in Fig. 91), 202 GENERAL PHYSICS. where G represents the centre of gravity of the body, and A the point of support. If we consider a bodv supported at a fixed point, round which the body is free to move in any direction (thus excluding the case of a body resting on a plane as at C in Fig. 91), it will be seen that the body is in stable equilibrium when its centre of gravity is vertically below the point of support, in unstable equilibrium when the centre of gravity is vertically above the point of support, and in neutral equilibrium when the centre of gravity is the point of support. Fig. 93. Thus, in Fig. 92 it will be seen that if a body in equilibrium with its centre of gravity G vertically below the point of support, A, receives a small displacement into the position indicated by the dotted outline in the figure, the moment of the weight of the body round A, the point of support, tends to restore the body to its original position of equilibrium. That is. the body is in stable equilibrium. Similarly, it can be seen from Fig. 93 that if a body in equilibrium with its centre of gravity G vertically above the point of support, A, receives a small displacement into the position indicated by the dotted outline in the figure, EQUILIBRIUM OF FORCES. 203 the moment of the weight of the body acting round A tends to displace the body further from its original position of equilibrium. This body is, therefore, in unstable equilibrium. When a body is supported at its centre of gravity it is obviously in neutral equilibrium, for the weight of the body must act through the point of support in all positions of the body. That is, the body is in equilibrium in any position. When a body rests on a plane which it touches at only one point, the point of support changes if the body is displaced, and the stability of its equilibrium depends upon the form of its base. When the base is spherical it can be shown that the body is in stable or unstable equilibrium, according as its centre of gravity is below or above the centre of the spherical base. Thus, in Fig. 94 it can be seen that if the body shown Fig. 94. in the figure is displaced by tilting it on its spherical base, so that the point at which it rests on the plane changes from A to B, the moment of the weight of the body round B, the new point of support, tends to restore the body to its original position if the centre of gravity is at a point Gj, below C, the centre of the base,* but tends to displace it further from its original position if the centre of gravity is at a point G 2 above C. That is, if the centre of gravity of the body is at a point below the centre of its base the body rests on the plane in stable equilibrium, but if the centre of gravity is above the centre of the base, the body will not in practice stand on this base, but may, in theory, be balanced on it in unstable * It should be noticed that C will always be on the normal to the plane at the point of support. 204 GENERAL PHYSICS. equilibrium. An egg, for example, will not stand in stable equilibrium on either end, for its centre of gravity is above the centre of the (approximately) spherical base in each case. It may, however, conceiveably be balanced in unstable equilibrium on either end, but the slightest disturbance would bring it down to its usual position of equilibrium. It should be noticed that this position is one of neutral equilibrium for displacements at right angles to its length, but in stable equilibrium for displacements parallel to its length. In the one case the centre of gravity coincides with the centre of the section of the base in the plane of the displacement, arid in the other it is below this point. It will be seen at once that if the centre of gravity of the body is at the centre of the base, as in the case of a sphere, then the body will rest on the plane in neutral equilibrium. It follows from what has been said above that when a heavy body, free to rotate about a fixed line as axis, is in equilibrium, the centre of gravity of the body must be in the vertical plane through the axis, and the equilibrium will be stable, unstable, or neutral, according as the centre of gravity of the body is below the axis, above the axis, or on the axis. 80. Equilibrium of a Heavy Body Standing" on a Base On a Plane. When a body rests on a plane the area enclosed by a fine thread stretched tightly round the body at the surface of the plane, is called the base on which the body rests. A body resting on a plane on any base will stand in stable equilibrium if the vertical line through its centre of gravity falls within the base. If. however, the vertical through the centre of gravity falls without the base the body will overturn. Thus, in the case of a cube of any material resting on an inclined plane, the cube will stand in stable equilibrium if the vertical line through its centre of gravity, G, falls within the base ABCD, as in Fig. 95, but will evidently overturn by EQUILIBRIUM OF FORCES. 205 rotating round the edge AD if this vertical line falls without the base as in Fig 96. It will be seen from what has been said that a body resting on a horizontal plane may be tilted up, or the plane on which it rests may be tilted up without overturning the body so long as the vertical line through the centre of gravity falls within the base on which the body rests. The stability of a body resting on any base may thus be considered to depend on the amount of tilting necessary to bring the vertical through the centre of Fig. 96. gravity of the body to the edge of the base as a limiting position. If the height of the centre of gravity of a body above the base on which the body rests is large compared with the dimensions of the base, the limits of stability for the body will 206 GENERAL PHYSICS. be comparatively narrow, for under these conditions a com- paratively small tilt of the body will cause the vertical line through the centre of gravity to fall outside the base. Thus, if a cart is loaded with* a very high load so that the centre of gravity of the cart and load is some distance above the ground, the comparatively small tilt caused by one wheel passing over a stone or a bank of earth may cause the vertical through the centre of gravity to fall outside the wheel base, and the cart may be overturned. On the other hand, if the centre of gravity of body is low that is, if its height above the base on which the body rests is small compared with the dimensions of the base, the limits of stability for the body may be very wide. When a body can rest on a plane on different bases it will be found, by comparing its stability in the different positions, that the lower its centre of gravity is, the wider are the limits of its stability. A brick, for example, can rest on a plane in stable equilibrium on three different bases, and it is easily seen that the limits of stability for these three positions become wider as the centre of gravity is lowered. 207 CHAPTER XIV. FRICTION. The Force Of Friction. When two plane surfaces are pressed together, and force is applied tending to make one surface move over the other, an opposing force is, in general, set up in the plane of contact of the surfaces in a direction tending to prevent the motion. This force is known as the force of friction between the surfaces in contact. It is due to the roughness of these surfaces ; the small inequalities on one surface engage with the corresponding small inequalities on the other surface, and in this way each surface is able to exert a force on the other in a direction tending to prevent any displacement of one surface relative to the other in the plane of contact. The force of a friction is thus a stress, as the term is used in Art. 41, acting between the surfaces in the plane of contact parallel to the surfaces. That is, the surfaces exert equal and opposite forces on each other in this plane, the force acting on each surface being directed so as to prevent its displacement relative to the other in the plane of contact. If either of the surfaces in contact is smooth there is no friction between the surfaces, and no force is exerted in opposi- tion to the displacement of one surface over the other. That is, a smooth plane surface is unable to exert force on any surface in contact with it in any direction parallel to itself in the plane of contact. It can, therefore, exert force on any body in contact with it only in a direction at right angles to itself. That is, a smooth surface cannot at any point exert force in a 208 GENERAL PHYSICS. direction tangential to the surface at that point, but only in a direction normal to the surface at that point. No real material surface can be so 2 )er f ec ^y smooth that it offers no resistance tangentially to the motion of another surface over it, but a surface may in practice be so smooth that the resistance, is negligibly small. The use of oil as a lubricant between metal surfaces in contact tends to reduce greatly the friction between these surfaces. 82. The Limiting Value of the Force of Friction between Two Plane Surfaces. Let a rectangular block of any material be placed as at A in Fig. 97, with one of its plane faces resting on the plane horizontal surface of a plate, B, of the same material. If a force P is now applied to A in a Fig. 97. horizontal direction, the block A will tend to move by sliding over the plate A. The friction between the surfaces in contact will, however, oppose this tendency to motion, and a force F acting in the plane of contact in a direction opposite to that of P will be established. If the force P is supposed to be at first very small, the force F will also be very small and equal to P. Then, as P is increased, the value of F also increases in such a way that the two forces are always exactly equal in magnitude and opposite in direction. The force P may be supposed to increase indefinitely, but the force F evidently cannot increase beyond a certain maximum limit determined by the nature of the surfaces in contact and the pressure exerted between them. This maxi- mum limit to the value of the force of friction between the FRICTION. 209 two surfaces, is called the limiting value of the force of friction between the two given surfaces under the existing conditions. When the force P is increased to this limiting value the block A will be on the point of slipping over the plate B, and when P is increased beyond this limit the block will move over the plate under the action of a force (P F) where F denotes the maximum limiting value of the force of friction between the two surfaces in contact. Experiment 8. Get a rectangular block of wood about 6" x 3" x 2" in size, and a plank of the same wood about 3 feet long, 1 foot wide, and 1 inch thick. The surfaces of the block and plank should be plane and even but not too polished or smooth. Fig. 98. Arrange these two pieces of wood on a table, as shown in Fig. 98, so that by means of the pulley at P and the cord carrying the scale pan at S, a force may be applied to the block A in a horizontal direction. This force is applied and measured by the weights placed in the scale pan and evidently tends, as the apparatus is arranged, to make the block A slide over the surface of the plank B. Begin the experiment by placing a small weight in the scale pan and then go on adding to this weight until the force of friction is no longer able to prevent slipping, and the block A begins to slide over the board on which it rests. The weight in the pan (together with the weight of the pan itself) when the block A is on the point of slipping is evidently the maximum or limiting value of the force of friction between the two surfaces. For when the block is on the point of slipping the force applied to A must be equal to the maximum force of friction 210 GENERAL PHYSICS. acting in the opposite direction. The force of friction must evidently be at its maximum value, or slipping would not be about to take place. The point at which slipping is about to take place is a little difficult to determine exactly. the board B should be gently tapped and the weight in the scale pan should be very gradually increased as the slipping point is approached. It will be found by repeating the experiment a number of times that the value obtained for the limiting fricton is fairly constant. It will generally be noticed in performing the foregoing experiment that the force, which is only just sufficient to bring the block A to the point of slipping over B, is sufficient, after slipping once takes place, to keep the block in motion and to give it a small acceleration. This shows that the friction between the two surfaces at rest, relative to each other, is slightly greater than the friction between the same two surfaces when in motion relative to each other. That is, the statical friction between the two surfaces is slightly greater than the dynamical friction between the same two surfaces. It is important to understand that the limiting value of friction found by an experiment similar to that described above is merely the limiting value in the particular case tested by the experiment. It is obviously to be expected that any change in the conditions under which the friction takes place will change the magnitude of the limiting value. 83. The Laws Of Friction. The limiting value of the friction between any two plane surfaces must obviously depend upon the material on which the surfaces are formed and on the degree of roughness of the surfaces. Although these conditions may be specified more or less definitely, they cannot be measured, and cannot, therefore, be involved in any quantita- tive law. The only measurable quantities on which the friction between the two surfaces may depend are the area of contact of the surface and the pressure exerted normally between the surfaces. The laws of friction deal, therefore, only with the relations FRICTION. 211 between the limiting value of the friction and these two measurable quantities for any two surfaces. These relations can be determined only by experiment. The relation between the limiting value of the friction for any two given surfaces, and the area of contact of the surfaces may be determined by the method of the following experiment. Experiment 9. Set up the apparatus of Experiment 8 and find the limiting value of the friction between the block and the plank when the block rests on the plank successively on each of the three sides of different area. In this way the area of contact between the surfaces is varied without altering the normal pressure between the surfaces, for the normal pressure is in each case equal to the weight of the block, and is, therefore, constant. It will be found that the limiting value of the friction is practically the same in each case, and is, therefore, independent of the area of the surface in contact, provided the normal pressure between the surfaces is constant. The same result will be obtained by using a block and plate of any given material, or a block of one material and a plate of another. That is, the result is true for any two specified surfaces. A block of the dimensions given in Exp. 8 is rather too small for use in this experiment. A block 8" x 6" x 4" will give more consistent results. The surfaces of the block and the plank should be very even and uniform. The result of this experiment shows, therefore, that the limiting value of the friction between any two surfaces is independent of the area of the surface of contact The relation between the limiting value of the friction and the normal pressure between the surfaces may be determined by the following experiment. Experiment 1O. Set up the apparatus of Exp. 8 with the block resting on the face of greatest area. The normal pressure exerted between the surfaces is, in this case, equal to the weight of the block, but if weights are placed on the block this pressure can evidently be adjusted to any required value without altering the area of the surface of contact, and without altering in any way (unless the pressure is made excessive) the nature of the surfaces in contact. 212 GENERAL PHYSICS. Hence, if we determine the limiting value of the friction, first using the unloaded block, and then the block carrying a number of different loads, we can obtain data from which we can determine the relations between, the limiting value and the normal pressure. It will be found, for any two specified surfaces that the ratio of the limiting value of the friction to the normal pressure between the surfaces is constant. Example. In an experiment of this kind the following data and results were obtained : Weight of Block. Load on Block. Total Weight of Block and Load Limiting Value of Friction Value of Ratio F R. F. R Grammes- Grammes- Grammes- Grammes- weight. weight. weight. weight. 1,000 1,000 210 210 500 1,500 325 217 1,000 2,000 425 212 1,500 2,500 549 216 2,000 3,000 645 215 3,000 4,000 850 212 4,000 5,000 1,080 216 F Average value of ^ = -214. It will thus be seen that for any two given surfaces the limiting value of friction is directly proportional to the normal pressure between the surfaces. That is, the ratio of the limiting value of friction to the normal pressure between the surfaces is constant. This is the important quantitative law of friction. The laws of friction may, therefore, be stated in the following terms.* * It would be more consistent with actual facts to state these laws as follows : The limiting value of the force of friction between any two surfaces is (1) Directly proportional to the normal pressure per unit area between the surfaces. (2) Directly proportional to the area of contact of the surfaces. FRICTION. 213 1. The limiting value of the force of friction between any two surfaces is directly proportional to the normal pressure exerted between the surfaces. 2. The limiting value of the force of friction between any two surfaces is independent of the area of contact of the surfaces. The dynamical friction between two surfaces in relative motion may evidently depend upon the velocity of the motion. If in an experiment similar to those described above the weight in the pan is made great enough to set the block in motion with acceleration, it is found that the acceleration is uniform. This shows that for the limited range of velocity possible in an experiment of this kind, the limiting value of the friction is constant, and practically independent of the velocity. 84. The Coefficient Of Friction. It has been explained in the foregoing article that the limiting value of the friction between any two surfaces is directly proportional to the normal pressure between the surfaces, and that the ratio of the limiting value of the friction to the normal pressure is, therefore, con- stant for two given surfaces. This ratio is called the coefficient of friction for the two given surfaces. That is, the coefficient of friction for any two specified surfaces is the ratio of the limiting value of the friction between these two surfaces to the normal pressure between the surfaces. Hence, if F denote the limiting value .of the force of friction, and R the normal pressure between the surfaces, we have F = Coefficient of friction, K or, as the coefficient of friction for any two surfaces is generally denoted by /x, we have F R = * or F = &. 214 GENERAL PHYSICS. The experimental determination of the coefficient of friction for any two given surfaces evidently involves the determination of F, the limiting value of the friction corresponding to any convenient value of K. the normal pressure between the surfaces. The determination can, therefore, be made conveniently by the method of Experiment 10. This experiment is, in fact, a determination of the coefficient of friction for the surfaces used, and the example given at the end of the experiment shows how the value of the coefficient of friction can be calculated from the data of the experiment. The values of the ratio F/R, given in the last column of the table in the example, are values of fji, the coefficient of friction for the surfaces to which the data apply, and the mean value of the ratio given at the bottom of the table is the mean value given by the experiment of the coefficient of friction for these surfaces. The determination may also be made by the method of the following experiment. Experiment 11. Take a block and plate similar to that used in Experiment 10, and place the blcck on the plate so that the surfaces for which the coefficient of friction is required are in contact. If necessary, one face of the block and the upper surface of the plate may be coated or covered with the surfaces to be tested. Place the plate and block in a table, and tilt the plate gradually by raising one end until the block is on the point of slipping down the inclined surface of the plate. The coefficient of friction for the two surfaces is then given by the tangent of the angle at which the surface of the plate is inclined to the horizontal when slipping is about to take place. For, let PQ in Fig. 99 denote the position of the plate when the block A is on the point of slipping down the plane. The forces acting on the block are its weight, W, acting vertically downwards through its centre of gravity, the limiting friction F acting up the plane, and the normal reaction of the plane R acting outwards at right angles to the plane. Let PN, the horizontal line through P, and QN, the vertical line through Q, meet at the point N. FRICTION. 215 The three forces, P, Q, and R, whose lines of action meet at the point 0, are in equilibrium by the triangle of forces, as explained in Example 2 in Art. 74, and we therefore have F R~ fJL = QN PN' QN PN' Hence, if the distances QN and PN are carefully measured for the position at which the block A is on the point of slipping, the ratio of the two distances, taken as above, gives the coefficient of friction for the two surfaces in contact. It will be seen that | = tan QPN. That is, if a denote the angle at which the plane PQ is inclined to the horizontal when slipping is about to take place, we have F = fj. = tan a. j\ Hence, if instead of measuring the distances QN and PN, the angle Fig. 99. NPQ or a is measured, the coefficient of friction, /n, is given by the relation, p. = tan a. For this reason, a, the angle whose tangent is /^, the coefficient of friction, is sometimes called the angle of friction. More accurate results can, however, be obtained in simple experi- ments by using a fairly long plane, PQ, and measuring the distances QN and PN to determine /u. , Example. In an experiment similar to that described above three separate determinations of the slipping point were made, and the distances QN and PN were measured for each determination. 216 GENERAL PHYSICS. The following data were thus obtained : QN (mm.^ PN (mm.). QN ^ = PN' 243 962 251 246 961 256 240 963 249 Mean value of /*, . "252 The values of the coefficients of friction in a few common cases are given below. Surfaces in Contact. Values of /*. Hardwood on hardwood ; with the grain, fibres parallel, 0'50 Hardwood on hardwood ; with the grain, fibres at right angles, 0'33 Hardwood on hardwood ; across the grain, . . . 0'26 Metal on hardwood, . . . . . . . . 0*55 Wrought iron on wrought iron, . . . . . 0'18 Cast iron on cast iron, ....... 0'15 It will be understood, however, that the coefficient of friction for surfaces of given material must vary within a somewhat wide range with the exact state of the surfaces. 85. The Friction Dynamometer. The friction dynamo- meter is a form of friction brake which can be applied to a pulley or flywheel driven by an engine, in order to measure the horse-power of the engine. A simple form of this type of dynamometer is shown in Fig. 100. It consists of a belt or rope, AB, passed over the pulley P in the manner shown in the figure. One end of the belt at A is attached to a strong spring-balance S, and the other end at B carries a weight W, which can be adjusted to any desired value. The pulley is driven by the engine in the direction indicated by the arrow, so that the friction belt is urged by the friction FRICTION. 217 between it and the pulley in the direction AB. The force of friction between the belt and the pulley acts everywhere along the tangent to the circumference of the pulley, and must, there- fore, act on the belt at every point in a direction parallel to its length and tending to pull it round from A towards B. It follows from this that the pull on the spring-balance S will be greater than the weight W by the force of friction exerted by the pulley on the belt. Hence, if P denote the pull exerted on the spring-balance, and F the force of friction between the belt and the pulley, we have P = F + W, or F = P - W. The value of F can thus be determined if the values of P and W are known. It can also be increased or diminished as may be required by increasing or decreas- ing the value of W. Thus, when W is increased P also increases, and as the pressure of the belt on the pulley is in this way increased the value of F must also increase. Similarly, when W is decreased, the values of P and F also decrease. In order to use this brake to measure the power of an engine or motor the pulley is driven by the engine, and the weight W on the brake is adjusted until the engine is found to be exerting its full power and working generally under the conditions under which it is to be tested. The reading of the spring-balance at S, and the rate of revolution of the pulley (given by a speed indicator), are then carefully noted and recorded. Fig. 100. 218 GENERAL PHYSICS. Let P denote the pull indicated by the spring-balance, and W the weight on the brake, then, as above, we have P - W = F. Now, during each complete revolution of the pulley this force, F, the force of friction exerted by the belt on the pulley, is overcome through a distance equal to the circumference of the pulley in the direction in which the force acts. Hence, if r denote the radius of the pulley, and n the number of revolu- tions made by the pulley per second, the work done against friction per second is given by F . 2 irnr or 2 irnr (P W). That is, the power absorbed and measured by the brake is given by 27TW(P-W). Example. In the determination of the power of a motor by means of a friction dynamometer the following data were obtained : Weight on brake, . . 10 pound- weights. Pull on spring-balance, . 16 ,, Speed of revolution, . . 1,200 revs, per minute. Radius of pulley, . . 6 inches. Find the power absorbed by the brake in horse-power. From the result given above, we have Power abstracted = 2 irnr (P - W), and in the English F.P.S. system we have n = 20 (revs, per second). r = -5 (foot). P = 16 (pound- weights). W = 10 ( ,, ). The power absorbed is therefore (2-r x 20 x *5 x 6) ft. -pounds per second, 1201T, or horse-power, 550 or '685 horse-power. 86. Reaction of a Rough Plane Surface. The reaction of a smooth plane surface against any body pressing on it can act FRICTION. 219 only in a direction normal to the surface. For, if we suppose the reaction at any point P, Fig. 101, on the surface to act in the direction PR, making an angle NPR, with the normal PN, we can resolve the reaction into a normal component along PN, and a component parallel to the surface along PM. But, if the surface is smooth its reaction at any point cannot have a component parallel to the surface, for a smooth surface cannot exert force in a direction parallel to itself The reaction of a smooth surface on any body can act, there- fore* only along the normal to the surface. The reaction of a rough plane surface may, however, act in a direction inclined at an angle to the normal, for it is, in general, the resultant of the normal reaction of the surface acting as in P M Fig. 101. A Fig. 102. the case of a smooth surface, and the force of friction acting parallel to the surface. Thus, when a heavy rod is set up against a wall, with one end resting on the ground, the reaction of the ground, as shown at A in Fig. 102, is the resultant of the normal reaction R, acting vertically upwards, and the frictional reaction F, acting parallel to the ground in the proper direction to prevent slipping. The magnitude of F will be only just sufficient to prevent slipping, and may, therefore, have any value between zero and its maximum limiting value. Whatever value F may have within these limits, the direction of the resultant reaction of the surface at A is such that it makes an angle with the normal at A whose tangent is F/R. 220 GENERAL PHYSICS. The size of this angle depends upon the value of F, and may have any value between zero, when F is zero, and a certain maximum value when F has its maximum limiting value. If the rod is on the point of slipping F will have its limiting value for the two surfaces in contact, and the ratio F/R will be the coefficient of friction for the surfaces. The greatest angle which the direction of the reaction at A can make with the normal is, therefore, the angle whose tangent is F/R, where F has its maximum limiting value, and F/R = /m, where JJL is the coefficient of friction for the surfaces in contact. That is, the greatest angle which the reaction of a rough surface can make with the normal to the surface is the angle of friction. 221 CHAPTER XV. THE BALANCE. 87. Theory of the Balance. The general construction of a simple form of balance has already been described in Art. 23. The elementary theory of its construction and action can now be considered. Let A, B, and C, in Fig. 103, represent sections of the knife- edges of the beam of a balance in a vertical section taken length- wise through the beam at right angles to the edges. The edges at A and B carry the scale -pans, and the edge at C is that on which the beam rests and turns. G Fig. 103. These knife-edges are fixed on the beam, so as to be exactly parallel to each other at right angles to the length of the beam, and they are usually set so as to lie truly in the same plane. The central knife-edge at C is fixed exactly midway between the two edges at A and B, so that the line AB is bisected at C ; and the two arms, CA and CB, are exactly equal. In order that the beam may set in stable equilibrium with the line AB horizontal when balanced, without the scale-pans, on the Itnife-edge at C, the centre of gravity of the beam and every- thing rigidly attached to it must be at a point G below C on the line CG, drawn through C at right angles to AB. 222 GENERAL PHYSICS. In the case of a beam constructed in this way, it is clear that if equal weights are suspended from the knife-edges at A and B their resultant must act at the knife-edge at C, and cannot, therefore, disturb the equilibrium of the beam. This can also be seen by taking moments round the knife-edge at C ; for, if the weights at A and B are equal, their moments round C must be equal and opposite, since the arms CA and CB are equal. Hence, if scale-pans of equal weight are suspended from the knife-edges at A and B, the beam will still set in stable equilibrium with the line AB horizontal as before, and it will always set in equilibrium in this position when the weights in the pans are exactly equal. A balance which fulfils this condition is said to be true. It should be noticed that it is essential in order that a balance may be true (1) that the arms of the beam should be equal, and (2) that the scale-pans should be of equal weight. For, let R and L denote the weights of the right-hand and left-hand pans respectively, and r and I the length of the right-hand and left- hand arms respectively, then if the balance is true the beam must set in equilibrium with AB horizontal with no load in the pans, and also when the same load W is carried by each pan. We must, therefore, have Rr = L/, and also (R + W)r = (L + W)/. Taking the difference of these two equations, we get and it follows from this, since Rr = L/, that R = L. Hence, in order that a balance may be true, so that it sets in equilibrium with the plane of the knife-edges horizontal when THE BALANCE. 223 the pans carry equal loads, the arms of the beam must be equal, and the pans must be of equal weight. Another important characteristic essential to a good balance is sensibility or sensitiveness. A balance must weigh truly, but it is even more essential that it should be sensitive to a small difference in the weights in the pans, and should indicate any very small difference of this kind by an appreciable deflection of the beam from its position of equilibrium. Suppose a balance to carry a weight W in one pan, and a weight (W -f x) in the other pan, and let it set in equilibrium when so loaded, with the line AB inclined at an angle a to the horizontal, as shown in Fig. 104. B ^ (W) Fig. 104. Since the beam is in equilibrium in this position the moments of the forces acting on it, taken round the central knife-edge at C,. must balance each other. The resultant of the two equal weights, W, acting at A and B must act at C, so that the algebraic sum of their moments round C is zero, and they need not be considered in taking moments round C. It follows, therefore, that for equilibrium we must have the moment of the weight x acting at A round C equal to the moment of the weight of the beam acting at G round C. That is, if X denote the weight of the beam, we must have 224 GENERAL PHYSICS. Now, if a denote the length of the arm CA, and h denote the distance CG, we have CE = a cos a, and CF = h sin a. - We therefore get x . a cos a = X/i sin a, a or tan a ^=-r- . x. X.h This result shows that, for a given value of x, the value of a depends upon the value of . That is, for a given small A./& difference, x, between the weights in the pans, the value of a, and the sensibility of the balance, may be increased by making the ratio a/h as large as possible, and the weight of the beam X as small as possible. It will be seen, however, that if the sensibility of a balance is made high by increasing a and decreasing h and X, the balance will be very slow in action, for its period of oscillation about any position of equilibrium will be very long. If we suppose the beam and the pans, carrying equal loads, to be in oscillation about its position of equilibrium, the moment tending to restore it to its position of equilibrium for any angular displacement, a, from this position is evidently X/i sin a, as explained above. The period of oscillation depends, therefore, as explained in Art. 43, upon the value of X/t sin a, and the moment of inertia of the oscillating system made up of the beam and the loaded pans. If X/i sin a is small, and the moment of inertia great, the period of oscillation may be very long, so that if X and h are small, and a is comparatively large, the time of swing might be so long that it would be impossible to make a weighing in any reasonable time. It thus appears that the very conditions which are necessary for high sensibility are those which make the balance impractic- ably slow in action. It follows, therefore, that in designing and THE BALANCE. 225 constructing a balance a compromise must be effected between sensibility and quickness of action. This compromise has led to the construction of balances of two different types: long beam balances of comparatively slow action, and short beam balances of quicker action. A short beam balance is generally provided with a somewhat long pointer moving over a finely divided scale, and in some cases a reading microscope is used for reading the position of the pointer on the scale. In this way very small deflections of the beam from its zero position can be detected, and the working sensibility of the balance is increased. In connection with what has been said above, it should be noticed that the stability of the balance beam in its position of equilibrium depends upon the centre of gravity of the beam, G, being below the knife-edge C, and upon the 'moment XA sin a being sufficiently great to make the beam come to rest always in the same position. It is very important that the zero or equilibrium position of the balance should be constant and invariable, and, for this reason only, it is necessary that the moment X/t sin a should not be too small. The reasons for setting the knife-edges of the beam all in one plane can now be considered. It is clear that when the pans carry equal loads the forces acting on the beam at the knife- edges A and B (Fig. 105) are equal, and that the resultant of these two forces acts vertically downwards at a point C midway between A and B. If the central knife-edge is set at this point in the same plane with those at A and B, as in Fig. 103, this resultant can have no moment round it, and it follows that the sensibility of the balance and the moment of the couple tending to restore it to its position of equilibrium (X.h sin a) are quite independent of the load carried by the balance. The time of swing of the balance will not, however, in this case be inde- pendent of the load ; for, although the moment, XA sin , is constant for all loads, the moment of inertia of the swinging 15 226 GENERAL PHYSICS. system increases with the load, and the time of swing will therefore increase as the load increases. If, however, the central knife-edge is set, as at C' in Fig. 105, in the plane bisecting AB at right angles, but above the point 0, the beam will set in equilibrium with the points C and G vertically below C', but the resultant force acting at C will have moment round the knife-edge at C' when the beam is displaced from its horizontal position of equilibrium. If E denote the magnitude of the resultant acting at C, the moment acting on the beam for a displacement a will evidently be X/i. sin a + Rti sin a, where h denotes the distance C'G and h' the distance C'C. From this result it will be seen that, under these condi- tions, the sensibility of the balance will decrease as the load 4i i* G Fig. 105. increases, and that the change in the time of swing as the load increases will be less marked than when C and C' are coincident. If the knife-edges were arranged so that C and G coincided, the time of swing would be practically constant. The case in which the central knife-edge is fixed below the point C need not be considered, for it can be seen that, except under certain evident conditions, a beam with the knife-edge so arranged would be in unstable equilibrium. It will be seen, therefore, that, in the case of a balance constructed in the usual way, with the knife-edges all in one plane, the sensibility will be practically independent of the load, but the time of swing will increase as the load increases. One very important essential in the construction of a balance is the rigidity of the beam. The beam should be rigid enough THE BALANCE. 227 to show no appreciable bending under the maximum load it is designed to carry. The girder beams in general use for Fig. 106. a, Central knife-edge ; b, b, end knife-edges ; c, c, stirrups for carrying pans ; d, graduated bar for rider ; /, milled head for lowering the pan rests and releasing the beam ; g, pan rests for arresting and supporting the pans ; h, spirit level ; i, levelling screws ; K, gravity bob ; M, I, I, arrangement for moving the rider. accurate balances are designed to give the necessary rigidity without making the weight of the beam unduly great. 228 GENERAL PHYSICS. The general details of the construction of an accurate balance are somewhat complicated, and are best learnt by studying the construction and action of a good balance practically in the laboratory, and by working with it. A good form of balance is shown in Fig. 106, and an enlarged view of the beam of this balance is given in Fig. 107. In connection with the theoretical discussion given above, it should be noted that the small screw vane V, shown in Fig. 107, K Back view. Fig. 107. is provided as a means of adjusting the position of the centre of gravity of the beam so that it lies, as explained above with reference to Fig. 103, below C on the line CD, which bisects AB at right angles at C. It will be seen that by turning the vane to the right or to the left,. the position of the centre of gravity is moved very slightly in the same direction. The gravity bob K, shown in the same figure, is provided for raising or lowering THE BALANCE. 229 the centre of gravity of the beam by screwing the bob up or down. It thus acts as a fine adjustment for adjusting the sensibility of the balance. 88. Practical Determination of the Sensibility of a Balance. The sensibility of a balance is usually determined by finding the difference in load which will deflect the pointer attached to the beam through one division on the scale over which it moves. Thus, if it is required to determine the sensibility of a balance when the load in each pan is 20 grammes, the method of the following experiment might be adopted. Experiment 12. Find the sensibility of the given balance for a load of 20 grammes in each pan. Set the beam of the balance free, and see if it swings freely and regularly, and is generally in proper adjustment. Then place a 20 gramme load in each pan, and find the division on the scale at which the pointer comes to rest with this load on the pans. This may be done by simply waiting until the beam stops swinging, and then reading off the division of scale at which the pointer comes to rest. It can, however, be done much more expedi- tiously by following the movement of the pointer over the scale as the beam swings, and reading the turning points for any three successive swings. If then we take the mean of the first and third readings, and then the mean of this mean and the second reading, we get the reading at which the pointer would come to rest. Thus, if the three observed turning points are at divisions 6, 17, and 8 on the scale,* the pointer would come to rest at the division marked 12 on the scale. [The reason for this method of taking the mean of the observed readings is readily understood. If the beam in swinging were quite free from friction and air resistance, its swing would obviously be of constant amplitude, and the resting point could at once be found by taking the mean of any two successive turning points. On account, however, of the damping effect of friction and air resistance, the swings gradually decrease in amplitude, and in order to find . * The scale is supposed to be numbered from one end, not from the middle, as it sometimes is. The need for plus and minus signs to distinguish between right and left readings is thus avoided. 230 GENERAL PHYSICS. the resting point by observing the turning points it is necessary to take the mean in such a way as to eliminate the decrement due to damping. If this decrement is small it may be supposed to be the same for several consecutive swings, so that if we denote it by x it will be seen that with no damping the pointer would swing between two constant turning points at, say, the pth and qih divisions on the scale ; but with damping, the successive turning points for a small number of swings would be approximately at the divisions p, (q-x), (p + 2x), (q-3x), (p + 4x), &c., on the scale. Now, if we take any three consecutive turning points, such as (q - x), (p + 2x), and (g-3a;), it will be seen that the mean of the first and third is (q - 2x), and that the mean of this mean and the second is the mean of (q-2x) and (p + 2x), or the mean of p and q which is obviously the true resting point. It will be seen that the same result is obtained by taking any odd number of successive turning points, and taking the mean, first of those for swings to the right, then of those for swings to the left, and, finally, the mean of the two means so obtained.] Having found the resting point with a load of 20 grammes in each pan, now place a small weight, say, a milligramme, in one pan, so as to make a difference of one milligramme in the weights carried by the pans, and find again the reading on the scale at which the point comes to rest. Suppose this resting point to be at 7 on the scale. It follows from these data that a difference of 1 milligramme in the loads on the pans changes the resting point of the pointer through 5 divisions on the scale. That is, a difference of *2 milligramme between the weights in the pans gives a deflection of 1 division on the scale. This is the sensibility of the balance when the load on the pans is 20 grammes. We can find in the same \vay the sensibility of the balance for different loads, from no load to the full load the balance can carry. It will be found, as a rule, that the sensibility is practically the same for all loads. It generally decreases slightly as the load is increased. From the data obtained a curve may be plotted showing how the sensibility varies, in the case of the given balance, with the load. The accuracy of this experiment depends upon the care with which the different resting points are determined. Each resting point should be determined as the mean of several consistent determinations. It will be understood that the sensibility of a balance found in the manner explained in this experiment, is a purely THE BALANCE. 231 empirical quantity, which depends for its absolute value on the length of the scale divisions, and also on the length of the pointer. When the sensibility of a balance is known the weight of a body may be determined without wasting time in making the final small adjustments of the weights which are usually necessary to bring the beam exactly into its position of equilibrium. If it is found when the adjustment is nearly complete that the resting point is n scale divisions from the balancing position, and if s is the sensibility of the balance in milligrammes per scale division, the weight of the body is evidently ns milligrammes greater or less than the weight in the pan, according as the weight pan is lighter or heavier than the other. This method of weighing is sometimes called the method of weighing by vibrations. 89. The Use Of Riders. Let the arm of a balance be divided into ten equal parts, and let the divisions be numbered from to 10 outwards from the centre, the division marked being at the central knife-edge, and the division marked 10 at a terminal knife-edge. The distance of any one of these divisions from the central knife-edge is thus equal to a certain number of tenths of the length of the arm, and it follows at once, by the principle of moments, that if a given weight is carried by the beam at that division, it is equivalent only to a certain number of tenths of its real weight placed in the scale-pan. Thus, if a centigramme is placed on the beam at the division marked 3, it is equivalent to 3 milligrammes in the scale-pan, or if placed at the division marked 8, it is equivalent to 8 milli- grammes in the scale-pan. Hence, if a piece of platinum wire, weighing exactly one centigramme, is bent into the form shown in Fig. 108, so that it can be placed as a rider at any point on the divided beam, the use of milligramme weights can be dispensed with, for any weight smaller than a centigramme can be obtained by adjusting the position of the rider on the beam. If each of 232 GENERAL PHYSICS. the ten divisions on the arm is further subdivided into ten divisions, the equivalent weight of the rider at any point on the arm can evidently be obtained in milligrammes and tenths of a milligramme, by simply reading the position of the rider on the scale marked along the divided arm. Thus, if the rider is placed at the 63rd division on this scale it is equivalent to 6*3 milligrammes in the scale-pan. The process of weighing with a rider thus resolves itself into the following procedure. The weighing is first made to the nearest centigramme by placing weights in the pan in the usual way. The position of the rider on the beam is then adjusted Fig. 108. until an exact balance is obtained. The required weight is then obtained from the weights in the pan and the position of the rider. Thus, if the weight in the pan is found to be 3 '5 4 grammes, and the rider is at division 36 on the beam scale, the weight required is 3*5436 grammes. It is sufficient for most purposes if the beam is divided only along one arm on the same side as the pan in which the weights are usually placed in weighing. It is usual, however, to divide the beam along the whole length between the terminal knife- edges, so that the weight equivalent of the rider may be added to, or taken from, the weight in either pan, as may be convenient. THE BALANCE. 233 In most balances a special bar is attached to the beam for the purpose of carrying the rider and special appliances are provided for putting on and taking off the rider without opening the balance-case or disturbing the beam. In Fig. 106 the rider bar is shown at RR, and the lever for lifting and carrying the rider is shown at L. The general details relating to the use of a rider on an accurate balance are best learnt by actual practice in weighing. 90. Special Methods Of Weighing. In cases where the accuracy of the balance is in doubt, any weighing may be tested by the following special methods. Place the body to be weighed in one pan, and counterpoise it exactly with fine shot or pieces of wire in the other pan. Then remove the body and put weights in its place until an exact balance is again obtained. By this method the weights must give truly the weight of the body, whether the balance is accurate or inaccurate, provided it is sufficiently sensitive for the purpose. The weights, it will be seen, are placed in the same pan as the body, and they balance the same counterpoise under exactly the same conditions, so that their weight must be exactly equal to the weight of the body. Another method of weighing which is useful for detecting and eliminating any error that may be caused by the arms of the balance not being exactly equal. The body is weighed in the ordinary way, first in one pan and then in the other, and the geometric mean of the two weights so obtained is taken as the true weight. When the two weights are very nearly equal, as they always would be in practice, a sufficiently accurate result is obtained by taking their arithmetic mean instead of the geometric mean. Let P and P' denote the two weights obtained by weighing the body first in the left pan and then in the right pan, and let I and r denote the lengths of the left and right arms respectively of the balance. Then, if W denote the true weight of the body, 234 GENERAL PHYSICS. and if we assume the balance to be in exact equilibrium when there is no load in the pans, we must have PI = Wr, and PY = W/. From these relations we at once get PP' - W 2 , or W - That is, the true weight W is the geometric mean of the false weights P and P'. When, however, P and P' are nearly equal, so that P' = P + , where S is small, we have w = Vp(PTiy=r(i + ) = P (1 + = P + That is, when P and P' are nearly equal, the true weight W is approximately equal to their arithmetic mean. 235 CHAPTER XVI. GENERAL PROPERTIES OP MATTER. 91. The Constitution of Matter. A piece of matter of any particular kind is supposed to be made up of minute ultimate particles or molecules which are assumed to be the smallest particles of that particular kind of matter which can exist independently. If the piece of matter is supposed to be divided and subdivided into smaller and smaller pieces, the ultimate particles into which it can conceivably be divided, and still continue to be matter of the same particular kind, are its molecules. As explained below, a molecule is generally divisible into component parts called atoms, but the molecule is the physical unit in the constitution of matter, and any particular kind of matter, whether it be an element or a compound, is supposed to be built up of its molecules and not of its atoms as constituent units. The molecules which make up any piece of matter are supposed to be aggregated together without being actually in contact. Force is exerted mutually between the molecules and the group of molecules which constitute any portion of matter are held together by these intermolecular forces. It is supposed also that the molecules of a body are not at rest, but in rapid vibratory motion. It is thus assumed that the molecules of a body are free and distinct from each other, that a stress of attraction or *236 GENERAL PHYSICS. repulsion * exists between each molecule and every surrounding molecule within its range,f and that every molecule is in rapid vibratory motion. A piece of matter considered as a system of molecules may thus possess molecular potential energy in virtue of the configura- tion of the system, and molecular kinetic energy in virtue of the motion of its molecules. The size of a molecule is almost inconceivably small. We have no exact knowledge of the actual form or size of a molecule, but approximate estimates can be made in several ways of the probable order of magnitude of the diameter of a molecule considered as a small spherical particle. Some idea of this magnitude may be obtained by considering that a piece of ordinary gold leaf, which is less than four- millionths of an inch in thickness, probably consists of more than a hundred layers of molecules. Lord Kelvin illustrates the size of a molecule by stating that if a drop of water were magnified to the size of the earth the molecules would be about the size of an orange or a cricket ball. Although the molecule is the unit in the physical constitu- tion of a piece of matter, the molecule of any substance is itself a group of component parts called atoms. Thus a molecule of water is a group of three atoms made up of one atom of oxygen and two atoms of hydrogen. Similarly, a molecule of chalk is a group of five atoms made up of one atom of calcium, one atom of carbon, and three atoms of oxygen. When the molecules of any substance are made of atoms of * It has been suggested that the law expressing the stress between two molecules is such that the stress is one of attraction or repulsion, accord- ing as the distance between them is greater than or less than a certain small limit. tlf the stress between two molecules decreases very rapidly with increase in their distance apart, it may become negligibly small beyond a certain small range. GENERAL PROPERTIES OF MATTER. 237 different kinds the substance is said to be a compound substance, but when the molecules of a substance are made up of one or more atoms of the same kind, the substance is said to be an elementary substance. Thus water and chalk are compound substances, but a substance, such as oxygen or hydrogen, whose molecules are made up of two similar atoms is an elementary substance. The atoms of an elementary substance are said to be atoms of that substance, although, strictly speaking, the substance is made up only of molecules. A molecule is thus supposed to be a group of atoms held to- gether by interatomic forces in much the same way as the molecules of a piece of matter are held together by intermolecular forces. Until quite recently an atom was considered to be an ultimate and indivisible particle of matter. There is now, however, abundant experimental evidence to establish the theory that an atom is really a group of component particles held together by the stresses existing between the particles. There are probably only two kinds of particles which enter in this way into the constitution of atoms, and it is probable that the atoms of different substances differ from each other only in the number and grouping of their component particles. It thus appears to be possible for the atoms of one substance to change by a process of disintegration and regrouping into the atoms of another substance. That is, any substance in which the atom is a large and complex group may possibly change into a substance in which the atom forms a smaller and simpler group. This process of change from one substance to another may take a very long time or a very short time, and may pass through many well-marked intermediate stages. The element radium derives the interest which at present attaches to it from the fact that its atoms are supposed to be in process of disintegration, and there is satisfactory evidence to show that the element helium is derived from radium, as a result of this disintegration. *238 GENERAL PHYSICS. 92. States Of Aggregation Of Matter. The three normal states of aggregation of matter are the solid state, the liquid state, and the gaseous state. In a piece of matter in the solid state the molecules are so aggregated together under the control of intermolecular stresses that their relative positions are fixed, and every molecule is able to offer resistance to displacement in any direction from the position it occupies in the body. That is, a molecule may be displaced slightly from the mean position it occupies without causing any rupture in its relations with the sur- rounding molecules, but the adjustment of the stresses between it and the surrounding molecules is disturbed by the displacement, and a resultant stress which opposes the displacement, and in- creases as the displacement increases, is thereby set up. Hence, if a force is made to act on any particle in a solid body, the particle may be slightly displaced in the direction of the force, but the displacement sets up in the material around the particle an opposing stress which resists the displacement, and tends to restore the displaced particle to its initial position. If the force applied to the particle is sufficiently great to displace the particle beyond a certain limiting position, the particle breaks away from the rest of the material, or becomes the starting point of a fracture in the material. It will be seen from what has been said, that if two equal forces acting in opposite directions along the same straight line, act on a solid body at points A and B, as in Fig. 109, the material of the body between A and B will be subjected to a tension or a pressure, and, in virtue of its properties as a solid, will be able to sustain and resist this stress unless it exceeds a certain limit. The body will be slightly stretched or GENERAL PROPERTIES OF MATTER. 239 compressed by the applied stress, but the internal stress set up in the material by this change in the molecular configuration of the body, will resist and balance the applied stress up to the yielding point of the material. When a block of any solid material rests on the ground under the action of its own weight, it is evident that any horizontal slice of the block must be subject, in this way, to pressure. The weight of the overlying portion of the block acts vertically downwards on the slice, and the resistance of the underlying portion on which the slice rests acts vertically upwards on it, and the slice is compressed or squeezed together under the action of these two forces. The internal stress set up in the material of the slice by this compression is, however, able to balance the external stress, and the slice, although slightly compressed, is able to sustain the pressure to which it is subject. It follows from what has been said that a given piece of matter in the solid state must, under given conditions, have a definite volume and a definite form, and that it is able to offer resistance to any change in configuration which involves a change of volume or a change of form. In matter in the liquid state the molecules are also aggregated together under the control of intermolecular stresses, but the relations between any moleculfe and those surrounding it are such that it is free to move about in any direction in the liquid. That is, a given molecule in any mass of liquid occupies no particular position in the mass, and offers no resistance to dis- placement in any direction from any position it may happen to occupy. This last statement is, however, subject to an important qualification. If force is applied to any molecule in a mass of liquid under such conditions that the molecule can be displaced only by forcing it nearer to, or further from, the surrounding molecules, it offers a very great resistance to displacement. That is, a definite mass of liquid possesses a definite volume, and offers 240 GENERAL PHYSICS. very great resistance to change of volume by compression or expansion. A mass of liquid * cannot, however, be said to possess a definite form or shape, and it is unable to offer even the smallest resistance to any change of shape which may be impressed on it. If, therefore, a mass of liquid is subject to the action of any force however small, which tends to change the existing shape of the mass, and is not prevented from doing so by the action of other constraints, the mass will undergo continuous and pro- gressive change of shape, and will extend or flow out in all directions in which it is free to move. Thus, if we imagine a cube of water, or any similar liquid placed as a cube on a plate, we know from ordinary experience that it would almost instantaneously spread out, or flow out into a thin layer covering the bottom of the plate. If we consider what takes place in this case, we can see that any thin hori- zontal layer of the liquid in the cube is subject to pressure due to the weight of the overlying liquid ; this pressure forces the upper and lower molecules of the layer in between the inner molecules in such a way that, while the layer is not in the least compressed, it is compelled to spread out, or flow out horizontally in all directions. Since the liquid is not compressed, the mole- cules offer no resistance to the displacements thus imposed on them, and the flow goes on freely under the action of the smallest force. This process goes on progressively in every layer until the liquid finally comes to rest as a thin layer covering the bottom of the plate. In the case of a mobile liquid, such as water, the process takes place too rapidly to be observed, but in the case of a thick viscous liquid, such as syrup, the process is a slow and gradual one, and may easily be observed. If a quantity of syrup r for example, is poured on a large plate, it at first forms an irregularly shaped heap in the middle of the plate. This * The effects of Surface Tension are not considered here. See Art. 118. GENERAL PROPERTIES OF MATTER. 241 heap, however, gradually spreads out horizontally, and ultimately the liquid flows all over the surface of the plate and comes to rest only when further extension in a horizontal direction is prevented by the sides of the plate. If a quantity of liquid were poured on a plane horizonal surface of indefinite extent it would spread out horizontally in this way until the film of liquid on the surface is so reduced in thickness that it begins to exhibit effects due to surface tension. It follows directly from what has been said above, that a liquid may be made to flow or may be poured from one vessel to another, and that when poured into any vessel it readily assumes the form imposed on it by the interior of the vessel. When a mass of liquid is at rest in any containing vessel the conditions are very different to those considered above. Any thin horizontal layer of the liquid is subject to the pressure due to the weight of the overlying liquid, and this pressure, as explained above, tends to make the layer flow outwards in a horizontal direction. This outward flow is, however, prevented by the pressure exerted inwards by the walls of the containing vessel on the edge of the layer. The layer will thus be compressed until the opposing stress set up in the liquid balances the external stress, and equilibrium is established. Every layer of the liquid is in this way supported in equilibrium, and the whole mass of liquid rests in equilibrium in the containing vessel. Force can be applied to the surface of a liquid only as a pressure or tension* uniformlyf distributed over the surface and acting normally or at right angles to the surface ; and a mass of liquid can be maintained in equilibrium within any given * The application of pressure is easily understood ; tension can be applied only under certain special conditions, and need not be further considered. t The weight of the liquid is here neglected. 16 242 GENERAL PHYSICS. boundary only by the action of a uniform stress of this nature all over the boundary surface. Thus, if a quantity of liquid, supposed to be without weight, is contained in any vessel open to the air, it will be subject to the atmospheric pressure over its free or exposed surface, and the walls of the vessel will exert everywhere an equal pressure per unit area acting normally over the whole of the surface with which they are in contact. Since the pressure which can be exerted by any surface on a liquid, or the pressure which a liquid can exert on any surface in contact with it, must, in a weightless liquid, be uniformly distributed over the surface, it is most conveniently measured as the pressure per unit area, for the pressure per unit area must be constant. The pressure exerted on a liquid or by a liquid is, therefore, usually measured and expressed as pressure per unit area. It will be seen that pressure acting at the surface of a liquid must be at all points normal or at right angles to the surface ; if we suppose the pressure acting on the liquid at any point to have a component parallel to, or tangential to the surface, the molecule at that point would be displaced in the direction in which the component acts, for a molecule in a liquid can be displaced in any direction by the smallest possible force. That is, the liquid can be in equilibrium throughout its mass only when the pressure over the surface of the liquid is at all points normal to the surface. It will be seen, too, that the pressure must be uniform if distributed over the boundary surface, for if the pressure per unit area is greater at one point than another, the liquid would flow from the region of I highest pressure to the regions of lower pressure. That is, the liquid would flow* through the boundary surface at the areas of low pressure, and equilibrium would be impossible. For example, if we attempt to compress a liquid in a vessel with * It must be remembered that the liquid is supposed to be without weight. GENERAL PROPERTIES OF MATTER. 243 holes in its walls, the liquid will be forced out through the holes if the external pressure acting on the liquid through the holes is less than that imposed on the liquid by the end-surface of the compressing plunger and the walls of the vessel. If, however, the external pressure over each hole is equal to the internal pressure impressed on the liquid, the mass of liquid in the vessel would be in equilibrium, and no flow would take place through the holes. It will readily be understood from what has been said that a mass of liquid in equilibrium under the action of a uniform normal pressure at its boundary surface is really subject to this pressure everywhere throughout its mass. That is, if we take, anywhere in the liquid, an imaginary surface separating any two portions of the liquid, the pressure exerted mutually between the two portions across this surface is normal to the surface, and equal to the pressure at the boundary surface. This pressure expressed, as explained above, as pressure per unit area is, therefore, appropriately called the pressure in the liquid. In the case of a real liquid possessing weight the conditions for the equilibrium of a mass of liquid are complicated by the effect of the weight of the liquid. It will be seen that the pressure impressed on a thin horizontal layer at any depth in the liquid must be greater than the boundary pressure impressed on the free surface of the liquid by the additional pressure due to the weight of the overlying liquid. It follows from this that the pressure at the boundary surface of the liquid cannot be uniform, but must increase with depth below the level of the free surface of the liquid. The properties of a liquid are mere fully considered in Chapter xx. The plastic state, which occurs in most substances during tj' transition from the solid to the liquid state, is dealt with in Art. 43 in Part iv. on Heat. 244 GENERAL PHYSICS. In matter in the gaseous state the molecules are supposed to be so far apart that the intermolecular forces are negligibly small. That is, the molecules are not aggregated together under the control of the intermolecular forces, but are free to move about in any direction quite independently of each other. It follows from this that a quantity of matter in the gaseous state cannot possess molecular potential energy of configuration, for if the intermolecular forces are negligibly small, no work is done against intermolecular force in effecting any change of configuration. It is assumed, in accordance with a theory known as the kinetic theory of gases, that the molecules of a gas move about in the space occupied by the gas with great velocity, and that they are constantly in collision with each other, and with the walls of the space in which they are enclosed. The path of any molecule between any two successive collisions is supposed to be a straight line, and although the mean or average length of this free path is really very short, it is long compared with the inter- molecular distances in solids and in liquids. The mean length of this free path from collision to collision is called the mean free path of the molecule. If a small quantity of any gas is introduced into any large space unoccupied by any other matter (a vacuum), we know from experience that it at once expands and fills the whole space. Or, if a small quantity of one gas is introduced into a large space already occupied by another gas, we know that it quickly spreads throughout the whole space, for after a very short time indications of its presence may be found in any part of the space. These results are evidently in accord with the kinetic theory for by this theory the molecules of a gas are free to extend their excursions in space outwards in all directions, and the only limit which can be set to the space which might be occupied by GENERAL PROPERTIES OF MATTER, 245 a given quantity of gas, is the mechanical limit set by the wall or boundary enclosing the space. It is a well-known experimental fact that a quantity of gas enclosed in any space exerts pressure on the walls of the enclosure. This pressure, which is exerted mutually between the gas and the walls of the enclosure, acts everywhere at right angles to the surface, and is measured by the pressure per unit of area. It is called the pressure of the gas, for it is found that it exists everywhere in the gas as a stress acting normally across any interface separating any two contiguous portions of the gas. The pressure which a given quantity of any particular gas exerts on the walls of the enclosure containing it, is found to depend on the capacity of the enclosure that is, on the volume occupied by the gas. If this volume is decreased the pressure increases, and if the volume is increased the pressure decreases. That is, if the gas is compressed into a smaller volume the pressure increases, but if it is allowed to expand and occupy a larger volume, the pressure decreases. These facts are explained on the kinetic theory by supposing that the pressure which a gas exerts on the walls of the enclosure containing it, is due to the continuous bombardment of the walls by the molecules of the gas. Every second a very large, and practically constant, number of molecules moving with high velocities strike and rebound from the walls of the enclosure, and by so doing exert a practically continuous and constant pressure on the walls. Example. A rain of small indiarubber balls, each weighing 1 gramme, falls vertically upon a plane horizontal surface, and, on an average, 1,000 balls fall upon every square metre of the surface every second. If the balls strike the surface with a velocity of 20,000 cms. per second, and rebound from it vertically upwards with the same velocity, find the average pressure exerted on each square centimetre of the surface. Every ball, by its impact with the surface, loses its downward 246 GENERAL PHYSICS. momentum, and gains an equal upward momentum. That is, every ball loses a momentum of (1 x 20,000) C.G.S. units in one direction, and gains (1 x 20,000) C.G.S. units in the opposite direction. The total change of mommtum which every ball undergoes by its impact on the surface is, therefore, (2 x 20,000) C.G.S. units. The number of impacts which take place in one second over a square metre of the surface is 1,000, so that the total change of momentum per second produced by the resistance offered by a square metre of the surface, to the rain of balls impinging on it, is (1,000 x 2 x 20,000) C.G.S. units, or, 4 x 10 7 C.G.S. units. But this rate of change of momentum measures the resistance offered by the surface in C.G.S. units of force. That is, the resistance offered by a square metre of the surface to the rain of balls impinging on it ; or, in other words, the pressure exerted by the rain of balls on every square metre of the surface, is 4 x 10 7 dynes. The average pressure exerted on one square centimetre will, therefore, be 4,000 dynes, or nearly 4 'OS gramme- weights. If this assumption as to the nature of the pressure exerted by a gas on the walls of the enclosure containing it is accepted, it is evident that the pressure must increase as the volume occupied by the gas decreases, for as the volume decreases the molecules become more crowded together, and the number of impacts per second on any given area of the walls must increase. Similarly, the pressure must decrease as the volume occupied by the gas increases, for as the volume increases the molecules become less crowded together, and the number of impacts per second on any given area of the walls must decrease. It can be shown, too, that the manner in which the pressure of a gas actually varies with its volume, as established by experiment, is the same as the manner in which it ought, theoretically, to vary in accordance with this assumption. It will be seen from what has been said that the aggregation of the molecules in the gaseous state differs essentially from that which obtains in the solid state or in the liquid state. A gas resembles a liquid in the fluidity which results from its GENERAL PROPERTIES OF MATTER. 247 mobility to resist change of form, but it differs essentially from a liquid in its indefinite compressibility and in its power of indefinite expansion. It should be noticed that resistance is offered by a solid or a liquid both to compression and to expansion, and that this resist- ance is due in each case to an opposing intermolecular stress set up in the material. In a gas there is no intermolecular stress opposing compression or expansion. Increase of pressure is necessary, as explained above, to produce compression, but expansion takes place freely when the pressure is decreased. The general properties of a vapour, and the distinction between a vapour and a gas, are dealt with in Chapter ix. of Part iv. on Heat. The conditions under which the liquid and gaseous states become continuous, and the critical state are also dealt with in the same chapter. 93. Inertia. The inertia of matter, as explained in Art. 35, in dealing with Newton's first law of motion, is one of its most characteristic properties. It is the property in virtue of which a body that is, a piece of matter continues in its state of rest or of uniform motion in a straight line unless acted on by the force. It will be understood, also, from what has been said in Chapter ix.. that it is the property which enables quantity of matter to be measured in units of mass. 94. Gravitation. The power which every piece of matter possesses of attracting every other piece of matter is one of the fundamental properties of matter. The force of attraction exerted mutually between any two pieces of matter is generally known as gravitation. The most familiar example of gravitation is the attraction exerted between the earth and bodies on its surface. The force of attraction exerted by the earth on any body at its surface is usualty called the force of gravity, and constitutes, as already explained, the weight of the body. 248 GENERAL PHYSICS. Gravitation is not, however, confined to the earth and bodies near it. Every piece of matter in the universe attracts, and is attracted by every othr piece of matter in the universe. That is, gravitation is exerted throughout the material universe, and if we wish to emphasise this fact we may use the term universal gravitation, instead of the simpler, general term. The law of gravitation was first correctly stated by Newton in the following form : The force of attraction between two particles of matter is directly proportional to the product of their masses, and inversely proportional to the square of the distance between them. That is, if two particles of masses m and rti are placed at a distance d apart, the force of attraction F, exerted mutually between them, is such that we have * 7 mm' or F = Ic . -jp , where k is a constant, known as the constant of gravitation. The formula F = k -^ applies primarily to the case of two particles of matter placed at a distance d apart, for in this case there is no ambiguity as to the distance denoted by d. It can be shown, however, that it applies also to the case of two spherical bodies of masses m and m respectively, placed with their centres a distance d apart. It will be understood, too, that the formula may be applied with approximate accuracy in the case of any two bodies whose dimensions are small compared with their distance apart. This law was deduced by Newton from a careful study of the available data relating to the motion of the heavenly bodies. These data had previously been studied by Kepler, and systema- GENERAL PROPERTIES OF MATTER. 249 tised by him into a few general results, known as Kepler's "Laws." Newton worked on the assumption that the general law governing the motion of all celestial bodies is the law of gravitation, and he showed that if the law takes the form given above, all Kepler's " laws " can be at once deduced from it, and that the laws stated in this form must, therefore, be in accord- ance with the data from which Kepler's empirical laws were derived. The truth of the law is now established beyond doubt. It has been since Newton's time, the basis of all astronomical calculations involving the forces acting between bodies moving in space, and the accuracy of the results obtained show that the law must be true. As an illustration of this Newton showed, by calculation from known data, that the force of attraction exerted by the earth on the moon in accordance with this law is exactly the force necessary to keep .the moon moving in its (approximately) circular orbit round the earth with the velocity it actually possesses. Example. Show that the acceleration of the moon moving in its circular orbit round the earth is the acceleration due to the force of attraction exerted by the earth on the moon. The following approximate data will be needed. Radius of earth, 4,000 miles. Radius of moon's orbit round the earth is approximately 60 times the radius of the earth. Time in which the moon makes one complete revolution round the earth is about 27 days 8 hours. If the velocity of the moon in its orbit round the earth be denoted by v, and the radius of the orbit by r, the acceleration of the moon towards the centre of its orbit that is, towards the earth is given by - , as explained in Art. 32. From the data here given the value of v in feet per second is 2ir x 60 x 4,000 x 5,280 656 x 60 x 60 and the value of r in feet is 60 x 4,000 x 5,280. 250 GENERAL PHYSICS. v 2 The value of in ft.-per-sec. per sec. is, therefore, 4-n- 2 x 60 x 4,000 x 5,280 (656) 2 x (3,600) 2 or -00897. That is, the acceleration of the moon in its circular motion round the earth is '00897 ft.-per-sec. per sec., and is directed towards the earth at the centre of its circular orbit. The acceleration of a body at the surface of the earth due to its weight that is, to the attraction of the earth on it is known to be 32'2 ft.-per-sec. per sec. The distance of a body from the earth (or from any spherical mass), is its distance from the centre of the earth, so that the distance of the moon from the earth is 60 times the distance of a body at the surface of the earth from the earth. The acceleration of the moon due to the attraction exerted on it by the earth will, therefore, in 32 '2 accordance with the law of gravitation, be T^TT?, ft.-per-sec. per sec., (oU)~ or '00895 ft. -per-sec. per sec. , and is directed towards the earth. The acceleration of the moon towards the earth, calculated from the data of its actual motion, is thus the same (within the limits of the error due to the use of approximate data) as the acceleration to which it is subject, as the result of the attraction exerted on it by the earth calculated in accordance with Newton's law of gravitation. If we consider the relation mm which expresses the law of gravitation, it will be seen that the law implies that the attraction between two particles depends only on their masses and their distance apart, and is quite independent of the material of which they are made. That is, in the formula F - Te mm ' " ~d r ' the gravitation constant k has the same constant value for matter of all kinds, and is not a specific constant having different constant values for different materials. GENERAL PROPERTIES OF MATTER. 251 The truth of this is established by the fact that the accelera- tion due to gravity is the same for all bodies whatever may be the material of which they are made. Thus, if M denote the mass of the earth, m the mass of a small body at the surface of the earth, and R the radius of the earth, we have where F is the force of attraction exerted by the earth on the small body. That is, F is the weight of the small body and is equal to mg, where g denotes the acceleration due to gravity at the point where the small body is situated. We may, therefore, write Mm kU or g . This result shows that if g is the same for all bodies at the same place, k must also be the same for all bodies, for =^ is necessarily constant. Newton and other experimenters investigated the constancy of the acceleration due to gravity for all bodies at the same place by a series of carefully-conducted experiments with the pendulum. The period of vibration of a simple pendulum has been proved to be given by the relation, IT where I denotes the length of the pendulum, and g the accelera- tion due to gravity at the place where the pendulum vibrates. From this relation we get 252 GENERAL PHYSICS. so that if g is the same for bodies of all materials at a given place, the period of vibration of a simple pendulum of a given length at that place, should be constant and quite independent of the material of the bob. It was found, after varying the material of the bob in many different ways, that provided the length of the pendulum, or the length of the equivalent simple pendulum, remained constant, the period of vibration was constant and quite independent of the nature of the material or materials which made up the bob. Experiment 13. Make three simple pendulums of exactly the same length by attaching small spherical bobs of lead, brass, and iron, to fine threads about two metres long. Suspend these so that they can vibrate one in front of the other in parallel planes at right angles to the plane in which they hang at rest. Set the three pendulums vibrating in the same phase with the same amplitude, and note that as long as they continue to vibrate they keep together in the same phase. This proves that each of the three pendulums has exactly the same period of vibration. That is, the period of vibration is independent of the material of the bob ; and it follows from this that the accelera- tion due to gravity at any place is independent of the material of the body subject to the force of gravity, and also that the gravitation constant is the same for all materials. The value of the gravitation constant /; can be found from the relation mm HF* by determining experimentally the value of F for known values of m, m f , and d. This determination \vas first made by Henry Cavendish in an historical experiment, now generally known as the Cavendish experiment. The full details of this experiment cannot be given here, but the general method of the experiment must be briefly indicated. Two small spheres of lead were attached to the ends of GENERAL PROPERTIES OF MATTER. 253 a light -wooden lever, and the lever was suspended by a long fine wire attached at its middle point so that it hung in a horizontal position. Suspended in this way the lever sets in a definite position of rest, and if deflected from this position in a horizontal plane, the suspension wire becomes twisted, and the moment of the couple due to the torsion on the wire tends to restore the lever to its position of rest. Thus, if T denotes the moment of the couple which is able to twist the wire through unit angle, then the moment of Fig. 110. the couple due to the torsion of the wire when the lever is deflected through an angle a from its position of rest is To in a direction tending to restore the lever to its initial position. This lever was suspended under conditions suitable to the experiment, and means were provided for observing its position and for measuring accurately any small deflection from that position. Let AB in Fig. 110 represent the plan of the suspended lever carrying the two small balls of lead at A and B. If now two large spheres or balls of lead are placed at C and D, as 254 GENERAL PHYSICS. shown in the figure, with their centres in the same horizontal plane as the lever, the large balls at C and D will attract the small balls at A and IJ respectively, and the two forces which thus act on the lever at A and B will tend to deflect it in the same direction from its position of rest. These two forces will in fact constitute a couple which deflects the lever into a new position of equilibrium at C'D', where the moment of the couple is balanced by the moment of the opposing couple due to the torsion on the suspension wire. In the same way if the large lead balls are removed from their position at C and D and placed in corresponding positions at E and F on the other side of the lever, the attractions exerted by them on the small balls carried by the lever will deflect the lever into a corresponding position of equilibrium at E'F. This was the method adopted by Cavendish. Two large spheres of lead were prepared and set up on a rotating stand which was so arranged that the spheres could be placed, relative to the lever, in the position indicated at C and D or at E and F, and could be moved from one position to the other by a simple movement of the stand. The positions of the lever at C'D' and E'F' were carefully observed, and the angle, C'OE', between these two positions was measured as accurately as possible. If this angle, C'OE', is denoted by 2a, and if the CD and EF positions of the large spheres are truly symmetrical, relative to the AB position of the lever, the angle through which the lever is deflected by the action of the attraction couple in either of the two positions is denoted by a. Hence, if M denote the mass of a large sphere, m the mass of a small sphere, and d the distance between the centres of the spheres when exerting attraction mutually on each other in a position of equilibrium at C and C', D and D', E and E', and F and F', the force of attraction between the spheres will be given by GENERAL PROPERTIES OF MATTER. 255 F = k =-, where k is the constant of gravitation, and in d each case this force will act along the line joining the centres of the sphere. The moment of the attraction couple acting on the lever in either position of equilibrium is given by F/ or Id p-, where / denotes the length of the arm of the couple. This moment must, however, be equal to the moment of the opposing couple due to the torsion of the wire, and as the twist on the wire is denoted by a when the lever is in either position of equilibrium the moment of this torsion couple will be Ta, where T denotes, as explained above, the moment of the couple able to twist the wire through unit angle. We must, therefore, have Mm w- = or k -^fj . a. Mml The value of k can thus be determined, for the values of T, M, m, /, d, and a can all be found experimentally. The value of T is most conveniently found by determining the time of vibration of the lever about its position of rest. It can be shown that if I denote the moment of inertia of the lever and balls round the suspension wire as axis, the time of /"T" A 2j vibration is given by t = STT^^ or T = 2 . The value found for k by Cavendish was about 6 '5 6 x 10~ 8 in C.G.S. units. A number of determinations of this constant have been made since the time of Cavendish by different experi- menters in England and in other countries. In England a determination was made by Poynting by a method in which 256 GENERAL PHYSICS. the ordinary balance was used instead of the torsion balance used by Cavendish. The value of k given by this method in one series of experiments was 6 '6 6 x 10~ 8 in C.G.S. units. A later determination was made in 1894 by Boys, who adopted the original torsion balance method, but used a very fine quartz fibre instead of a fine metal wire for the suspension of the lever. The value given by Boys for the constant is 6-6576 X 10- 8 in C.G.S. units. It will be seen that if the value of k is known it is at once possible to calculate the mass and the mean density of the earth. Thus, if m denote the mass of any small body at the surface of the earth at a place where g is the true acceleration due to gravity, M the mass of the earth, and R its- radius, we have,* JM 9 = R' or M = l , and M can be calculated from this relation for g and R are known. When M and R are known the mean density of the earth can be found, for we evidently have D = _M_ 3M or D = 47TR 3 ' where D denotes the mean or average density of the earth. If we take 6 '65 7 6 X 10~ 8 , given by Professor Boys, as the value of k, the value obtained for the mean density of the earth is 5'527 grammes per cubic centimetre. 95. Elasticity. Elasticity is a property of matter, in virtue of which a body is able to resist change of size or change * The form of the earth is assumed to be truly spherical. GENERAL PROPERTIES OF MATTER. 257 of shape, and in virtue of which it tends, while resisting the change, to recover its original size or shape, and is able, when the force causing the change is removed, to recover completely its original size or shape, provided the change has not exceeded certain small limits which differ for different materials. A body is thus said to offer elastic resistance to change of size or change of shape when the stress set up in the material by the change not only opposes the change, but at the same time tends to restore the displaced particles of the body to their original positions. Thus, if a soft ball of clay is squeezed flat between the finger and thumb the resistance it offers to the change of shape is not elastic resistance, for although the friction between the particles of clay opposes the displacement of one particle relative to another, it does not at any stage in the process tend to restore the displaced particles to their original positions. The resistance in this case is frictional in character, and resembles the resistance offered by a viscous liquid to change of shape, as explained below. In general a body is able to offer elastic resistance to change of size or change of form, only within certain narrow limits of change. These limits for any material are called the limits of electricity for that material, and are found to differ considerably for different materials. The substances of highest elasticity are those which, like steel, glass, ivory, and most liquids, offer very great resistance to change of size or change of shape. Certain substances, such as indiarubber, are commonly called elastic substances, because the limits of elasticity for these substance are unusually wide. A piece of indiarubber, for example, offers elastic resistance to change of size and shape through a very wide range of change, and is able to recover its original size and shape after undergoing very large changes of this kind. A solid substance is able to offer elastic resistance to change 17 258 GENERAL PHYSICS. of size, and also to change of shape. That is, a solid substance possesses elasticity of bulk or elasticity of volume, as well as elasticity of shape orform. A liquid substance, on the other hand, is able to offer elastic resistance only to change of size, and offers no elastic resistance whatever to change of shape. That is, a liquid possesses only elasticity of volume, and is devoid of elasticity of form. This, in fact, constitutes the essential difference between a solid and a liquid : a solid possesses elasticity of form in a verv marked degree, but a liquid has no trace of this property. A gas like a liquid has elasticity of volume, but no elasticity of form. When force is applied to a body in order to produce change of volume or change of form, it is generally assumed to be applied as a pressure or a tension exerted uniformly over the whole surface of the body, or over a portion of the surface, and is supposed to act either normally or tangentially to this surface. The force applied is, therefore, generally measured as the force per unit of area, and when so measured is known as the stress to which the change it produces is due. The change of size produced in any body by a suitable stress is not measured by the actual change of volume produced, but by the ratio of this change to the initial volume. Similarly, change of shape is measured, as explained below in Art. 98, by the ratio of a linear displacement to a length definitely associated with the displacement. Change of size or change of shape, measured in this way as a ratio or a proportional change, is called a strain, and any body in which a change of this kind is produced is said to be strained. When a body is strained within the limits of elasticity for the material of which it is made, it is found by experiment that the strain produced is directly proportional to the stress applied, That is, for small strains within the elastic limits of the GENERAL PROPERTIES OF MATTER. 259 material considered, stress . the ratio . s~ is constant. strain This constant ratio for any material is the modulus of elasticity for the particular case of stress and strain to which it applies. 96. Density and Specific Gravity. The mass of any volume of a given uniform material* is obviously proportional to the volume. Thus, the mass of n cub. cms. of pure water at C is n times the mass of 1 cub. cm. of pure water at the same temperature. It follows from this that the mass per unit volume for any definitely specified material is constant for that material. Further, if the mass per unit volume for different materials is compared it is found that, although it is constant for a given material, it differs widely for different materials. Thus, the mass of 1 cub. cm. of gold is about 19*3 grammes, the mass of 1 cub. cm. of silver is about 10'5 grammes, the mass of 1 cub. cm. of copper is about 8'9 grammes ; while the mass of 1 cub. cm. of pure water at 4 C. is almost exactly 1 gramme. The mass per unit volume of any substance is thus a characteristic or specific constant of the substance, and is called the density of the substance. Hence, if the mass of a body of any uniform material is denoted by m, and its volume by v, the density, d, of the material is given by the relation m a = v This formula may be written in the form, m = cd^ and establishes a very important relation between mass, volume, and density, for any uniform material. When a body is not of uniform material throughout, its * The material is here supposed to be uniform with regard to the distribution of its mass throughout its volume. '260 GENERAL PHYSICS. density is not uniform, but varies from point to point, and is measured at any point by the mass per unit volume for a very small volume yf the material taken at that point. This case need not, however, be further considered. It will be seen from what has been said that the experimental determination of the density of any material involves the measurement of the mass and volume of a selected portion of the material. Thus, to find the density of silver by a direct experimental method it would be necessary to take a suitable piece of silver, and to find its mass by weighing it, and its volume by measuring it. Then, if m and v denote respectively the mass and volume thus determined, the density of silver could be calculated from the relation m d= v It is explained below, however, that although this direct method may be adopted, and is adopted in the case of gases, it does not give such accurate results as the indirect methods, dealt with in Chap. xix. The inaccuracy of the method depends upon the fact that, although the mass of a body can be determined by weighing with the highest accuracy, the volume of a body cannot be determined by direct measurement with anything like the same degree of accuracy. Density, as defined above, is sometimes called absolute density, Instead of expressing the density of a substance absolutely, as explained above, it may evidently be expressed relatively with reference to the density of some well-defined substance as a standard. The density of a substance, expressed relatively to the density of a specified standard substance, is called the relative density of the substance. The standard substance selected for reference is pure water at 4 C. It will be seen that the relative density of a substance is thus the ratio of the density of the substance to the density of water at 4 C., and is merely a number expressed without units. GENERAL PROPERTIES OF MATTER. 261 Thus, the absolute density of gold is 19'3 grammes per cub. cm., and the absolute density of water at 4 C. is 1 gramme per cub. cm., so that the relative density of gold is 19'3. It is one of the advantages of the C.Gr.S. system that the relative density of 'a substance is expressed by the number which measures its absolute density. This is not the case in the English system. For example, the absolute density of gold in English units is about 1207 pounds per cubic foot, and the absolute density of water is about 62*5 pounds per cubic 1207 foot, so that the relative density of gold is - or 19*3. It is to 2i 'D important, however, to note that the relative density of a material must be the same in all systems of units. The specific gravity of a substance is essentially the same at its relative density, and may be defined as the ratio of the weight of any volume of the substance to the weight of the same volume of water at 4 C. Thus, if W denote the weight of any volume of a given substance, and W the weight of the same volume of water at 4 C., then W' w =: S) where s denotes the specific gravity of the substance. It will be seen here that if V denote the volume of the substance, d' its density, and d the density of water at 4 C., we have W = Vd', and W = Vd, so that w/ = Yi' _ & W == Vd ~ d' That is, s = , or the specific gravity, s, of any substance is a essentially the ratio of the density of any substance to the density of water at 4 C., and is, therefore, the same as the relative density of the substance. 262 GENERAL PHYSICS. The relative densities or specific gravities of a few of the commoner substances are given in the following table. Table of Relative Densities OP Specific Gravities. Solid- 1 *. Aluminium, . 2-6 - 2-8 Carbon (graphite), . 1-9 - 2-2 Copper, . . 8-8 8-95 Carbon (diamond), . 3-5 - 3-6 Gold, . 19-25 - 19-35 Sulphur, . 2-0 Iron (wrought), . 7'8 7-9 Brick, . 2-0 - 2-2 Iron (cast), . 7-1 - 7' 7 Chalk, . 1-9 - 2-8 Steel, . 7'8 - 7'9 Coal, 1-2 1-8 Lead, . 11-35 Flint, 2-65 Nickel, . . 8-3 - 8-8 Granite, . 2-4 3-1 Platinum, . . 21-3 - 21-6 Glass, 2-4 - 2-8 Silver, . 10-4 - 10-6 Marble, . 2-6 - 2-8 Tin, . . . 7-3 Porcelain, 2'4 2-6 Zinc, . 7-1 - 7'2 Quartz, . 2-65 Slate, . . . 2-6 2-7 Brass, . 8-4 - 8-7 Pitch, . 1-1 Bronze, . 8-7 8-9 Paraffin wax, . 88 92 German silver, . 8-3 - 8-5 Sand, 1-5 - 1-7 Ash, . '7 9 Ebonite, . 1-1 1-2 Oak, . . -6 9 Indiarubber, 9 1-0 Beach, . '7 9 Ivory, 1-8 1-95 Box, . . -9 - 1-1 Bone, 1-5 - 2-0 Deal, . . -4 6 Sugar, 1-6 Willow, . -4 6 Rock salt, 2-3 2-4 Liquids. Alcohol, . . "79 Petroleum, 8 - '9 ,, (methyl), . "81 Sea water, 1-025 Benzene, . . -90 Chloroform, . 1-48 ; Ether, . "74 i Sulphuric acid, 1 -85 Glycerine, . 1 -2G Nitric acid, . 1-56 Linseed oil, 94 Hydrochloric acid, . 1-27 Olive oil, . -92 Water at 4 C., 1 Mercury at C., . . . . 13-596 GENERAL PROPERTIES OF MATTER. 263 The density of gases is considered in Art. 126. The absolute density of dry air at C., and under normal atmospheric pressure, is T293 grammes per litre, or '001293 gramme per cubic centimetre. The density of a substance generally decreases as the tempera' ture rises,* for as the substance expands with rise of temperature, the volume occupied by any given mass must necessarily increase, and the density must, therefore, decrease. The variation of density with change of temperature has been very carefully studied f experimentally for water and mercury. A short tabular statement of the results obtained for temperatures within the ordinary range is given below. Water. The absolute density of pure water free from air, and under normal pressure at 4 C., is found to be '999955 gramme per cubic centimetre. Specific Gravity of Water at Different Temperatures Relative to Water at Jf C. Temperature. Specific Gravity. Temperature. Specific Gravity. 0C. 999868 16 C. 998970 2 999968 18 998622 4 1-000000 20 998230 6 999968 22 997796 8 999876 24 997322 10 999727 26 996810 12 999525 28 996259 14 999271 30 995672 Mercury. The density of mercury at C. is usually taken as 13*596 grammes per cubic centimetre. * See Art. 17 in Part iv. on Heat. t See Art. 28 in Part iv. on Heat. 264 GENERAL PHYSICS. Density of mercury at different temperatures calculated from Regnaulfs value of the coefficient of cubical expansion of mercury. Temperature. Density. Temperature. Density. oc. 5 10 13-596 13-584 13-571 15 C. 20 25 13-559 13-547 13-534 265 CHAPTER XVII. PROPERTIES OF SOLIDS. 97. Volume Elasticity. When a body is strained in such a way that it undergoes change of volume without change of shape, the elasticity which enables it to resist the change is known as volume elasticity or bulk elasticity. A body can be strained in this way only by subjecting the body to a uniform pressure acting normally all over its surface, and then increasing or decreasing this pressure according as it is required to increase or decrease the volume of the body. Pressure may conveniently be applied in this way by immersing the body in a suitable liquid and then applying pressure to the liquid. The pressure on the liquid is exerted on the surface of the immersed body, and acts everywhere at right angles to the surface. The piezometer apparatus, described in Art. 116, has been used for applying pressure to solid bodies in this way. It is, however, difficult to obtain in any way reliable measurements of the compressibility of a solid : most solids offer very great resistance to compression, so that the changes of volume, even under very great changes of pressure, are very small, and in any piezometer method they are subject to correction for the compression of the liquid used. By using a compressible substance, such as cork, the meaning of change of volume, without change of shape, may be illustrated in a striking manner by means of the piezometer apparatus. If a ball of cork of uniform structure is subjected to pressure, it may be compressed to a ball of much smaller volume 266 GENERAL PHYSICS. without losing its spherical form, and when the pressure is removed it recovers its original volume, retaining its spherical form throughout the process. A body may be subjected to very great hydrostatic pressure by sinking it to a great depth in the sea. A piece of cork may in this way be made so dense by compression that it sinks. Let a body occupy a volume V when subjected to a uniform pressure of P units per unit area, and let the volume be decreased to (Y v), without change of form, when the pressure is increased to (P + p) units per unit of area. The stress to which the change of volume in this case is due is measured by p, and the volume strain produced is given by the ratio ^- The modulus of volume elasticity, usually denoted by k, is given therefore by As explained above, it is very difficult to measure v in this relation with any accurracy, so that the value of h cannot be determined accurately by any direct experimental method. It can, however, be determined indirectly, as explained below. 98. Simple Rigidity. The elasticity which enables a solid body to resist change of shape or form, without change of volume, is known as form elasticity, or, more generally, as simple rigidity. The fundamental strain in any change of shape without change of volume is the displacement in its own plane of any very thin plane layer of the material, relative to an adjacent parallel layer. Thus, if the thin plane layer AB, in Fig. Ill, is displaced in its own plane through a distance A A' or BB', relative to the adjacent parallel layer CD, the material made up of the two layers is subject to a simple form strain, without attendant change of volume, and the strain is measured by the PROPERTIES OF SOLIDS. 267 ratio of the displacement, AA', to AC, the perpendicular distance between the layers. Similarly, if a plane layer AB, Fig. 112, is displaced in its own plane (through a distance d) relative to a plane parallel layer CD, at a perpendicular distance D from it, the material between the layers is subject to simple change of form without change of volume, and the form strain thus produced is measured d by the ratio This form of strain is known as a shearing strain, or a simple shear. The forces by which this strain is produced must evidently be applied tangentially over the surfaces of the plane layers AB and CD, and must act on these layers in opposite A (d)A' 3 B' (D) A' B B' C * D Fig. 111. Fig. 112. directions in the line of displacement. The force per unit area thus applied to either surface is taken as the measure of the applied stress, and the ratio of this stress to the strain d/D is called the modulus of simple rigidity for the material under strain. This modulus is usually denoted by n, so that if P denote the tangential stress and s the shearing strain then It can easily be seen that a shearing strain involves no change of volume. Thus, if a rectangular block of any material be sheared by the relative displacement of any two parallel faces, the volume of the block must remain unchanged ; for, if 268 GENERAL PHYSICS. we consider the block made up, like a pack of cards glued together, of a very large number of thin layers parallel to the two faces considered, any one layer is simply displaced slightly in its own plane, relative to the adjacent layers, and the length, breadth, and depth of the block must remain unchanged. A block of any material might conceivably be sheared by applying to any two parallel faces uniform stresses acting in opposite directions and parallel to the surface in each case. Thus, in Fig. 113, if we suppose the block ABCD to have its upper and lower faces glued to the planes PQ and RS, the block could be sheared by applying equal forces to displace the planes PQ and RS, each in its own plane, in opposite directions, as indicated by the p ^ Q arrows in the figure. It can be shown, however, that when a cylinder is sub- jected to twist or torsion round 5 its axis, the strain in any thin Fig. 113. cylindrical shell is really a shearing strain which increases as the radius of the shell increases. Thus, let ABCD, in Fig. 114, represent a cylinder, and suppose the upper face of this cylinder to remain fixed, while the lower face is twisted through a small angle a relative to the upper face in the direction of the arrow. If we consider the effect of this twist on the outermost shell of the cylinder, it will be seen that any strip of the shell, such as abed, taken parallel to the axis before twisting, is displaced by the twist into the position abed/ That is, the strip is subjected to a simple shear, and the shearing strain produced in it measured by the ratio - From this ac result it will be seen that the shearing strain in any cylindrical PROPERTIES OF SOLIDS. 269 shell of radius x, when the lower face of the cylinder is twisted in its own plane through an angle a relative to the upper face, is measured by the ratio y , where I denotes the length of the cylinder. Since stress = n, strain nxa it follows in this case that, stress = n . strain = j-. That is, the I c d Fig. 1.14 stress or force per unit area applied to the lower face of the shell of radius x in order to twist it through an angle a relative nxn to the upper face is = , where / is the length of the cylinder. This force is applied uniformly over the surface and acts every- where parallel to the surface and tangential to the section 270 GENERAL PHYSICS. of the shell, so that if S denote the thickness, radially, of this shell, the moment of the force causing the torsion of the shell is <* nxa lTrny?$* 27rxc . i . x, or -- - - . in the same way the moment ot the I v force causing the torsion in every one of the thin cylindrical shells, into which the cylinder may be supposed to be divided, can be determined, and by taking the sum of the moments for all the shells we can get the value of the moment of the couple which produces the torsion of the cylinder as a whole. It can be shown in this way, by summing the moments for all the shells by a suitable mathematical method, that the moment of the couple which will twist one end of a cylinder of length I and radius r through an angle a. relative to the other end, is -^y a, where n is the simple rigidity of the material of the cylinder. It follows from this result that the moment of the couple which will twist the cylinder through unit angle (circular measure) is given by _ 21 This moment is sometimes called the modulus of torsion for the particular cylinder (usually a rod or wire) to which it applies. This result at once suggests a method of determining n, the modulus of simple ragidity, experimentally, for any material which can be drawn into a wire or thread of truly circular section. A heavy bob, in the form of a metal sphere or cylinder, is attached to one end of a convenient length of fine wire made of the material for which the modulus of simple rigidity is to be determined and then suspended, as shown in Fig. 115, by fixing the other end of the wire in a suitable clamp. If the bob used is cylindrical in form it is best to attach it to the wire, so that it hangs (as shown in the figure) with its axis PKOPERTIES OF SOLIDS. 271 in the same vertical line as the axis of the wire. The arrange- ment thus set up is called a torsion pendulum, and may evidently be set in vibration under the influence of the torsion of the wire by twisting the bob round its axis through any convenient angle, and then letting it go. If the vibrations of a torsion pendulum are studied experi- mentally, it will be found that the period of vibration is per- fectly constant, and quite independent of the angular amplitude of the vibrations, whether this amplitude be large or small.* Experiment 14. Set up a torsion pendulum similar to that shown in Fig. 115. Attach a very light wire pointer horizontally to the lower end of the bob, and arrange a circular scale immediately below, so that the pointer indicates the zero of the scale when the bob is at rest. Now determine the period of vibration under torsion for different amplitudes. The amplitude may be varied from two or three complete revolutions down to a few degrees. The period should be found by determining the time occupied by a sufficient number (as many as possible) of complete vibrations within a certain range of amplitude. The complete vibrations should be counted, as in Experiment 3, as the intervals between successive transits of the bob in the same direction through its zero position. This position is p. j^ easily observed by means of the pointer and scale. It will be found that the period is quite constant for all amplitudes. The period of vibration of a torsion pendulum can thus be easily and accurately determined by experiment. It can be shown, however, that the period of vibration of the pendulum is given by where T is the modulus of torsion of the wire and I the * This result shows, incidentally, that the moment of the couple due to the torsion of the wire is directly proportional to the angle of the twist. This is in accordance with the theoretical result obtained above. 272 GENERAL PHYSICS. moment of inertia of the vibrating system round the axis of rotation. From this relation we get T = ^ That is, A A OT irnr* ^TT^L _ SirU where ? and I denote respectively the radius and length of the wire, and n the modulus of simple rigidity for the material of the wire. Now, in any experiment, such as that described above, the quantities t, r, and / can be measured directly, and I can be calculated from mass, dimensions, and form of the bob and wire. The value of n for the material of the wire can thus be determined with considerable accuracy by direct experiment. 99. Stretching 1 , When a wire or rod is stretched by a tension in the direction of its length the strain produced is not a pure volume strain or a pure form strain, but involves a change both of volume and of form. The elastic resistance which a material offers to stretching involves, therefore, both the volume elasticity and the simple rigidity of the material. It can readily be shown that stretching a rod produces a true change of form in the material of the rod. When any portion of the rod is stretched its length increases, but the dimensions at right angles to the length decrease. Hence, if we consider a spherical portion of the material of the unstretched rod, taken in the interior of the rod as at A in Fig. 116, it will be seen that this spherical portion must, in the stretched rod, take the ellipsoidal form indicated at B in the figure. PROPERTIES OF SOLIDS. 273 Stretching is thus accompanied by true change of form. It is accompanied also by change of volume for experiment shows, as explained below, that a rod or wire when stretched increases in volume. When a wire of length L is elongated by stretching by an amount /, the elongation of each unit length of the wire is obviously the same, and is measured by the ratio =-. This Lt ratio which may be defined as the elongation per unit length, or the ratio of the total elongation to the initial length of the wire, is taken as the strain due to stretching. Also, if W denote the stretching force or tension applied to the wire in stretching it, and a denote the area of cross section of the wire, the stress to which the stretching is due W is evidently given by for the B Fig. 116. tension is necessarily distributed uniformly over the cross section. Experiment shows that for all stretching strains within the limits of elasticity of the material, the ratio of stress to strain, where each is measured as explained above, is constant. This ratio may, therefore, be considered as the modulus of elasticity for stretching, and is known as Young's modulus of stretching, or simply as Young's modulus. Hence, if Young's modulus for the material of the wire considered above is denoted by M, we have W / al L The value of M for the material of any wire or rod can 18 274 GENERAL PHYSICS. thus be determined experimentally by stretching the wire under such conditions that W, L, a, and I can be accurately measured, and then calculating out the value from the relation given above. A convenient form of apparatus for carrying out a determination of Young's modulus by this method is shown in Fig. 117. The wire to be stretched is suspended, as shown at A in Fig. 117, from a vice clamp CC in which its upper end is fixed, and carries at its lower end a vernier V and a scale pan P, as shown in the figure. The free length of the wire for stretching should be about 2 metres, and the stretching force should be applied gradually by placing weights in the scale pan. In most cases it is convenient to use a bucket as a scale pan, and to apply the stretching weight by pouring measured quantities of water slowly into the bucket. The scale, SS, on which the elongation is measured is carried from the suspension clamp by the two side wires shown in the figure, and is held in a fixed position relative to the vernier by means of the heavy tubular weight of lead shown at LL. When weights are placed in the pan so as to stretch the wire, the vernier V is pulled downwards in the groove in which it moves freely relative to the scale, and the difference in the vernier readings before and after the addition of any weight gives the F - jjy elongation due to that weight. The elongation produced by any weight can thus be measured with considerable accuracy by using a good scale and an accurately divided vernier reading to a small fraction of a scale division. PROPERTIES OF SOLIDS. 275 The initial length of the wire, between the clamp and the vernier, can be measured with sufficient accuracy by direct application of a metre scale; and the area of cross section can be found by taking a number of careful measurements of the diameter with a screw gauge, and then calculating the area from the mean value of the diameter. The stretching weight to which the measured elongation is due is given by the weights placed in the scale pan. All the data necessary for the determination of Young's modulus from the relation, M = WL al ' can thus be found by direct experiment. Example. In an experiment for determining Young's modulus for copper by stretching a copper wire, the following data were obtained : Initial length of wire, 206 '2 cms. Mean diameter of wire, '0914 cm. Load on Wire. (In addition to starting load necessary to straighten wire.) Kil ogram me - weights. Elongation due to Additional Load. (Cms.) Elongation per Kilogramme -weight. (Cms.) 5 s 20 155 320 470 635 0310 0320 0313 0317 Average elongation per kilogram me- weight, 0315 Substituting in the relation, M = WL 276 GENERAL PHYSICS. from these data we have W = 1 kilogramme-weight = 1,000 x 981 dynes. a = -00657 sq. cm. I = -0315 cm. L = 206 "2 cms. We, therefore, have 1,000 x 981 x 206-2 00657 x -0315 = 9-77 x 10 U dynes per sq. cm. That is, Young's modulus for copper is given by this experiment as 9'77 x 10 U in dynes per square centimetre. The behaviour of a wire when stretched beyond the limits of elasticity for the material illustrates the general behaviour of a material when strained in any way beyond the limits of perfect recovery from the strain. The data obtained by stretching a wire beyond the elastic limit serve also to determine the breaking stress at which the wire breaks under tension. This breaking stress is taken as a measure of the tenacity or tensile strength of the material of the wire. The breaking stress for a thin wire may be determined experi- mentally by an extension of the method, described above, for the determination of Young's modulus. The apparatus shown in Fig. 117 may be used, but a very long scale suitable for measuring considerable elongations should be used. The wire is stretched by gradually increasing the load in the scale pan, but when the elastic limit is reached the increments of load must be made smaller and smaller, and must be added very gradually preferably by pouring water or shot into a bucket used as a scale pan. This process of gradual stretching is continued until j the wire breaks. If the data obtained in such an experiment are examined it will be found that the elongation produced by a given increment PROPERTIES OF SOLIDS. 277 of load is practically constant up to the load at which the elastic limit for stretching is reached. Beyond this limit, however, the elongation for a given increment of load steadily increases until a point known as the yield point of the material is reached. At this point the wire " yields " or " gives " under the applied load, and a considerable permanent elongation is produced without any further addition to the load being made. Beyond this point a stage is reached in which the load must again be increased in order to obtain further elongation, and in this stage, as in that immediately preceding the yield point, the elongation for a given increment of load rapidly increases until ultimately the breaking point of the wire is reached. During this stage the material of the wire appears to be in a more or less plastic condition, for it is found that the elongation due to a given increment of load increases with the time for which the load is applied. It is found, too, that as the breaking point is approached the wire is drawn out more at some points than others, and its cross section ceases to be of uniform area. If a curve is plotted showing how the elongation of the wire increases w 7 ith the stretching stress, it will have the form shown in Fig. 118. The straight portion OA of the curve shown in the figure represents the relation between the elongation and the stress within the limits of elasticity where the elongation is directly proportional to the stress. At the point A, where the continuation of the straight portion OA leaves the experimental curve, the elastic limit is reached, and the limiting stress at which this takes place is represented by OP. The portion AB represents the relation between the elongation and the stress between the elastic limit at A and the yield point at B, and it can be seen from the form of the curve that in this portion of it, the elongation due to any given increment of load, rapidly increases as the load increases, until the point B is reached, where a considerable elongation takes place under the 278 GENERAL PHYSICS. constant load represented by OQ. The portion BC represents the relation between the elongation and the stress between the yield point at B*and the breaking point at C, and in this portion, too, it will be seen that the elongation increases very rapidly with the load until the breaking point is reached. The stretching stress corresponding to this breaking point, represented in the figure by OK, is the breaking stress for the material of the wire. It must be noted, however, that the breaking of the wire is often due to some small flaw in P q Stretching Stress Fig. 118. the wire, and may, therefore, take place with the same wire under widely different stresses. It is necessary, therefore, in determining the breaking stress to eliminate accidental breakages, and to find the maximum constant stress at which breakage takes place. It will readily be understood that the whole process of stretching a wire, as represented by the curve OABC in Fig. 118, can be followed out experimentally only when the material of the wire is of a very ductile and tenacious character. In the case of brittle material the stages represented by the portion ABC of PROPERTIES OF SOLIDS. 279 the curve are practically non-existent, and the breaking point is reached very quickly after the elastic limit is passed. Experiment 15. Find the breaking stress for soft iron by the method indicated above. Use about a metre length of No. 22 wire. A serious difficulty which presents itself in all experiments dealing with strains beyond the limits of electricity for the strained material, is the effect of time on the strain due to any given stress. In an experiment, such as that described above, it is found, even before the yield point is reached, that the elongation produced by any load beyond the elastic limit is not constant, but increases slightly with the time for which the load is allowed to act. The experimental laws of stretching may be expressed concisely by saying that, within the limits of elasticity the stretching stress and the stretching strain, as defined above, are directly proportional the one to the other. They may be expressed more formally from the relation M WL M = =- al which gives = JL wk ~ M' a ' That is, the elongation of a wire of given material due to stretching within the limits of elasticity for the material, is directly proportional to the stretching force, directly proportional to the initial length of the wire, and inversely proportional to the area of cross section. It should be noted that compression in one direction is essentially the same strain as stetching in one direction. That is, whether a rod is stretched or compressed in the direction of its length the modulus of elasticity involved in either case is Young's modulus of stretching. 280 GENERAL PHYSICS. Young's modulus is, like the modulus of simple rigidity, a very important elastic constant for any material. These two moduli are the moduli which can be determined most accurately by direct experiment, and as other moduli of elasticity, such as the modulus of volume elasticity, can be derived from them theoretically, they serve to determine indirectly the values of these moduli. 100. Bending". When a rod or beam is bent in any way, as shown in Fig. 119, the layers on the convex side are obviously stretched, while those on the concave side are compressed in the direction of their length. A certain neutral layer, indicated in outline in the figure, is neither stretched nor compressed, but all layers on the concave side of this layer are compressed, and all on the convex side are stretched. Fig. 119. The modulus of elasticity involved in bending is, therefore, Young's modulus of stretching, and this modulus may be determined experimentally by data derived from the bending of a beam. The theory of bending strains is too complicated to consider here, but the experimental laws of bending for light beams of rectangular section loaded at one point, may be established by a few simple and interesting experiments, such as those indicated below. For the purpose of these experiments a beam may be bent by fixing it securely in a horizontal position with one end firmly clamped and the other end free, and then applying a weight at the free end, as in Fig. 120, or by supporting it in a horizontal position on two knife-edges, placed one under eacli end, and then applying the weight at a point midway between the supporting edges as shown in Fig. 121. In either case the PROPERTIES OF SOLIDS. 281 bending is conveniently measured by the downward deflection of the point at which the weight is applied. The bending deflection produced in a light beam of rect- angular section of given material when loaded at a particular point, must evidently depend upon (i) the load applied ; (ii) the length of the beam ; (iii) the width of the beam ; and (iv) the depth or thickness of the beam. It is necessary, therefore, to find by experiment how the deflection depends upon each of these four factors separately when the other three are kept -constant. Experiment 16. Find how the bending deflection of a light beam loaded at its middle point depends upon the load to which the bending is due. Set up a light wooden beam about a metre long and about 2 cms. x 1 cm. in cross-section, in the manner indicated in Fig. 121, and Fig. 120. Fig. 121. arrange a scale and pointer to measure as accurately as possible the downward deflection of the middle point of the beam. Measure carefully the deflections produced by a series of gradually increasing loads, taking care not to pass the elastic limit for the beam, and then arrange the results in a tabular form similar to that given in the Example in Art. 99 above. It will be found that the bending deflection is directly proportional to the load, and that the average deflection per unit load is, there- fore, constant for all loads within the elastic limit. Note that in this experiment the load only is varied. Experiment 17. Find how the bending deflection of a light beam loaded at its middle point depends upon the length of the beam. Set up a wooden beam as in the foregoing experiment, and find the deflections produced by the same load for different lengths of the same beam. The length of the beam subjected to bending is very easily varied by simply varying the distance between the knife-edges which support the beam. 282 GENERAL PHYSICS. It will be found, on tabulating and reducing the results obtained, that the bending deflection is directly proportional to the cube of the length. Note that in this experiment the length of the beam only is changed. Experiment 18. Find how the bending deflection of a light beam loaded at its middle point depends upon the width of the beam. Obtain a number of beams cut from the same piece of wood of the same length and thickness, but of different widths. Set up each beam in turn, as in the foregoing experiments, and measure the deflection produced by the same load for the same length of beam in each case. It will be found on tabulating and reducing the results that the bending deflection is inversely proportional to the width of the beam. This result might have been deduced on first principles without appealing to experiment. Noj:e that in this experiment the width only is varied. Experiment 19. Find how the bending of a light beam loaded at the middle point depends upon the depth of the beam. Obtain a number of beams which differ in depth, but which are exactly alike in material, length, and width. For this experiment the beams must be accurately made of very uniform material, and the depth should vary within comparatively narrow limits. Five beams, varying in depth from 8 mm. to 10 mm., in steps of half a millimetre, would answer the purpose. Set up each beam in turn, and find the deflection of the same length under the same load. It will be found, on tabulating and reducing the results, that the bending deflection is inversely proportional to the cube of the depth or thickness of the beam. Note that in this experiment the depth only is varied. If the foregoing experiments are repeated for a beam loaded at one end, as in Fig. 120, exactly similar results will be obtained. It will be seen from the results of these experiments that the experimental laws of bending for a light beam of rectangular section loaded at its middle-point or at one end may be stated as follows : The deflection of the point at which the load is applied is directly proportional to the load, directly proportional to the PROPERTIES OF SOLIDS. 283 cube of the length, inversely proportional to the width, and inversely proportional to the cube of the depth or thickness of the beam. That is, if I denote the deflection, W the load, L the length, a the width, and b the depth of the beam, we have WL 3 Ufl' WL 3 or I = K ab 3 where K is a constant. The value of this constant depends upon the value of Young's modulus for the material of the beam. It can be shown that its 4 i value is when the beam is loaded at the middle, and M when the beam is loaded at one end. The value of Young's modulus for the material of a beam can thus be determined from the relation given above. In the case of a beam loaded at the middle, for example, we have 4 WL 3 / = > M ab 3 4WL 3 or M = - * Hence, if in any experiments, such as those outlined above, the values of W, L, a, b, and / are carefully and accurately measured, the value of M can be at once obtained by means of this formula. Experiment 20. Find Young's modulus for glass by the method of bending. Use a fairly long strip of plate glass, of uniform width. The bending deflection may conveniently be measured by a sphero- meter or screw gauge suitably arranged for the purpose. 101. Torsion. The theory of torsion in the case of a cylindrical rod or wire has already been considered in Art. 98, as far as it can be dealt with here. 284 . GENERAL PHYSICS. It has been shown that a twisting strain is essentially a shearing strain, and that the modulus of elasticity involved in torsion is, therefore, trtie modulus of simple rigidity. The laws of torsion for a uniform cylindrical rod are expressed concisely by the relation, T = -> given in Art. 98. They may, Fig. 122. however, be established experimentally in a simple manner by means of the apparatus shown in Fig. 122. A long cylindrical rod is mounted in a horizontal position, so that one end is rigidly fixed in a socket, while the other end is free for twisting, and carries a wheel or pulley by means of which a twisting couple can be applied. This couple is applied PROPERTIES OF SOLIDS. 285 by hanging a weight from the rim of the wheel. The cord carrying the weight is attached at one end to a point on the rim, and is then coiled once or twice round the grooved rim, so that the free end to which the weight is attached hangs vertically downwards at one side of the wheel. The weight thus acts vertically downwards, in all the positions of the wheel, at the extremity of a horizontal radius, and the moment of the twisting couple thus applied to the rod is, therefore, measured by the product of the weight into the radius of the wheel. The angle of twist produced in any length of the rod by a given twisting moment is measured by means of the movable pointer and the circular scale shown in the figure. This pointer and scale can be set up at any distance from the fixed end of the rod, and the twist produced in this length of the rod is indicated directly by the deflection of the pointer on the scale. It should be noted that the moment of the couple which produces torsion in a rod, or the moment of the couple which a rod is able to exert in virtue of torsion imposed on it, is usually called the torque applied to, or exerted by, the rod. The following experiments indicate, in a general way, how this piece of apparatus may be used to determine the experi- mental laws of torsion for a cylindrical rod. The angle of twist produced in a circular rod of given material must evidently depend upon (i) the torque applied, (ii) the length of the rod, and (ill) the radius of the rod. It is necessary, therefore, to find by experiment how the twist depends upon each of these three factors separately when the other two are constant. Experiment 21. Find how the twist produced in a given rod varies with the torque applied to the rod. Set up the pointer and scale at a point on the rod near the free end, and measure carefully the angle of twist produced by a number of different torques. The torque should be varied by gradually increas- ing the weight carried by the torque wheel, and care should be taken not to exceed the limit of perfect recovery for the rod. 286 GENERAL PHYSICS. It will be found, on tabulating and reducing results, that the twist produced is directly proportional to the torque applied. Experiment 22. Find how the twist produced depends upon the length of the rod. Apply a convenient fixed torque to the rod. Then measure the twist produced by this torque in different lengths of the rod by setting up the pointer and scale at different distances from the fixed end of the rod. It will be found, on tabulating and reducing the results, that the twist produced by a given torque is directly proportional to the length twisted. It should be noted that this means that the twist per unit length in a twisted rod is constant for a given torque. Experiment 23. Find how the twist produced depends upon the radius (or diameter) of the rod. For this experiment it is necessary to obtain a number of rods of exactly the same material but of different diameters, or to turn the same rod down so as to obtain from it in succession rods of smaller and smaller diameter. The diameters of the rods should vary within comparatively narrow limits an extreme ratio of 4 : 5 or 5 : 6 is small enough and should be measured with the greatest possible accuracy by means of a good screw gauge. Measure the twist produced for the same length by the same torque for each rod in turn. It will be found, on tabulating and reducing the results, that the twist produced is inversely proportional to the fourth power of the radius of the rod. It follows from the results of these experiments that the twist produced in a cylindrical rod of given material is (i) directly proportional to the torque applied, (ii) directly proportional to the length of the rod, and (iii) inversely proportional to the fourth power of the radius (or diameter) of the rod. These may be taken as the experimental laws of torsion for a cylindrical rod of given uniform material. Hence, if Q denote the torque applied to a cylindrical rod of length / and radius r, and denote the twist produced in PROPERTIES OF SOLIDS. 287 the rod, we may write '- e = K-J r 4 where K is a constant. The value of this constant depends upon the material of the rod, and also upon the fact that the rod is of circular cross section. If this relation is written in the form it will be seen that the torque, which will produce unit angular twist, is given by Q = r * Kl But -~ is the quantity denoted by T in the relation, as explained in Art. 98. It follows, therefore, that r 4 7T7W 4 TQ " 2J ' o or K = - 7T/1 Experiment 24. Set up the apparatus shown in Fig. 122, and measure for a given rod the value of 6 for a measured torque Q. Then measure I and r for the rod, and calculate from the KQZ relation, B = -J-, the value of K for the rod. Then from the 2 relation, K = , find the value of n, the modulus of simple rigidity for the material of the rod. 288 GENERAL PHYSICS. The laws of torsion may be established experimentally by experiments with wires if the apparatus shown in Fig. 123 is used. The method* of applying torque to the wire is plainly indicated in the figure. It is frequently necessary in engineering practice to measure the torque transmitted by a cylindrical shaft. It will be seen, from what has been said above, that this can be done for a shaft of given material and diameter, by simply measuring the twist on a measured length of the shaft. Thus, if is found to be the twist on a length I of the shaft, we have Q =~- Fig. 123. where Q denotes the torque on the shaft, r the radius of the shaft, and n the modulus of simple rigidity of the material of the shaft.* 102. Poisson's Ratio. When a solid is stretched in any direction it undergoes elongation in that direction and contrac- tion in every direction at right angles to it within the limits of elasticity. The elongation per unit length may be taken as a measure of the elongation strain, and, in the same way, the contraction per unit length may be taken as a measure of the contraction strain. The magnitude of each of these strains is proportional to the stretching stress, but their ratio is constant whatever the stress, and the ratio of the contraction strain to the elongation strain measured in this way is known as Poisson's ratio. Thus, if a cylindrical rod of length L and diameter d undergo, when stretched by any load within the * It must be remembered that d in this formula is in circular measure. PROPERTIES OF SOLIDS. 289 limits of elasticity, an elongation I in the direction of its length, and a contraction 8 laterally at right angles to its length, the elongation per unit length is T , the contraction per unit length L g ^\ is y, and Poisson's ratio is given by the ratio __. / L Hence, if the elongation per unit length, which accompanies stretching, be denoted by a, and the corresponding lateral contraction per unit length by ft, and if Poisson's ratio be denoted, in the usual way, by for W is the weight of a volume of water exactly equal to the volume of the solid substance of weight W. This result must, if necessary, be corrected for the tempera- ture of the water as explained above. The process of making a determination by this method is fairly obvious. A convenient quantity of the solid substance is taken about sufficient to fill the bottle half full and its weight, W, is determined in the usual way. The weights Wj and W., are then determined as explained above. The bottle is then half filled with water and the solid substance is run into it slowly in such a way that every particle of the substance is wetted by the water, and no air bubbles are entangled in the mass. The substance is then allowed to settle, and when this is over the bottle is, if necessary, filled up with water and the stopper inserted. The weight W 3 can then be determined, and the specific gravity of the substance calculated from the data obtained. Example. Find the specific gravity of sand from the following data : Weight of sand taken - 96 '4 grammes. Weight of bottle = 23 '1 Weight of bottle full of water = 123 "3 Weight of bottle containing the sand and filled up with water = 170 '3 ,, From these data we get Weight of water which fills the bottle = (123-3 - 23-1) grammes = IGO'2 grammes. Weight of water in the bottle when the sand is placed in the bottle = 170-3 - (90-4 + 23-1) grammes = 50-8 ,, DETERMINATION OF SPECIFIC GRAVITY AND DENSITY. 315 Hence, weight of water displaced by the sand is (100 '2 - 50 '8) grammes or 49 '4 grammes. The weight of the sand is 96*4 grammes. 96 "4 The specific gravity of the sand is, therefore, ^-^ or l'~- The correction for the tenfperature of the water is not here made. It is at most a very small correction, and need only be made in very accurate determinations. [No attempt should be made to work an example of this kind by the iise of a formula. Every step should be set out from first principles, but the student must know definitely, from the theory of the method, the successive steps to be taken.] It will be seen that a determination of the specific gravity by means of a specific gravity bottle involves weighing only, and may, therefore, be made with great accuracy. 113. Hydrometers. An hydrometer is an instrument designed to float vertically in a liquid and constructed to indicate the specific gravity of the liquid, either by the depth to which it sinks when floating in the liquid, or by the weight necessary to make it float immersed to a certain fixed depth in the liquid. In the former case the hydrometer is of the type known as variable immersion hydrometers, in the latter case it is called a constant immersion hydrometer. A common form of variable immersion hydrometer is shown in Fig. 136. It consists of a glass bulb and stein weighted with mercury so that it floats with the stem vertical. When floating in any liquid the hydrometer is immersed to a depth which depends upon the specific gravity of the liquid, and since the weight of the displaced liquid is, in any liquid, equal to the weight of the instrument, the depth to which it is immersed must increase as the density of the liquid decreases. It is, therefore, possible by graduating the stem to provide a scale which will indicate the specific gravity of the liquid in which the instrument floats within a range which depends upon the construction of the instrument. Thus, if an hvdrometer 316 GENERAL PHYSICS. floats immersed up to a mark at the bottom of the stem in water at 4 C. and up to a mark near the top of the stem in a liquid of specific gravity 0'7, it is evidently possible by graduating the stem between these two marks so as to indicate Fig. 136. Variable Immersion Hydrometer. any specific gravity between '1 and 1. Similarly, if an instru- ment floats immersed up to a mark near the top of the stem in water at 4 C. and up to a mark at the bottom of the stem DETERMINATION OF SPECIFIC GRAVITY AND DENSITY. 317 in a liquid of specific gravity i'2, it may evidently be graduated to indicate any specific gravity between 1 and T2. In this way by means of a set of hydrometers with con- secutive or overlapping ranges it is possible to determine the specific gravity of a liquid by simply finding the instrument which floats in it and then reading the graduation which marks the depth to which it floats. A number of somewhat empirical scales of graduation have been used for various technical purposes ; the simplest for general use is that on which the specific gravity of water at 4 C. is marked 1,000, and specific gravities higher and lower than this by correspondingly higher and lower numbers. On this scale the specific gravity corresponding to any division is evidently obtained by dividing the number of the division by 1,000. Thus, if an hydrometer floats in a liquid immersed to a point marked 969 on the scale of the stem, the specific gravity of the liquid is '969. The only constant immersion hydrometer in common use is Nicholson's hydrometer. A convenient form of this instrument is shown in Fig. 137. It consists of a central hollow cylinder or barrel, C, carrying, in the same axial line, an upper pan at A Nicholson's and a lower pan at B. The lower pan is weighted Hjdro- with lead so that the instrument may float with its axis AB vertical, and a fine line on the straight wire stem which carries the upper pan marks the fixed point to which the instru ment is immersed in all liquids. The hydrometer is so constructed that it floats in any liquid with only a portion of the central barrel immersed. It can, however, be sunk in any liquid to the fixed mark on the stem by placing weights in the upper pan, until the mark is seen at the surface of the liquid. This adjustment is best made by first sinking the instrument a little too deeply so that the 318 GENERAL PHYSICS. mark is below the surface of the liquid, and then gradually reducing the weight in the pan until the image of the mark, seen from below by toteal reflection at the surface, coincides with the mark itself. When the hydrometer floats in any liquid immersed exactly to the mark on the stem the weight of the instrument together with the weight in the pan is, by the principle of Archimedes, equal to the weight of the displaced liquid. Hence, if the hydrometer floats immersed up to the mark in any given liquid, and then in water at 4 C., it displaces exactly the same volume in* each case, and the ratio of the weight of the loaded instrument as it floats in the liquid to its weight as it floats in the water is, therefore, the specific gravity of the liquid. That is, if W denote the weight of the hydrometer, Wj the weight necessary to sink it to the index mark in the liquid, and W the weight necessary to sink it to the mark in water at 4 C. (or at t C. if the necessary correction is afterwards made), then, s, the specific gravity of the liquid is ^iven by Wi + W - w, + w The weights W and \V X may evidently be determined once for all as working constants for the hydrometer for which they are determined. Nicholson's hydrometer may also be used to determine the specific gravity of a solid substance. The method of using the instrument for this purpose is as follows : The hydrometer is floated in water in a tall wide jar as shown in Fig. 138. The weight necessary to sink it to the index mark is then found as explained above. The weights are then removed and the piece of solid whose specific gravity is to be found is placed on the pan. The weight of the solid must not, however, be sufficiently great to sink the instrument to the index mark, and weights must, therefore, be added to effect DETERMINATION OV SPECIFIC GRAVITY AND DENSITY. 319 this adjustment. When this adjustment is made and the weight in the pan noted, the solid is removed from the upper pan and placed in the lower pan where it displaces its own volume of water. Since the apparent weight of the solid is less in water than in air the hydrometer will now float less deeply immersed, and it is necessary, in order to sink it again to the index mark, to add to the weight in the upper pan. This is done and the weight in the pan again noted. The specific gravity of the solid can now be calculated from the data obtained in this way. Let W t denote the weight necessary to sink the hydrometer to the index mark in water, W 2 the weight necessary to sink it when the solid is in the upper pan, and W 3 the weight necessary for the same purpose when the solid is in the lower pan. Then, a very little con- sideration will show that the weight of the solid is (Wj W. 2 ), and the weight of water which it displaces when in the lower pan is (\V 3 W.,). The specific gravity of the solid is, therefore, given by W, - W, w, - w; Fig. 138. The method is not accurate enough to make it necessary to consider the correction for the temperature of the water. It will be seen that the hydrometer serves as a balance for determining the weights to be observed in using the instru- ment. This is convenient for some purposes, but it will be found in practice that the necessary weights cannot be found with certainty to much less than a decigram. This hydrometer is mainly of theoretical interest and is used more in the laboratory than anywhere else. It is usually made of brass for use in water for the determination of the specific 320 GENERAL PHYSICS. gravity of solids. It can, however, be made of glass and platinum wire, and may then be used for liquids or solids. In using the instrument a guard disc should be placed over the mouth of the jar containing the liquid to prevent the weights getting into the liquid by falling in accidentally, or by overloading the hydrometer. The disc may be of cardboard, glass, or sheet metal, and must be cut with a narrow radial slot for the stem of the hydrometer. This disc serves also to keep the instrument in the middle of the liquid and prevents it clinging to the sides of the jar. 113. Liquid Column Methods. The specific gravity of a liquid may be determined by balancing a column of the liquid hydrostatically against a column of water, and then comparing the heights of the two balancing columns. Thus, let a quantity of mercury be poured into the bend of a U-tube, and then let water be poured on the mercury in one limb of the tube till the column of water, AB, in this limb is balanced by a column of mercury, CD, in the other limb, as shown in Fig. 139. It has been shown in Art. 108 that in a liquid at rest the pressure in the liquid is the same at all points in the same horizontal plane. Hence, if we take a horizontal plane through the junction of the water and the mercury at A, in the limb AB, it follows that the pressure at A, in the limb AB, is the same as the pressure at C, in the limb CD.* Now, if /&]_ denote the height of the column AB, and d^ the density of the water of the column, the pressure at A due to * In applying this principle in a case of this kind it is very important to notice that the principle applies only when the liquid betoiv the horizontal plane considered is one and the same liquid throughout. For example, if we take any horizontal plane between C and D in the figure, the pressure is not the same at points in this plane in both limbs of the tube. The essential point is" that the pressure will be the same at two points in the same horizontal plane only if there is free communication between the points by a path below the plane which passes from one point to the other through one and the same liquid. DETERMINATION OF SPECIFIC GRAVITY AND DENSITY. 321 the weight of the overlying water is h^g, as explained in Art. 108. Similarly, if h, 2 denote the height of the mercury column CD, and d. 2 the density of the mercury, the pressure at C due to the weight of the overlying mercury is h 2 d 2 {j.* If, therefore, the pressure at A is equal to the pressure at C, we must have h^g = h. 2 d 2 g, or Ji^d^ = k. 2 d.,. That is, y 1 = Fig. 140. But is the ratio of the density of mercury to the density of water, and is, therefore, the relative density or specific gravity of mercury. That is * The external atmospheric pressure impressed on the columns at B and D is the same for both columns, and need not here be considered. 21 322 GENERAL PHYSICS. where s denotes the specific gravity of mercury at the tempera- ture of the experiment relative to water also at the temperature of the experiment. If this temperature is observed the usual reduction to 4 C. can be made, but the method is not, under ordinary conditions, sufficiently accurate to make it worth while doing so. This method may be employed for determining the specific gravity of any liquid, or for comparing the densities of any two liquids, by using a tube of the form shown in Fig. 140 instead of a simple U-tube. One liquid is poured into one branch, ABC, of the tube, and the other liquid into the other branch, DEF. The air in the bend AD separates the two liquids, and prevents them coming into contact with each other. The tube is filled by pouring a little of one liquid into one branch, and then a little of the other liquid into the other branch, and continuing this process until a sufficient quantity of liquid has been introduced into each branch. The air in the bend AD is compressed as the quantity of liquid in each branch increases, and the increased pressure which it thus exerts on the liquid surfaces at A and D forces the liquid to stand higher in the outer than in the inner limb in each branch. Suppose the tube to be filled, as shown in the figure, with water in the branch ABC, and with a liquid whose specific gravity is to be determined, in the branch DEF. Then, if AB is the horizontal through A, the pressure in the tube at A is the same as the pressure at B, and the pressure at B is the pressure due to the column of water BC, together with the atmospheric pressure impressed on the surface of the water at C. That is, if A t denote the vertical height BC, ^ the density of the water in the tube, and P the atmospheric pressure impressed on the surface at C, the pressure at B is (P + hidtf), as already explained ; and this is also the pressure at A when the water is in contact with the air in the bend AD. In exactly the same way, if DE represents the horizontal DETERMINATION OF SPECIFIC GRAVITY AND DENSITY. 323 through D. it follows that if h. 2 denote the vertical height EF, <1. 2 the density of the liquid in the branch DEF, and P the atmospheric pressure on the liquid surface at F, the pressure at E in the liquid is (P -f- h. 2 d z g), and that this is also the pressure at D where the liquid is in contact with the air in the bend AD. The pressure at A is, however, practically the same'" as the pressure at D, for it is the pressure of the air in the bend AD. We may, therefore, write P + hfy = P + Maft or h^ = h. 2 d. 2 . h. do That is, h,=^ s > as obtained above. It may be noticed that the most direct method of obtaining the above result is to note that the pressure due to each of the liquid columns BC and EF gives the difference between- the pressure of the air in the bend AD and the external atmospheric pressure. The pressure due to these two columns are, therefore, equal, and we may at once write h^g ~ W 2 <7, and so obtain the value of s, as above. Another application of this method is found in the hydro- meter usually known as Hare's hydrometer. The essential parts of this hydrometer are shown in Fig. 141. It consists of two straight lengths of glass tubing, BC and EF, about a metre long, set up a short distance apart in a vertical position, and joined at their upper ends by rubber connections, RR, to the T-piece T. The lower ends of the tubes dip into the beakers A and D, and the whole piece of apparatus is supported by a suitable wooden stand. A rubber tube, S, is attached to the stem of the T-piece, and is used to draw air out of the tubes. * Pressure due to the weight of the air may be neglected except in cases where, as in the atmosphere, very great differences of level have to foe considered. 324 GENERAL PHYSICS. The two liquids whose densities are to be compared are placed in the beakers A and D. Then, on withdrawing air from the tubes, by means of the tube S, the pressure of the air in the tubes decreases; that is, the pressure on the liquid surfaces inside the tubes decreases, and the liquid is forced up into each tube by the excess of the external atmospheric pressure on the Fig. 141. liquid surfaces in the beakers outside the tubes, over the pressure at the same level inside the tubes. When the liquid rises in each tube to such a height that the pressure in the liquid in the beakers is the same at the same level outside and inside the tube the mass of liquid in each beaker and its tube will again be in equilibrium. Hence, if the tube S is closed with a screw-clip after a quantity of air is withdrawn from the tubes, a column of liquid will be left standing in equilibrium in each tube. DETERMINATION OF SPECIFIC GRAVITY AND DENSITY. 325 In an experiment with this hydrometer for determining the specific gravity of a liquid, suppose water to be placed in the beaker A and the liquid in the beaker D, and let BC and EF represent the columns of water and liquid respectively, which stand in the tubes after air has been withdrawn by the tube S, and the tube closed. Then, if 7^ denote the vertical height of the column of water BC, measured from a point, B, inside the tube at the level of the water outside the tube to the upper level at C, ^ the density of the water, and p the pressure of the air in the bend CRKF, the pressure in the water at the point B is (p + h^g). Similarly, it can be shown that the pressure in the liquid column EF at the point E inside the tube at the level of the liquid outside the tube is (p -f htfl*g), where A., denotes the vertical height EF and d 3 the density of the liquid. Now, the pressure at B and the pressure at E must be equal, for each is equal to the external atmospheric pressure on the surface of the liquid in the beaker. We may, therefore write p -f- h^y = p or That is, where s is the specific gravity of the liquid relative to the water in the beaker. As in the case considered above, this result may be obtained more directly and concisely by stating that the pressure due to each of the columns BC and EF measures the difference between the pressure of the air in the bend CRRF and the external atmospheric pressure. We may, therefore, at once write h } d } y = hiLy, or y 1 = s, as before. /i.i This method of comparing densities by balancing liquid columns is not in ordinary practice a very accurate method. 326 GENEKAL PHYSICS. It is, however, capable of considerable accuracy when specially accurate methods of measuring the heights of the columns are adopted. A very notable instance of the use of this method is the method of Dulong and Petit for determining the coefficient of absolute expansion of mercury,* by comparing the density of mercury at different temperatures with its density at C. The same method was also ernplo} r ed by Thiesen in one of the best of the more recent determinations of the density of water at different temperatures relative to water at 4 C. It is important, too, to note that the usual method adopted in physical measurements for measuring pressure or difference of pressure is to balance the pressure to be measured by a liquid column, and then to calculate the magnitude of the pressure from the height of the column and the density of the liquid. 114. Absolute Density of Water. It will be understood from what has been said above that the best method of determining the absolute density of any liquid is to determine its specific gravity relative to water at 4 C. Then, if the absolute density of water at 4 C. is known, the absolute density of the liquid can be calculated. Thus, if s denote the specific gravity of the liquid, and d the absolute density of water at 4 C., sd is the absolute density of the liquid. The absolute density of water at 4 C. is thus an important physical constant, and a vast amount of careful and laborious research has been expended on the experimental determination of this constant. The results of several important determinations made in recent years agree in placing the absolute density of pure, air- free water, at the normal atmospheric pressure, and at the temperature of its maximum density (4 C.) between 0'99995 and 0*99996 in grammes per cubic centimetre. 115. Indirect Methods of Measuring Capacity and Volume. The most convenient and most accurate method of * See fffnt. Art. 25. DETERMINATION OF SPECIFIC GRAVITY AND DENSITY. 327 finding the capacity of a vessel is to find, by direct weighing, the weight of water or mercury which fills the vessel at a known temperature. The volume of the mass of liquid can be at once calculated from its mass and density at the observed temperature of filling, and this gives the capacity of the vessel at this temperature. Thus, if m denote the mass of the liquid which fills the vessel, and d its density at the observed temperature, then is its volume, and this is also the internal volume or capacity of the vessel. Water is used when the capacity to be measured is large, and mercury when the capacity is small. Experiment 28. Find the capacity per cm. length of the bore of the given tube of fine bore. Clean and dry the tube care- fully. Draw a long thread of clean, dry merciiry into the tube and measure the length of thread as accurately as possible. Observe the temperature at which this measurement is made. Then run the mercury out of the tube into a weighed porcelain crucible, and find its weight by careful weighing on the balance. From the data thus obtained calculate out the required capacity. Example. In an experiment of this kind the following data were obtained. Calculate the capacity per cm. length, and also mean diameter, of the bore of the tube. Length of mercury thread at 15 C., . 15 '43 cms. Weight of mercury, .... 0'2134 gramme. Take density of mercury at 15 C. as 13 '59. The weight of mercury which fills 1 cm. length of the tube is 0-2134 15^43 and the volume of this mass at 15 C is 0-2134 15-43 x 13-59 or 0-00102 c.c. This is the capacity of 1 cm. length of the bore of the tube. 328 GENERAL PHYSICS. If r denote the radius of the bore in cms. then (1 x ?-) is the capacity of 1 em. length of the bore. We, therefore, have * <* = 0-00102, 0-00102 That is, ,>, = That is, the diameter of the bore of the tube is O'OIS cm. A similar method based on the principle of Archimedes may be employed for finding the volume of a body. The body is weighed first in air, and then in water ; the apparent los.s of weight thus observed is the weight of the displaced water, and the volume of this water is exactly equal to the volume of the body by which it is displaced. The volume of the body can, therefore, be calculated from its apparent loss of weight in water and the density of the water at the temperature of weighing. Thus, if w denote the apparent loss of weight in water at f C., and d the weight of unit volume of water at t C., the volume of the body at t C. is - units of volume. Experiment 29. Find the diameter of a fine wire by the method described above. Measure off, as exactly as possible, a metre of the wire, coil it up, and weigh it in air. Then wash it in dilute caustic soda and water, and weigh it in water. From the apparent loss of weight, and the density of the water, calculate out the diameter of the wire. Example. In an experiment of this kind the following data were obtained. Calculate the diameter of the wire. Weight of 1 metre of wire in air, . . 1 '8642 grammes. Weight of wire in water, . . . 1'6537 ,, Temperature of water, . . . . 15 C. Density of water at 15 C. is 0'999 gramme per c.c. The apparent loss of weight in water is here (1-8642 - 1-6537) grammes, or 0*2105 gramme. The volume of water displaced by the wire at 15 C. is, therefore, ^ c.c., and this is also the volume of the wire at 15 C. 0-999 DETERMINATION OF SPECIFIC GRAVITY AND DENSITY. 329 If r denote the radius of the wire in cms. the volume of the wire is 100 x irr 2 cub. cms. We, therefore, have 100 ^,2 _ _ '~1P_ 5 0-2105 100 x TT x 0-999' and r = J"~ ' 2105 - - 59 \ 100 x TT x 0-999 That is, the diameter of the wire is 0*0518 cm. The methods illustrated by the examples given in this article are in very general use, and, when carefully carried out, they are probably the most accurate methods of measuring capacity or volume. 330 CHAPTER XX. PROPERTIES OF LIQUIDS. 116. Compressibility of Liquids. It has already been explained that although a liquid possesses no elasticity of form or rigidity, it possesses elasticity of volume in a marked degree. The volume elasticity of a liquid can, however, be exhibited, under ordinary conditions, only by the resistance it offers to compression, and the compressibility of a liquid is, therefore, the only elastic property by which the volume elasticity >of the liquid can be determined experimentally. The general method which has been adopted in studying the compressibility of a liquid has been to place the liquid in a glass tube made up of a bulb and a graduated stem, and then to subject the liquid to pressure, either by immersing the open tube in water under pressure, or by putting the interior of the tube in communication with a reservoir of compressed air. The compression of the liquid was then indicated by the graduations on the stem of the tube, and the pressure applied was measured by a suitably arranged manometer. A tube of this kind was first used for this purpose by Oersted, and is usually known as a piezometer. One form of the tube is shown in Fig. 142; it is a glass tube of sufficient strength to resist considerable pressure, and consists, as shown, of a long cylindrical bulb and a long fine-bore stem graduated in divisions whose volume relative to the capacity of the tube up to the zero of the scale has been accurately determined by calibration with mercurv. PROPERTIES OF LIQUIDS. 331 One of the simplest and most satisfactory methods of making a determination of the compressibility of a liquid by means of the piezometer is that originally adopted by Regnault. The tube is filled to a suitable point in the graduated stem with the liquid whose compressibility is to be determined, and is then immersed in water to which great pressure can be applied. The water is contained in a vessel specially fitted for the application and measurement of the necessary pressure, and capable of withstanding the pressure so applied. The liquid in the open piezometer tube is in this way subjected to known pressure and suffers cor- responding compression ; at the same time, the tube is subjected to the same pressure internally and externally, and is, therefore, compressed to the same extent as a solid piece of glass of the same external volume would be compressed. It follows from this that the compression of the liquid, as indicated by its apparent change of volume in the piezometer tube, is really the difference between the real com- pression of the liquid and the compression of the tube, and that the real compressibility of the liquid can be found from the data of the experiment only -pig. 142. if the compressibility of the material of the tube is known. This fact makes the accurate determination of the compressibility of a liquid very difficult, but satisfactory methods have been devised for finding the compressibility of the material tube, and the determination of liquid compres- sibilities can now be made with some accuracy. It will be seen from what has been said above that if V denote the volume of liquid in the piezometer tube, v the real decrease of volume produced by compression under a hydrostatic pressure p, then v/\ measures the volume strain produced in the liquid, and : is the modulus of rnhime elasticity for the liquid. 332 GENERAL PHYSICS. Instead of calculating out the modulus of volume elasticity in this way it is more usual to give what is called the coefficient of compressibility of the liquid. This coefficient is merely the decrease in volume per unit volume per atmosphere increase of pressure. That is, if p in the notation used above be taken to denote the increase of pressure in atmospheres, then -- r is the coefficient of compressibility which is thus seen to be the reciprocal of the modulus of volume elasticity when p is measured in atmospheres. The following coefficients of compressibility for water, sea- water, and mercury were given by Professor Tait, and serve to indicate the order of magnitude of this constant for liquids : Liquid. Coefficient of Compressibility. Water, . . . 0-0000047 Sea water, ..... 0*0000041 Mercury, . . . 0'0000036 This result means, for example, in the case of water that an increase in pressure of one atmosphere will compress any given volume of water by a little less than five millionth.? of its initial volume. 117. Viscosity. Although a liquid offers no elastic resist- ance to change of form, it is found that most liquids resist this change in some degree as the result of molecular friction between adjacent layers of the liquid. This resistance merely acts as an opposing force while the change of form is in progress, and does not tend at any stage in the process to restore the mass to its initial form. It is of zero value at any instant when the rate of change is zero, and it is found that its value at any instant while the change is in progress is proportional to the rate at which the change takes place. The property of a liquid which enables it to offer this frictional resistance to change of form is called viscosity. Any PROPERTIES OF LIQUIDS. 333 liquid, such as glycerine, which possesses this property in a marked degree is called a viscous liquid, while a liquid such as water or alcohol, which is of comparatively low viscosity, is called a mobile liquid. It will be seen from what has been said above that if a viscous substance changes form very slowly the resistance which it offers to the change will be very tmuH. That is, a liquid substance, whatever may be its viscosity, will undergo change of form under the action of the smallest force ; but, if the viscosity of the substance is great, the rate at which the change of form goes on may be very slow. If we adopt this fact as the criterion of a liquid it is found that a number of substances which appear to be solids are essentially liquids of very great viscosity. Thus, substances such as pitch, sealing-wax, cobbler's wax, and other substances are found to undergo continuous and progressive change of form under the influence of a small deforming force. A mass of pitch, for example, if placed on a table, will, in time flow over the surface of the table and cover it with a thin sheet of pitch. This flow takes place at constant temperature below the melting point, so that it is not due to melting, but the viscosity increases very rapidly as the temperature falls below the melting point. Substances which behave in this way are probably at the ordinary temperature in the semi-plastic state which precedes melting and which may extend, in some sub- stances, over a considerable range below the melting point. The method of defining and measuring the viscosity of a liquid is based upon the following considerations. Let ABCD, in Fig. 143, represent at a given instant a horizontal rectangular plate of liquid, of very small thickness, in a mass of liquid flowing horizontally in the direction of the arrow, and suppose the rate of flow in any horizontal layer to decrease with the depth of the layer below the surface of the liquid, and to be slightly greater, therefore, in the layer CD than in the layer AB. Now, imagine the flow to continue for one 334 GENERAL PHYSICS. unit of time from the instant considered, and let ABCD, in Fig. 144, represent the form of the plate at the end of this unit of time. Since the rate" of flow in the layer CD is greater than in the layer AB, the upper face CD of the plate advances in unit time through a slightly greater distance than the lower face AB and the plate thus becomes sheared into the form shown in the figure. The shear thus produced in the plate in unit time Fig. 143. Fig. 144. is evidently measured, as indicated in the figure, by the ratio . That is, if v denote the difference in the velocities of AJN the upper and lower faces of the plate, and d the thickness of the plate, the rate at which the plate is being sheared is measured by T . Now, the stress to which this continuous shearing is due is the friction exerted by the adjacent liquid on the upper and lower faces of the liquid plate. The liquid layer immediately .above the upper face flows a little more quickly than that face, and the viscous friction between the two liquid surfaces acts, PROPERTIES OF LIQUIDS. 335 therefore, on the surface of the plate in the direction of the flow. Similarly, the liquid layer immediately below the lower face Hows a little more slowly than that face, and the viscous friction between the liquid surfaces acts, therefore, on the surface of the plate in a direction opposite to the direction of flow. The liquid plate is thus sheared by the opposing frictional stresses on its upper and lower faces ; these stresses act parallel to the faces of the plate, and if the plate is very thin then the liquid friction per unit area of surface may be taken to be the same for both faces. If this frictional stress on the faces of the liquid plate v be denoted by F, the ratio of the stress, F, to , the shear produced in the plate per unit time is taken as a measure of the viscosity of the liquid, as is called the coefficient of viscosity for the liquid. That is, if m denote the viscosity of the liquid, we have 7 - F ^ m = F /-=-.-. / d c It is to be noted that the stress F exists only while the shearing is going on in the liquid, so that m must be defined in relation to the shearing strain per unit time* When a liquid which wets glass flows through a glass tube of fine bore a stationary film of liquid adheres to the tube, and the rate of flow, which is zero for this film, increases from zero at the outer surface of the stream to a maximum at the axis of the stream. It follows from this that any infinitely thin cylindrical shell of the liquid stream flows a little more rapidlv than the * In comparing m, the coefficient of viscosity for a liquid, with n, the modulus of simple rigidity for a solid, it is to be noted that the frictional stress between the layers of the liquid while shearing is going on con- tinuously corresponds to the elastic stress set up between the layers of the solid in opposition to the existence of any given shear. Hence, in defining m, the shear produced per unit time must be taken for the shearing strain ; whereas, in defining n, only the magnitude of the shear produced by the applied stress lias to be considered. 336 GENERAL PHYSICS. similar shell immediately surrounding it, and that there must, therefore, be viscous friction between the two shells of liquid. That is, the flow of tlfe liquid within any cylinder taken inside the bore, and co-axial with it, is retarded by the viscous friction of the liquid surrounding it, and the rate of flow of liquid through the tube must, therefore, depend upon the viscosity of the liquid. It can, in fact, be shown that the rate of flow under given conditions through a given tube is inversely proportional to the viscosity of the liquid, as defined above. The viscosity of different liquids may, therefore, be compared by comparing the times in which the same quantity of each liquid flows through the same capillary tube under the same conditions of flow. It is a matter of common observation that a viscous fluid such as glycerine becomes less viscous with rise of temperature, and experiment shows that it is generally true for all liquids that viscosity decreases as the temperature rises. Approximate values of the coefficients of viscosity at C. and 20 C. are given below for a few typical liquids : T tmiirl Coefficient of Viacosity. At C. At 20 C. Water, 00181 0-0102 Alcohol, 0-0018 00013 Ether, 0-0030 0-0026 Mercury, 0-0170 0-0150 Glycerine, . 40-0000 5-0000 (At 4C.) 337 CHAPTEK XXI. PROPERTIES OP LIQUIDS (Continued). 118. Surface Tension and Capillarity. Certain phe- nomena, which may be observed at the boundary surface of a mass of liquid, seem to indicate, and may be explained by assuming, that the thin surface layer or skin which forms the boundary surface of the liquid acts as a stretched membrane under uniform tension in all directions. Thus, if a large drop of oil is allowed to form on the surface of water, it rests on the water as a lens-shaped mass, and the surface of separation between it and the water is distinct and clearly marked, just as it would be if each mass of liquid were contained in a very thin elastic skin or membrane. In the same way if a light object, such as a needle, is slightly greased and placed very carefully on the surface of water, it will be supported by the surface, just as it would be by a thin-stretched mem- brane. The surface shows a slight depression at the point where the object rests, and the weight of the object may be supposed to be supported by the vertical components of the surface tension round the edge of the depression. When a small quantity of mercury is spilt on the table or on the floor, it usually breaks up into a large number of small globules, which are nearly spherical in form. This is readily explained by supposing that the outer surface layer of each globule is subject to uniform tension, and that it, therefore, assumes a spherical form in which its surface area is a minimum. The same effect may be exhibited in a striking way by one of 22 338 GENERAL PHYSICS. the many beautiful experiments due to Plateau. If a quantity of olive oil is passed gently from a pipette into a mixture of water and alcohol of the same density as the oil, the effect of gravity in preventing the formation of large drops is eliminated, and the oil takes the form of a large spherical drop, separated by a clearly marked boundary surface from the surrounding liquid. A similar effect may be observed in the formation of drops at the end of a glass rod or pipette which has been dipped into a liquid. The forms which the drop assumes during its formation, and the manner in which it breaks away from the rod to which it is attached, are all consistent with the supposition that the surface layer of the drop acts as a stretched membrane under uniform tension in every direction. The fact that the surface of a liquid behaves as if subject to tension is further illustrated by the behaviour of liquid films. When a soap bubble is blown from soap solution, the double film which forms the wall of the bubble is extended by increasing the pressure inside the bubble, and if this pressure is removed the wall of the bubble contracts and drives out the air which has been forced into it. That is, the bubble behaves as if it were a very thin elastic membrane under uniform tension. Numerous other experiments show that a liquid film always behaves in this way. When a flat wire ring is dropped into soap solution and withdrawn, a thin film of the solution will be found stretched across it. If a loop of thread which has been dipped in the solution is then placed gently on this film it will retain any irregular form that may be given to it, but if the film is broken at a point within the loop the portion of the film between the wire ring and the loop at once contracts, and the loop takes a circular form as the inner boundary of the film. The tension which is thus supposed to exist in a very thin layer or skin at any boundary surface of a liquid is called the surface tension of the liquid. It is supposed to have the same value for all directions in the surface, and is measured for any PROPERTIES OF LIQUIDS. 339 surface by the tension at right angles across unit length of a line taken in any direction on the surface of the liquid. That is, the surface tension is the tension in a strip of the surface film of unit width, taken in any direction on the surface. The magnitude of the surface tension at the boundary surface of a liquid is found to depend upon the substance from which the liquid is separated by the surface. Thus, the surface tension of water at the boundary surface between water and air differs from the surface tension of water at the boundary surface between water and oil, or between water and glass. It must be understood, too, that the surface tension of a liquid for any boundary surface is not increased by extending the sur- face. The surface film of a liquid cannot be stretched in the sense that an elastic membrane is stretched ; it may be extended by the transfer of molecules from the deeper layers into the surface layer, but the tension in the surface remains constant during the process of extension. Thus, in the case of a soap bubble, the wall of the bubble consists of an inner or outer surface film enclosing a very thin layer of liquid between them ; as the bubble is blown the inner and outer films are extended at the expense of the liquid between them until all their liquid is transferred to the surface films and the bubble bursts. The films thus remain in the same state and subject to the same tension throughout the process of extension. It will be seen from what has been said that if a strip of liquid surface of width (a) is extended in length by a distance (6), the work done during the process of extension is given by T(ab), where T denotes the surface tension for the liquid surface. That is, the work done in extending a liquid surface by an area (ab) is T(ak), or, in general, the work done in extending a liquid surface by an area A is TA, where T denotes the surface tension of the surface. It follows from this that the energy of a liquid surface per unit of area is numerically equal to its surface tension, .and that the energy which a liquid possesses as surface energy 340 GENERAL PHYSICS. increases when the surface area is extended, and decreases when the surface ajea is diminished. The energy which a liquid surface possesses in virtue of its surface tension is called the surface energy of the surface. The interesting phenomena which are usually considered under the head of capillarity or capillary action are due also to surface tension. When a rod or plate of any solid is dipped into a liquid which does not wet the solid, it is noticed that the liquid surface is convex and slightly depressed round the line of contact with the solid. Thus, if a rod or strip of glass is dipped into mercury, the liquid surface round the line of contact takes the form shown in section in Fig. 145. If the mercury is Fig. 145. Fig. 146. contained in a glass vessel the same effect may also be observed round the line of contact of the mercury surface with the glass. Further, if one end of a capillary glass tube, about 1 mm. or less in bore, and open at both ends, is dipped into mercury so that a column of mercury enters the bore, it is found that the surface of the mercury column is not only convex, owing to the convex curvature round the line of contact with the glass, but that it stands at a lower level than the general surface of the mercury outside the tube. This effect is shown in section in Fig. 146; it will be seen that the mercury surface is convex and slightly depressed round the lines of contact of the surface with the glass inside the tube, outside the tube, and round the PROPERTIES OF LIQUIDS. 341 inner surface of the containing vessel, and also that the level of the mercury inside the tube is slightly lower than the general level outside. If tubes of different bores are used in this Experiment it is found, too, that the capillary depression inside the tube increases as the bore diminishes, and measurement shows that it is inversely proportional to the diameter of the bore of the tube. Experiment 30. Take three or four lengths of capillary glass tubing of different bores, place them in succession with one end dipping in mercury, and measure the capillary depression in each case. This may be done by bringing the tube close to the side of the glass vessel containing the mercury ; the depression of the thread of mercury in the tube can then be measured directly with a scale outside the tube, or read off on a scale attached to the tubing. Measure also the diameter of the bore of each tube and establish the relation between the capillary depression and the diameter. Again, when a rod or plate of a solid is dipped into a liquid which wets the solid, it is seen that the liquid surface is con- Fig. 147. Fig. 148. the other limb of the tuj)e at a point A' at the same level as A r and that the pressure of the air in the flask is, therefore, equal to the atmospheric pressure at B plus the pressure due to the mercury column A'B. Similarly, in Fig. 155 the atmospheric pressure on the mercury surface at B is equal to the pressure in the other limb at a point B' at the same level as B, and that n M M Fig. 153. Fig. 154. Fig. 155. the pressure of the air in the flask is, therefore, equal to the atmospheric pressure minus the pressure due to the mercury column AB'. It thus appears that the pressure of the air in the flask is, in the general case, equal to the atmospheric pressure, plus or minus the pressure due to a column of mercury whose length is equal to the difference in the levels of the mercury at A and B. That PROPERTIES OF GASES. 357 is, if P denote the atmospheric pressure, p the pressure due to the column of mercury of height equal to the difference of the levels at A and B, and P' the pressure of the air in the flask, then P' = P p. If the atmospheric pressure P is known in dynes per square centimetre, the pressure denoted by p must be similarly expressed by means of the relation hdg (Art. 108), where h is the height of the column in cms., d the density of the mercury in grammes per cub. cm., and g the acceleration due to gravity at the place of observation. If the atmospheric pressure is known to be equal to the pressure due to a column of mercury of height H, and the difference of the mercury levels at A and B is denoted by Ti, then the pressure of the air in the flask is conveniently expressed as that due to a column of mercury of height (H h) as the case may be. It must be noted, however, that if the pressure of a gas is expressed in terms of the height of a column of mercury (or other liquid), the density of the mercury must be specified, usually by giving its temperature, and the value of g, the acceleration due to gravity at the place of observation, must be known. A U-tube filled with mercury, and constructed for the purpose of measuring pressure in the manner indicated in Fig. 153, is generally called a manometer or pressure gauge. 122. The Atmospheric Pressure: The Barometer. The atmosphere surrounds the earth as a spherical layer of air which extends, as an appreciable atmosphere, to a height of from two hundred to three hundred miles above the surface of the earth. We thus live at the surface of the earth at the bottom of a deep sea of air, and subject to the pressure produced in the air surrounding us by the weight of the overlying air. This pressure in the air around us is called the atmospheric pressure. We. are not sensible of the pressure existing in the air around us, because, at every point, it is exerted equally in all directions. 358 GENERAL PHYSICS. That is, the air exerts pressure on any surface with which it is in contact in a direction at right angles to the surface, and the magnitude of this pressure at any point is the same for all positions of the surface. It follows from this, as in Art. 109, by the principle of Archimedes, that the resultant effect of the air pressure on a body completely surrounded by air is an upthrust equal to the weight of the displaced air, and acting vertically upwards on the body through the centre of gravity of the dis- placed air. At the same time, the body being subject to a normal pressure at all points on its surface, is compressed to an extent which depends upon its elasticity of volume. Hence, if the hand is held out in the air with the palm horizontal, it is not forced downwards by the downward air pressure on the upper surface, or upwards by the upward pressure on the lower surface, but experiences merely a very small upward thrust equal to the weight of the air which it displaces. At the same time the tissue and blood-vessels of the hand are subject to the compressing or supporting effect of the pressure to which it is exposed. Both these effects are inseparable from the conditions of life at the surface of the earth, and we are not sensible of them because we are always subject to them. The upthrust due to the buoyancy of the air is too small to have any special relation to the structure of the body, but the supporting effect of the air pressure on the tissues of the body is one of the conditions to which the structure of the body is specially adapted, and which cannot be altered without danger to life. In the same way if a thin glass flask is exposed to the air it is not crushed by the pressure of the air on it. The outward pressure of the air inside it on its inner surface practically balances the inward pressure of the external air on the outer surface, and the final result is merely that the glass of the flask is very slightly compressed, and that its pressure on the table on which it stands is less than its true weight by the weight of the air which it displaces. PROPERTIES OF GASES. 359 The fact that the air around us does exert a very considerable pressure on all surfaces with which it is in contact may be demonstrated in a simple and striking manner by the familiar experiment described below. The experiments with an air pump, described in Arts. 128, 129, also illustrate this fact in an interesting way. Experiment 32. Get a cylindrical flask made of thin tin plate for the purpose of this experiment. Put a small quantity of water in the flask, and boil the water until all the air in the flask is expelled by the steam. Then, while the water is still boiling, cork up the flask with a good, well-fitting cork, and stand it in a suitable trough to cool it by pouring cold water over it. As the flask cools, the steam inside it condenses, and a partial vacuum is produced inside the flask. The walls of the flask are now subject to the atmospheric pressure externally, and to a pressure diminished almost to zero value internally. They are not strong enough to withstand this excess of external pressure, and are consequently crushed violently inwards soon after the cooling begins. A thin glass flask may be broken in the same way, or, if the flask is stout enough to stand the pressure, it will be found that the cork is forced inwards so as to fit very tightly into the neck of the flask. The pressure in the air at any level being due to the weight of the air above this level must evidently be greatest at the surface of the earth, and must decrease from layer to layer as the height above the surface increases. It will be understood from . this that, as the air is a gas, and easily compressible under pressure, the density of the air is greatest at the surface of the earth, where the pressure is greatest, and decreases as the height above the surface increases, and the pressure diminishes until, at a height of from two hundred to three hundred miles above the sea level, the density and pressure become inappreciably small. . It follows from this that the pressure of the air at any level could not be found by applying the relation p = hdg, as explained in Art. 108, even if the value of h were definite and accurately known, for the value of d varies from layer to layer, 360 GENERAL PHYSICS. and is, therefore, not a constant. The value of g, also, varies from layer to layer, fof it decreases as the height above the sea level increases. For the same reasons it will be seen that the difference of pressure for a vertical difference of level denoted by h is not given by hdg. If, however, h is small, so that both d and . That is, V = k . p , where k is a constant, and, therefore, PY = k. This form of the relation between P and V is conveniently expressed graphically; for if a curve is plotted, as in Fig. 164, so that the abscissae represent the volumes, arid the ordinates the corresponding pressures of a given mass of gas at a constant temperature, the form of the curve obtained is the characteristic rectangular hyperbola shown in the figure. 374 GENERAL PHYSICS. It will be seen that Boyle's law may be stated in terms of the pressure and density of the gas. For, if a given mass of a gas has a volume Vj and density DJ under a pressure P 1? and a volume V 2 and density D 2 under a pressure P 2 , then since , _ and , _ 2 D. " V' a 3 V " P we must have D 2 P 2 ' That is, the density of a gas at constant temperature is directly proportional to its pressure. C c' VOLUME Fig. 164. Fig. 165. Boyle's law may be studied experimentally, in the case of air, by the following simple method. A large U-shaped tube with a long open limb, CD, and a shorter closed limb, AB, as shown in Fig. 165, has a quantity of mercury poured into the bend AC, so as to separate a quantity of air in the closed limb AB from the external air. The volume of this air can be measured either by means of suitable graduations on the tube or simply by measuring the length of the tube occupied by the air. This latter method assumes, however, that the bore of the tube PROPERTIES OF GASES. 375 is uniform, and the unit in which the volume is measured is evidently the capacity of the bore per unit length. The pressure to which the air in AB is subject is evidently given by the atmospheric pressure exerted on the surface of the mercury at C, plus or minus the pressure due to the difference of the mercury levels at A and C, according as the level at C is higher or lower than the level at A. Thus, in Fig. 165, the pressure of the air in the closed limb is the atmospheric pressure at C, plus the pressure due to the short column CO' of mercury. It will be seen that this pressure may be increased to an extent which is limited only by the height of the limb CD by pouring mercury into this limb. It is possible, therefore, by increasing the pres- sure in this way, step by step, and observing the corresponding values of the pressure and volume at each step, to determine experimentally the relation between the pressure and volume of air in the tube AB at constant temperature. The tube shown in Fig. 165 is similar to that used by Boyle in the experiments by which he established the law. It usually takes the form shown in Fig. 166, and is generally known as Boyle's tube. Both limbs of the tube are usually graduated in the same way in length divisions (preferably mms.) from a zero at the same hori- zontal level on both tubes. Experiment 33. Take a Boyle's tube, similar to that shown in Fig. 166, pour a quantity of mercury into the bend, and adjust the level of the mercury in the closed limb to the zero level of the scales on the tube, by tilting the tube and allowing air to enter or escape from the closed limb as may be necessary. Read the height of the barometer, and also the height of the level Fig. 160. 376 GENERAL PHYSICS. of the mercury in the open limb above the zero level. The sum of these two readings "(in mms. or inches) is the pressure of the air in the closed limb in mms. or inches of mercury. The volume of the air in the closed limb may be recorded in terms of the linear divisions of the scale, but it must be remembered, as explained above, that the unit of volume thereby adopted is the capacity of the bore of the tube per division. If it is found that the level of the mercury in the closed limb is not exactly adjusted to the zero level, the actual level must be read, and the necessary corrections applied to the pressure and volume readings. Now pour mercury into the open limb, step by step, so as to raise the level above 50 mms. at each step, and read the pressure and the volume of the air in the closed limb at each step. Arrange the readings taken in tabular form, and verify that the product PV for the air in the closed limb is practically constant. Plot a curve showing the relation between P and V, as in Fig. 164. Example. In an experiment, similar to that described above, the observations and results set out in the table given below were obtained. Heading of Level of Volume of Mercury in Height of Barometer. Pressure of Air in Closed Limb. Air in Closed Limb (Scale extends from Value of Product. T>TT Closed Limb. Open Limb. P. - 200 mm.). XT . mms. mms. mms. mms. of mercury. Scale units. 0-5 2-5 756 758 199-5 151,200 10-5 52-5 798 189-5 151,200 25-0 133-0 864 175-0 151,200 36-5 205-5 925 163-5 151,200 50-0 302-0 1,008 150-0 151,200 61-5 397-5 1,092 138-5 151,200 74-0 518-0 1,200 126-0 151,200 85-0 644-0 1,315 115-0 151,200 100-0 856-0 1,512 100-0 151,200 It will be seen from these results that the value of PV is practi- cally constant, and it may be noted from the first and last readings that when the pressure of the air in the closed limb is doubled, the volume is halved. The constant in the last column is given only to the fourth signifi- PROPERTIES OF GASES. 377 cant figure. It will generally be found, in a rough experiment of this kind, that the agreement be- tween the different values does not go beyond the second or third sig- nificant figure. The agreement is, however, sufficient to indicate the general truth of the law. The type of tube described above is not suitable for accurate work. A better form of tube is shown in Fig. 167. In this form the closed limb is fitted with a stopcock at the closed end, and is graduated, like a burette, in cubic centimetres and tenths of a cubic centimetre. It is also arranged, as shown in the figure, so that mercury can be ad- mitted into the tube from below instead of being poured in from above at the upper-end of the open limb. The mercury supply is con- tained in a reservoir connected to the bend of the tube by a length of stout rubber-tubing, and is carried by a metal holder which slides up and down the stand as required. The pressure to which the gas in the closed limb is subject can thus be varied by raising or lowering the mercury reser- voir. If the stopcock on the closed limb is opened at the beginning of any set of observations the initial volume of the air in the closed limb can be adjusted to any Fig. 167. 378 GENERAL PHYSICS. desired value, and the initial pressure will be the existing atmospheric pressure. For exact work it is necessary that the air or gas experimented on should be perfectly dry. The inner surface of the tube and the mercury should, therefore, be thoroughly dried, and the closed limb should be filled by drawing in the air or gas at the stopcock through a range of drying tubes connecting the limb with the gas supply. This may easily be done by first filling the limb with mercury up to the stopcock, and then drawing in the air or gas, through drying tubes, by lowering the mercury reservoir, so that the dry gas enters the limb as the mercury leaves it. In the experiments described above, the volume of gas in the closed limb is supposed to be varied only by increasing the pressure to which it is subjected. That is, the law is tested by these experiments only for pressures greater than the atmospheric pressure. The experiments may, however, be extended to pres- sures less than one atmosphere by arranging the tube, shown in Fig. 167, with the closed limb on a level with the top instead of with the bottom of the open limb. If the initial volume of the gas in the closed limb is then adjusted to about half the capacity of the limb at atmospheric pressure, the pressure can be reduced, and the volume increased by lowering the mercury reservoir. As the reservoir is lowered the gas in the closed limb expands, and, step by step, readings of the volume and pressure can be taken as explained above. More accurate results may, however, be obtained at low pressures by the method adopted by Regnault. A graduated tube, similar to a barometer tube, is nearly filled with clean dry mercury, and then inverted, as shown in Fig. 168, in a tubular cistern filled with mercury, so that the small quantity of air left in the tube at filling rises to the upper part of the tube, and occupies the space AB above the mercury in the tube. The volume of this small quantity of air in AB can be read off PROPERTIES OF GASES. 379 B on the graduations of the tube, and its pressure is evidently the atmospheric pressure on the surface of the mercury in the cistern, minus the pressure due to the column BC of mercury. The volume and pressure of the air, which can thus be de- termined for any position of the tube in the cistern, can evidently be varied by raising or lowering the tube in ^\. the cistern. As the tube is lowered, for ex- ample, the volume decreases, and the pressure increases and becomes equal to the atmos- pheric pressure when the volume of the air in AB reaches its initial value at the time of filling the tube. The volume and pressure of the air can, therefore, be observed for a series of successive positions of the tube, and the truth of Boyle's law, for pressures less than one atmosphere, can be tested and established. Experiment 34. Take a clean dry barometer tube and pour clean dry mercury into it until the level of the mercury rises to within a few inches of the mouth of the tube. Measure carefully the length of part unoccupied by mercury : this gives the volume of the air which fills the space, and, as the air is at the atmospheric pressure, its volume and pressure are both known. Now place the thumb, or a flat rubber-pad, over the mouth of the tube, and invert it in an ordinary cistern of mercury, so that the air rises to the upper part of the tube. Measure the length now occupied by the air, and determine its pressure by subtracting the height of the column of mercury standing in the tube from the barometric height. Verify that the product PV for the quantity of air in the tube is the same for the values of P and V observed before inverting the tube, as for the values observed after inversion. Repeat the experiment for a number of different quantities of air. Note that the value of PV is constant for any one quantity of air, but the value of the constant is different for the different quantities. Fig. 168. 380 GENERAL PHYSICS. After Boyle's law was first established by Boyle and Mariotte it became the subject of much careful experimental research with the object of discovering whether the law was strictly exact and applicable to all gases. The first result which was conclusively established was the fact that different gases, such as air, oxygen, hydrogen, nitrogen, carbon dioxide, and sulphur dioxide, when subjected to the same increase of pressure, were compressed to different extents. This showed at once that the law could not apply exactly to every gas even if it applied exactly to some one gas. Dulong and Arago, in 1826, and Regnault, in 1847, carried out very careful and laborious investigations of the truth of the law by the simple method described above, but with much more accurate and elaborate apparatus. The open limb of the tube was extended in Regnault's apparatus to over thirty metres in height, so as to extend the range of the experiments to high pressures, and the mercury was pumped into the tube from below instead of being poured in from above. An important improvement introduced by Regnault in the method of the experiment enabled him to avoid the comparatively large percentage error which attends the measurement of the volume of the air under high pressure. As the pressure is increased the volume of the air decreases, and at high pressures the volume becomes so small that the inevitable error made in observing it may become a very large fraction of the observed value. Regnault avoided this error by taking the same initial volume of gas at each pressure, and reducing it by increase of pressure to the same final volume (about half the initial volume) in each case. Every time the volume of the gas in the closed limb was reduced to this final volume, more gas was pumped into the limb until the initial volume was again reached, and this volume was then, in turn, reduced again to the final volume by applying the necessary increase of pressure. Neither Dulong and Arago, nor Regnault were able to deduce PROPERTIES OF GASES. 381 any general result from their investigations. It was shown, however, conclusively that different gases deviate to different extents from exact conformity to the law. It was found, for example, that all gases, except hydrogen, are slightly more compressible under increase of pressure than they would be if they obeyed the law exactly, and that this deviation wa& more marked at ordinary temperatures in the case of gases, such as carbon dioxide arid sulphur dioxide, which are not far removed from their temperatures of liquefaction, than in gases such as oxygen and nitrogen. Hydrogen, on the other hand, was found to be less compressible under increase of pressure than is required by the law. These results, although well established by experiment, are evidently of an empirical, rather than a general nature, and it was not until 1870 that the general nature of the deviation of all gases from strict obedience to Boyle's law was formulated. In this year Amagat published the result of a very complete research on the truth of Boyle's law, and the nature of the deviations from it in the cases of a number of different gases. It was found that in all gases the value of PY, instead of being a constant for all values of P, first decreased to a minimum and then increased as the value of P increased. The value of P, for which PV reached its minimum value, and the rates of decrease and increase before and after reaching the minimum were found to vary within somewhat wide limits for different gases, but the general nature of the variation of PV with P was the same for all gases. These results are most effectively exhibited by means of curves showing the relation between PV and P for different gases. The curves for hydrogen, nitrogen, and carbon dioxide at several different temperatures are given in Figs. 169, 170, and 171. It will be seen that the minimum value of PV occurs at very different values of P for different gases, and that it is a much more marked and characteristic feature of the curve for an easily 382 GENERAL PHYSICS. liquefiable gas, such as carbon dioxide, than for the more " permanent " gases, sucn as hydrogen and nitrogen. It will be seen, too, from the carbon dioxide curves in Fig. 171, 44 42 40 38 36 34 32 30 28 26 46 44 42 40 38 36 34 32 30 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 Fig. 169. Hydrogen. Pressure in. ATM. 20 40 60 80 tOO 120 f40 160 180 200 220 240 260 280 300 320 Fig. 170. Nitrogen. that the deviation from Boyle's law, and the minimum value of PV, become more and more marked as the temperature approaches the temperature of liquefaction. PROPERTIES OF GASES. 383 In the case of hydrogen it is to be noted that the minimum value of PV corresponds to a very low value of P, so that the value of PV for this gas appears at all ordinary pressures to increase with P. If we consider the variation of PV with P for the gases 2-00 100 600 700 200 300 400 500 Fig. 171. Carbon dioxide. for which curves are given in the figure, it will be seen that for a limited range extending, say, from one to ten atmospheres, the value of PV decreases with P for all gases, except hydrogen, and increases for hydrogen. These are obviously the results 384 GENERAL PHYSICS. given by Eegnault, which are thus seen to be included in Amagat's more complete and general result. 124. Dalton's Law. When a mixture of gases, which have no chemical action on each other, occupy the same space, it is found that each gas is distributed uniformly throughout the space, and that it exerts exactly the same pressure as if it alone occupied the space. It is also found that the pressure exerted by the mixture is the sum of the individual pressures exerted severally by the constituents of the mixture. The atmospheric pressure, for example, at any point is the sum of the pressures exerted individually and separately by the various constituents of the atmosphere. The general law, that the pressure in a mixture of gases which have no chemical action on each other is equal to the sum of the individual pressures of the constituents of the mixture, was established experimentally by Dalton and is known as Dalton's law. The pressures exerted individually by any constituent of a gaseous mixture, for which Dalton's law holds, is sometimes called the partial pressure due to that constituent. 125. Correction for the Buoyancy of the Air in Weighing. It has already been explained that the principle of Archimedes applies in a gas as in a liquid. The apparent weight of a body in air must, therefore, be less than its true weight in a vacuum by the weight of the air which it displaces. It is necessary, therefore, to correct all weighings in air for the buoyancy of the air by adding the weight of the displaced air to the apparent weight in every case. This correction is in general very small, but in the case of bodies of large volume and small mass the weight of the displaced air may be comparable with the weight of the body, and the correction becomes an important one. The value of the correction in any given case is usually found in the following way. When a body is weighed in air in the PROPERTIES OF GASES. 385 usual way the balance obtained evidently indicates that the apparent weight of the body in air is equal to the apparent weight of the weights in air. Now, the nominal value, or the value marked on any accurate weight, is usually its true weight in vacuo, so that, if W denote the observed weight of the body, as indicated by the nominal value of the weights in the scale pan, and d denote the density of the material of the weights, the apparent weight of the weights in air is ( W . SJ, where S denotes the density of the air. Similarly, if W denote the true weight of the body in vacuo, and d' the density of its material, its / W x apparent weight in air is ( W y \ Cv / We, must, therefore, have and AV = W f. The true weight W can thus be calculated from the observed weight W if d, d' and 8 are known. The value of B varies from day to day with the temperature, pressure, and humidity of the air, so that for accurate work its value in any particular case would have to be calculated from observations of these atmo- spheric data at the time of weighing. For ordinary purposes, however, the value of 8 may be taken as 0'0012 gramme per c.c., and, as the weights in general use are made of brass, the value of d' is about 8'4 grammes per c.c. If we substitute these 25 386 GENERAL PHYSICS. values in the relation given above, we get W' = W Q- 1 " 0-0012\' ~~jr) where W and W are expressed in grammes. In the calculation given above it is assumed that the body weighed is of uniform density throughout its mass. When this is not the case as, for example, in the case of a large glass globe completely or partially exhausted of air the volume of the displaced air must be found by determining the external volume of the body. This may be evidently done by finding directly the weight of water which the body displaces, or by finding its apparent loss of weight in water. The relation given above will then become or W = W !-- where V denotes the volume of the displaced air. 126. Determination of the Density of Gases. The density of a gas is usually determined by finding the mass of unit volume of the gas directly. The mass of a known volume of the gas is found by finding the weight of the gas which fills a glass bulb of known capacity at a known temperature and pressure. A spherical glass bulb or globe provided with a capillary glass stopcock is completely exhausted of air, and weighed ; it is then filled at known temperature and pressure with the gas whose density is to be determined, and again weighed. The increase in weight gives the mass of the gas which fills the globe at the temperature and pressure of filling. In order to eliminate the correction due to the buoyancy of the air in these weighings PROPERTIES OF GASES. 387 the globe is usually counterpoised by a similar globe of exactly equal external volume ; the air displaced by each globe is thus of the same weight, and the buoyancy correction need not be considered. The capacity of the bulb is found by determining, in exactly the same way, the mass of water which fills it at a known temperature. From these weighings the absolute density of the specific gravity of the gas may be determined. In some cases the density of the gas is found relative to some standard gas, such as oxygen, instead of its density relative to water. In order to do this it is necessary to find the mass of the standard gas which fills the density globe at a known temperature and pressure. The capacity of the globes used for the determination of the density of a gas need not be more than 50 c.c., except in the case of a gas of very low density, such as helium or hydrogen, when a capacity of nearly 200 c.c. is desirable. In accurate determinations of the density of gases it has been found necessary to make a correction for the decrease in the volume of the density globe when exhausted of air. The -decrease due to the compressing effect of the external pressure .affects the accuracy of the result obtained to an extent which is far from negligible. The absolute densities of a few of the common gases at normal temperature and pressure are given below in grammes per cubic centimetre : Air, T2932 grammes per c.c. Nitrogen, 1 '2571 grammes per c.c. Oxygen, 1-4293 ,, ,, Hydrogen, 0'0899 ,, ,, 127. Diffusion. The process of diffusion in gases is of a very similar character to that of diffusion in liquids. If a vessel contains two or more gases, none of which is uniformly -distributed throughout the available space in the vessel, mole- 388 GENERAL PHYSICS. cules of each gas will ^>ass from points of higher to points of lower density or pressure for that particular gas, until it is in hydrostatic equilibrium at all points throughout the space occupied by the gases. This process of mixing of gases by the migration of the molecules from points of higher to points of lower density for each constituent is called diffusion. The process is much more rapid in gases than in liquids, and takes place in every direction irrespective of the action of gravity and the relative density of the gases. The uniform distribution * of the constituents of the atmo- sphere throughout the space which it occupies is due to diffusion, and the fact that the composition of the atmosphere i& the same at all points and at all levels, shows that diffusion takes place equally in all directions for all its constituents. The quantitative treatment of diffusion in gases follows the same line as that adopted for diffusion in liquids. Imagine each gas in a mixture of gases to be arranged in parallel plane layers- of uniform density, and let $ denote the difference in density between two adjacent layers a very small distance x apart, then g - is the gradient of density between these layers, and the $/ quantity of gas which diffuses across unit area in unit time from one layer to the other, is proportional to this gradient, and is, g therefore, denoted by k -, where k is a constant. This constant k for a given gas in a given mixture is the coefficient of diffusion of the gas under the given conditions. Experiments for the determination of k for a given gas have been made only for mixtures of that gas with one other gas, and the coefficient is then known as the coefficient of inter-diffusivity of the two * Uniform distribution must here be understood to mean that the ratio of the partial pressure due to any constituent at any point, to the atmospheric pressure at that point, is the same for all points in the atmosphere. PROPERTIES OF GASES. 389 gases. In one method of experiment for the determination of this coefficient, a long cylinder is divided into two halves by a -central partition, and one half is filled with one gas and the other half with the other gas. The partition is then removed without disturbing the gases, and mixing by diffusion allowed to go on for a known time. The partition is then replaced, and the quantity of each gas which has diffused from one half into the other is determined by analysis. From results obtained in this way the coefficient k can be calculated. It can be shown that if the coefficient of diffusion for a gas is defined as above, and if at any point in the mixture TT denote the difference of partial pressure corresponding to the difference of density S, then the rate at which the gas diffuses at that point is directly proportional to TT, and inversely proportional to the square root of the density of the gas. The passage of a gas through a porous partition is usually considered under the head of diffusion. It is, however, important to understand that a gas may pass through a porous partition by three different processes. If the holes or pores in the partition are not very fine, and are large compared with the thickness of the partition (as would be the case, for example, in a partition of metal foil perforated by very small holes made by a needle point), the gas passes through the partition by the process of effusion. This process is the same as that by which a liquid flows out through a hole made in the wall of the vessel containing it. The theory of the process cannot here be con- sidered, but it may be stated that the rate of flow of the gas through the partition is proportional to the difference in the pressure of the gas on the two sides of the partition, and inversely proportional to the density of the gas. If the holes or pores in the partition are not very fine, but are small compared with the thickness of the partition, the gas passes through by the process of transpiration. This process is the same as that by which a fluid passes through a capillary 390 GENERAL PHYSICS. tube, and the rate at which it takes place depends, as explained in Art. 117, on the viscosity of the fluid. If, however, the pores of the partition are fine enough to be of molecular dimensions, the gas passes through by a process which is practically identical with diffusion as considered above. The rate at which a gas passes through a partition of this kind will, therefore, be proportional to the difference of the partial pressures due to the gas at the two faces of the partition, and inversely proportional to the square root of the density of the gas. Thus, if two gases diffuse through a partition under exactly the same conditions of pressure and temperature, the rate at which they severally pass through the partition are inversely proportional to the square roots of their densities. For example, if two gases, at the same pressure and temperature, are separated by a porous partition through which each gas can pass by diffusion, the rates at which the gases begin to diffuse through the partition, are inversely proportional to the square roots of their initial densities. As diffusion proceeds, however, the difference between the partial pressures at the opposite faces of the partition ceases to be the same for each gas, and the ratio of the rates of diffusion is determined by the more general rule given above. The law that the rates of diffusion of different gases under the same conditions of pressure, gradient, and temperature are inversely proportional to the densities of the gases, is Graham's law of diffusion, and was established by Graham experimentally from observation of the rates of diffusion of gases through partitions of porous materials such as meerschaum, compressed graphite, and plaster of paris. When a mixture of gases passes through a partition by effusion or transpiration, no separation or partial separation of the constituents of the mixture takes place ; the mixture passes through as a mixture, and its composition is practically un- changed by its passage through the partition. When, however, PROPERTIES OF GASES. 391 a mixture of gases is allowed to diffuse through a partition, the different constituents in general diffuse at different rates, and a partial separation of the constituents may thus be produced. The gas which diffuses through the partition will obviously be richer than the initial mixture in those constituents which diffuse most rapidly, while the gas which has not passed through at any stage in the process will be richer in those constituents which diffuse most slowly. This process of partial separation of the constituents of a gaseous mixture by diffusion through a porous partition is called atmolysis. In practice the mixture is usually passed through a long stem of a clay tobacco pipe enclosed in a tube from which the air is exhausted. A slow current of the mixture is passed through the stem, and diffusion of the several con- stituents takes place through the wall of the stem into the vacuum in the surrounding tube. The composition of the mixture may thus be appreciably changed by its passage through the pipe stem, for the mixture which emerges from the stem will be richer than the initial mixture in those con- stituents which diffuse most slowly through the stem wall. A notable instance of the application of this method is found in its use by Sir William Ramsay and Lord Rayleigh in the partial separation of nitrogen and argon. A mixture of these gases, containing only a small percentage of argon, was passed through a considerable length of pipe stem as explained above. The nitrogen being greatly in excess, its partial pressure in the mixture was much greater than that of the argon, and its density being also less than the density of argon, it diffused through the wall of the stem much more rapidly, so that the gas collected at the other end of the stem, after passing through its whole length, was much richer in argon than the initial mixture. 128. Mechanical Air Pumps. An air pump is an instru- ment constructed for the purpose of pumping air or any similar 392 GENERAL PHYSICS. gas out of a closed vessel, and so producing a more or less com- plete vacuum in the vSssel. Pumps of this kind are of two distinctly different types; one type being known as the mechanical pump, and the other as the mercury pump. The general construction and action of the mechanical pump are indicated in Fig. 172. It consists of a cylindrical metal barrel, AB, in which a lightly fitting piston, C, can be worked up and down by the handle D at the upper end of the piston Fig. 172. rod. This barrel communicates through the tube EF with the vessel to be exhausted of air. or, as shown in the figure, with a bell-jar receiver, R, which fits as an airtight cover on the flat circular plate GH. The piston C is fitted at a with a valve, which opens upwards only,* and a similar valve is fitted also at b, at the mouth of the tube EF, which passes from the barrel to the receiver. * A small hole covered with a stretched strip of thin sheet rubber forms a simple value of this kind. PROPERTIES OF GASES. 393 It will be seen that when the piston is forced downwards from the top to the bottom of the cylinder, it tends to compress the air in the barrel below it. This at once closes the valve at b, and when the pressure in the cylinder becomes greater than the atmospheric pressure, the valve at a opens, and the air is forced out through it into the upper part of the cylinder and thence to the outer air. In this way practically the whole of the air in the cylinder is driven out through the valve a by the downstroke of the piston. Then, when the piston is raised for the upstroke, the valve at a at once closes, and the air in the receiver expands into the cylinder as the piston is raised, so that at the end of the upstroke the air which filled the receiver only at the beginning of the stroke, now fills the receiver and cylinder. At the next downstroke the whole of the air in the cylinder will again be expelled, and at the following downstroke the air left in the receiver will again expand so as to fill both receiver and barrel. This process of exhaustion goes on, stroke after stroke, until a fairly low vacuum is produced in the receiver. The theory of this process is comparatively simple. Let V denote the volume of the receiver and tube up to the valve at b, and v the volume of the cylinder from the valve at b to the valve at a when the piston is at the top of its upstroke. Then, since at each double stroke, consisting of a downstroke, followed by an upstroke, the air occupying a volume V in the receiver expands, and occupies a volume V + v in the receiver and cylinder, it follows that if D and d denote respectively the density of the air at the beginning and at the end of the stroke, we must have or 394 GENERAL PHYSICS. That is, the density d at the end of the stroke is equal to V ^ - times the density at the beginning of the stroke. Hence, if D denote the density of the air in the receiver at the beginning of the process of exhaustion, the density of the air at the end of the first double stroke will be given by Similarly, the density at the end of the second double stroke will be given by V / V ^ and it will be seen by continuing this line of argument, that the density at the end of the rz th double stroke is given by This result shows that the value of d n can never become zero (indicating a perfect vacuum) no matter how many strokes are made, but it is obvious that if the pump is mechanically perfect the value of d. n may be made negligibly small by a comparatively small number of strokes. Pumps of the pattern indicated by Fig. 172 are, however, very far from being mechanically perfect, and are rapidly going out of use in modern practice. There is always a certain amount of leakage in action, and it is impossible to make the bottom of the piston fit so closely to the bottom of the cylinder as to drive out all the air in the cylinder at the end of the downstroke.* Another difficulty arises out of the fact that when the pressure of the air in the receiver becomes very low it is unable to raise the valve at b during the upstroke of the * That is, there is always a small clearance or space between the bottom surfaces of the piston and cylinder. PROPERTIES OF GASES. 395 piston. This difficulty can be remedied by arranging for the valve to be opened and closed mechanically by the action of the piston, but this complicates the construction of the instru- ment, and adds to the risk of leakage. One of the best of the mechanical pumps of the piston and valve pattern is the Tate pump, shown diagrammatically in section in Figs. 173 and 174. The barrel or cylinder AB is fitted with a double piston, CD, of which the length is rather less than half the length of the cylinder. This cylinder communicates at its middle point, E, with the receiver. Two valves, both opening outwards, are ru Lrx M\ A I <( L I 1 B r^^J C D E pS\ [ E jr Figs. 173 and 174. provided at a and b at the ends of the cylinder, and the piston is arranged to be worked backwards and forwards in the cylinder by means of the handle shown at H. Imagine the piston to be drawn out the full length of its stroke from the extreme position at the end A, as shown in Fig. 173, to the corresponding extreme position, indicated at the end B, as shown in Fig. 174. It will be seen that as the piston is drawn out from A towards B, the communication between the receiver and the half DB of the cylinder is cut off, and the air in this half is forced out through the valve at b. At the same 396 GENERAL PHYSICS. time the valve at a closes, and a vacuum is formed in the space CA, between the piston .and the end of the cylinder, until the end C of the piston passes the point D. and the air in the receiver is free to expand into this space. At the return stroke of the piston the process just described is reversed ; the air in the half DA is driven out at the valve a, and a vacuum is formed in the space DB until the piston is pushed home, and the receiver is again put in communication with the half DB of the cylinder. Fig. 175. It thus appears that every stroke of the piston is a double stroke, for each half of the cylinder acts as a separate cylinder. In the case of this pump the volume of the receiver, denoted by V in the general theory given above, is evidently the volume up to the point E, and the volume of the cylinder, denoted by /, is the volume of the space between the piston and either end of the cylinder when the piston is in its extreme position at the other end. PROPERTIES OF GASES. 397 A general view of a good form of this pump is shown in Fig. 175. A more satisfactory form of mechanical pump which has come into general use in recent years is the Fleuss pump. This pump is practically free from the defects due to valve action and piston clearance, and is found to give a much better vacuum than the older forms described above. The general construction of this pump is shown diagrammatically in Fig. 176. The cylinder AB is fitted with a piston, C, and a fixed partition, D, through which the piston-rod works. This partition is a little below the cover of the cylinder, and is provided with a value at a, which opens out- wards. The cylinder contains a quan- tity of oil sufficient to fill it up nearly to the level of the side tube E when the piston is at the bottom. The tube E communicates through the tube G with the receiver or vessel to be ex- hausted, and the tube F is provided to allow the oil to flow round from the under to the upper side of the piston at the end of each downstroke. The safety bulb at H is intended to prevent the oil finding its way into the tube G- if the piston is forced down too suddenly at the end of its stroke. Fig. 176. If we imagine the piston to be raised from the bottom to the top of its stroke it will be seen, from the description given above, that as it rises it carries up some of the oil with it as a layer on its upper surface, and after cutting off the communica- tion between the cylinder and the receiver through the tube E r 398 GENERAL PHYSICS. it forces out the air between it and the partition D, through the valve at a. The oil On the piston also passes through the valve, and sweeps out every trace of air from the space between the piston and the partition. Then, at the downstroke, a vacuum is formed in the space between the piston and the partition until the tube at E is cleared, and the receiver is put Fig. 177. in communication with this space. The air in the receiver then expands into the cylinder, and the air thus withdrawn from the receiver is swept out of the cylinder at the next upstroke. During the formation of the vacuum between the piston and partition during the first part of the downstroke, the oil which found its way through the valve a, at the end of the PROPERTIES OF GASES. 399 upstroke, effectually prevents leakage at this valve, or at the point where the piston-rod passes through the partition. In this form of pump the motion of the piston is comparatively free from friction, and the pump is very easily worked. When in good working order it readily gives a vacuum in which the pressure is less than that due to a fifth of a millimetre of mercury. A convenient form of the pump is shown in Fig. 177. A mechanical compression pump or syringe, such as may be used for compressing air into a reservoir, is shown diagrammati- cally in Fig. 178. The action of the pump can readily be Fig/ 178. followed from the figure ; the valve at a opens inwards, and the valve at b outwards from the cylinder. 129. Mercury Air Pumps. The most satisfactory form of air pump for obtaining a very low vacuum is a mercury pump. In this form of pump the piston is replaced by mercury which is made to rise or fall in the tube or barrel of the instrument, and the use of valves is dispensed with entirely. A mercury pump is called a lift pump or a fall pump, according as the air is driven out of the body of the pump by the rise or fall of the mercury. In both cases the principle of action is the principle of 400 GENERAL PHYSICS. 13 S Toricelli's experiment, and the vacuum produced by the pump is essentially the Toric^llian vacuum. The typical fall pump is the Sprengel pump, first described by Sprengel in 1865. In its original and simplest form this pump is arranged as shown in Fig. 179. The essential part is a long capillary tube, AB, about 900 mm. long, and with a bore of about 1*5 mm. diameter. This tube is connected at A by a stout rubber tube connection with a funnel or reservoir, F, which can be filled with mercury, and a screw clip on the rubber at S enables the fall of mercury from the reservoir into the tube to be controlled and arrested. At its lower end, B, the tube dips into a small cistern of mer- cury which is arranged so that any excess of mercury which falls into it through the tube from the upper reservoir overflows into the small vessel C, and can be returned to the reservoir. At a point, D, about 100 mm. below A, the side tube branches off, as shown in the figure, from the tube AB. This tube com- municates with the vessel which is to be exhausted of air by the action of the pump. When the reservoir F is filled with mercury, and the clip at S opened, the mercury falls through the tube AB into the cistern below. The tube being, however, a capillary tube, the Fig. 179. PROPERTIES OF GASES. 401 mercury does not fall in a continuous stream, but in a sequence of drops which fall through the tube as a sequence of short columns or threads of mercury separated by longer columns of air, as indicated in the figure. Hence, as the mercury continues to fall through the tube, air is continuously removed from the tube and the vessel communicating with it through the side tube. This air is carried down the tube between the successive mercury drops into the cistern, and finds its way thence into the outer air. When the space between any two falling drops comes opposite the opening of the tube at D, the air in the vessel V expands into it, and some of the air from the vessel is carried down between the drops as they fall below D. This process goes on continuously as the successive air spaces pass the point D, until, ultimately, a vacuum is produced in the vessel. As a vacuum forms in the vessel and the upper part of the tube, mercury rises as a continuous column in the tube above the level of the mercury in the cistern, and when the process of exhaustion is complete, and a perfect vacuum is formed, this column stands at the barometric height in the tube with its upper level a little below the point D. The clip at S is closed,* and at this stage the pump is practically a simple barometer, as in Toricelli's experiment with the Toricellian vacuum in the closed space above the mercury in the tube. In the simple form described above Sprengel's pump is subject to a serious defect due to the fact that the mercury falling from the reservoir at F carries down air with it and makes it impos- sible to obtain a good vacuum. This defect is most satisfactorily * When the vacuum is nearly complete before S is closed, the drops of mercury fall from the reservoir on to the mercury and glass below with a sharp metallic clink due to the absence of the air, which serves, when present, as a cushion or buffer between the drop and the surface on which it falls. 26 402 GENERAL PHYSICS. removed by the arrangement shown in Fig. 1 80, which represents the form in which the jftimp is now used. The capillary tube AB is arranged much as in the earlier form of the pump. The reservoir F is not, however, con- nected directly to the head of the tube, but communicates with it through a long _r. U-tube, G, a bulb, K, which is exhausted of air, and a second long U-tube, H. The side tube DE communicating with the vessel to be exhausted branches off Fig. 180. Fig. 181. Topler Pump. in this form of the pump at the head of the tube AB at or near the bend where the tube joins the U-tube H. PROPERTIES OF GASES. 403 The presence of the vacuous bulb at K is found to prevent any air from finding its way with the mercury into the pump. The mercury flows from the tube G into the bulb K as a fine jet which strikes on the side of the bulb; from the bulb K it passes through the U-tube H into the head of the pump at A. Here, as in the capillary tube, it breaks into drops, and carries the air from the vessel V down with it, as explained above. The U-tubes G and H are evidently needed to protect the vacuum at K and the vacuum in V when it is formed. The reader will find it a good exercise to draw diagrams showing the levels of the mercury in the pump at the beginning and at the end of the process of forming a vacuum in V. One of the most convenient forms of mercury pump is, how- ever, the lift pump in the form devised by Topler, and known generally as Topler's pump. The general arrangement of this pump and the relative proportion of its parts is shown in Fig. 181. The barrel or body of the pump is formed by the cylindrical reservoir A, which is about 200 mms. long and 50 mms. in diameter. This cylinder is continued downwards into the long tube B, about 00 mms. in length and 13 mms. in diameter. Upwards to the point C, for a short length of about 50 mms., the cylinder is continued by a similar tube, but at this point it joins the long capillary tube D, and the bore narrows gradually and evenly to