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7 
 
 queen's UNIVBBSITT 8BBIB8. 
 
 INTRODUCTION TO THE SCIENCE 
 
 ov 
 
 DYNAMICS. 
 
 BY 
 
 D. H. MARSHALL, M.A., F.R.S.E., 
 
 PROKR88()K OF PHYSICS IN (^UKBN'B UNIVERSITY. KINOBTON, OHT. 
 
 KINGSTON . ONT. : 
 
 PRINTRD BY WnjJAM BAILIR. 
 
 1886. 
 
lot 
 Ch 
 
 F 
 V 
 
 Si 
 
 H 
 
 L 
 A 
 V 
 F 
 
TABLE OF CONTENTS. 
 
 Introduction 
 
 Chapter I. 
 
 II. 
 
 ni. 
 
 IV. 
 
 V. 
 
 VI. 
 
 VII. 
 
 VIII. 
 
 IX. 
 
 X. 
 
 XI. 
 
 XII. 
 
 XIII. 
 
 Page. 
 
 T. 
 
 Matter. Extension 1 
 
 Motion. Velocity 8 
 
 Acceleration ^^ 
 
 Uniformly Accelerated Motion 31 
 
 Inertia. Mass 31 
 
 Momentum. Force 40 
 
 Weight 48 
 
 Archimedes' Principle 57 
 
 Weight of Gases 66 
 
 Exact Specific Weights 73 
 
 Energy. Work 81 
 
 Action and Reaction 94 
 
 Dimensional Equations 104 
 
 TABLES OF MEASUREMENT. 
 
 French and English units of measurement vii. 
 
 Values of g at different latitudes 53 
 
 Specific weights of solids, liquids, and gases 71 
 
 Regnault's maximum pressures of aqueous vapour, for dew-points 
 
 from 0° to 20° C 76 
 
 Length, Area, Volume 107 
 
 Angle, Mass, Density, Time 108 
 
 Velocity, Momentum 109 
 
 Force, Pressure-intensity, Work and Energy 110 
 
 Answers to the Exercises 116 
 
INTRODUCTION 
 
 Mathematien mav be definod ad the rtcit^nce of measure- 
 ment. It is divided generally into (1) Pure Matliematicrs, 
 and (2) Applied Matlieniaties. In the former, measure- 
 ments of ffparc and time alone are considered. In the latter, 
 besides spaee and time, the pro])erties and conditions of 
 matter^ such as mass, weight, energy, temperature, poten- 
 tial, are measured. In a wider sense Applied Matiiematics 
 is known as Natural PhUomphy or PhysicK. Natural 
 Philosophy is the science which investigates and measures 
 the properties of matter as discovered by direct observation 
 and experiment and deduces the laws connecting these 
 properties. So extensive, however, has our knowledge 
 of the properties and conditions of matter become, that 
 different branches of Natural Philosophy are cimveniently 
 separated from the parent stem. Chemistry, Astronomy, 
 Geology, Physiology, &c., though originally branches of 
 Natural Philosophy, have put forth roots like the branches 
 of the banyan tree and become themselves trees of know- 
 ledge, sending forth their own branches, and these in their 
 turn new roots. But the same I'italforre permeates trunk 
 and branches alike, and it is this vital force, under its new 
 name energy^ which now forms the subject-matter of phy- 
 sical science. Natural Philosophy or Physics is thus the 
 science of energy and is divided into the following princi- 
 pal divisions: (I) Dynamics, which treats of plainly visible 
 energy, (2) Sound, (3) Heat, (4) Light, (5) Electricity anc' 
 Magnetism. 
 
i!i 
 
 ! I 
 
 ▼I. 
 
 Before any measurements can be made, certain units of 
 measurement must be fixed upon. Tlius, the navigator 
 measures the run of his ship in knots, the surveyor his 
 land in acres, and states of heat are measured in iher- 
 mometdc degrees. Now, not only in different countries, 
 but even in the same country, different units, bearing no 
 simple relations to one another, are constantly used in mea- 
 surements of the same kind. In order to avoid all un- 
 necessary calculations in the comparison of different obser- 
 vations, scientific men have agreed to adopt a uniform 
 system of units. This is founded on the French system 
 of units and is known as the Ctntimeli'e-GmmSccond or 
 C. G. S. system. In the following pages the student is 
 therefore exercised in the use of the C. G. S. as well as the 
 English units. 
 
 When for special measurements it is desirable to use 
 larger or smaller units than the standards, these are formed 
 in the C. G. S. system quite uniformly, except in measure- 
 ments of time, hy prefixing the words deca, hecto, kilo, 
 mega, to the name of the standard to indicate multiples of 
 10, 102, |03j 10^, times the standard unit, and by prefixing 
 deci., cefitf', tnilli, to indicate submultiples of 10"*, 10"*, 
 10~3. Taken in connection with the decimal notation in 
 the writing of numbers, such a system of forming the mul- 
 tiples and submultiples saves all unnecessary calculations 
 in reducing to the standard unit. In the English system 
 of forming multiples and submultiples, the numbers seem 
 to have been chosen with a view of containing as many 
 prime factors as possible, an imaginary advantage which 
 has occasioned a very great amount of unnecessary calcu- 
 lation. In conformity with the scientific system of units, 
 all temperatures in the following pages are given in degrees 
 centigrade. 
 
VII. 
 
 nits of 
 vigator 
 ^or his 
 11 llier- 
 iintries, 
 'ing no 
 in mea- 
 all un- 
 t obser- 
 miforni 
 system 
 'Olid or 
 (lent is 
 II as tlie 
 
 to use 
 formed 
 leasure- 
 to, kilo, 
 tiples of 
 refixing 
 
 ition in 
 he mul- 
 nlations 
 system 
 Ts seem 
 18 many 
 e which 
 •y calcu- 
 )f units, 
 I degrees 
 
 Unit's of TA'ugth. 
 
 10^ rnillimetres = 10* centimetres = 10 decimetres = 1 
 metre = 10 ' decametre = 10-» hectometre = 10 ^ kilo- 
 metre = 10"' megainetre. 
 
 3 feet = I yard, 6 feet = I fathom, 100 links = I cluiin 
 = 22 yards, 5280 feet -• 1760 yards = 80 chains = I inilo. 
 
 UidU of Surfac*'. 
 
 I are = I square decametre = 10* square metres = 10« 
 square centimetres. 
 
 1 acre = 10 square chains, fi40 acre^ = 1 square mile. 
 
 UnitH (if Volume. 
 
 1 litre = 1 cubic decimetre = 10' culiic centimetres. 
 
 1 gallon = 277 cubic inches nearly, and holds 10 lbs. 
 avoir, of water at 62 ° F. 
 
 Unlt^ of Mass. 
 
 1000 milligrams = 100 centigrams = 10 decigrams = 
 I gram = 10~^ decagram = 10-* hectogram = 10~* kilo- 
 gram = 10~* tonne. 
 
 1 pound avoirdupois = 7000 grains, I ton =2240 pounds. 
 
 The numerlnil relations heiioeen the C. G. S. and Eng- 
 lish units are given on pp. 107-110. 
 
5 
 
 i ^> 
 
INTRODUCTION TO THE SCIENCE OF DYNAMICS. 
 
 Chapter I. 
 
 Matter. 
 
 Exteyision. 
 
 1. The student of elementary dynamics is not concerned 
 with tlie ultimate structure of matter, of which various 
 theories have been advanced by scientific men, but only 
 with its properties. The principal of these which we shall 
 consider are extension, inertia, mass, weight, and energy. 
 
 2. Any portion of matter is called a body. The grains 
 of sand on the sea-shore, our own bodies, houses, the whole 
 earth, the planets, the fixed stars, &c., arc examples of 
 l)odies. The expression of the fact that two or more bodies 
 cannot at the same time occupy the same portion of space 
 is known as the principle of irnpenetrahility. 
 
 3. Extension is that property of matter in virtue of 
 which every body occupies a limited portion of space. This 
 implies that every body has/orm or shape. The volume 
 of a body is the measure of its extension. The term hulk 
 is often used in the same sense. The internal volume of a 
 body, e.g. that of a cup or of a hollow sphere, is the amount 
 of space enclosed by the body, and is called its capacity. 
 
 4. Before any measurements can be made it is necessary 
 to fix upon units or definite quantities of what we desire 
 
to measure. In terms of these units we express by num- 
 bers any other quantities of tlie same kind. In measure- 
 ments of extension four units are used, viz., units of length, 
 area, vohime, and angle. Of these one may be taken as a 
 fundamental unit, and the others made to depend upon it, 
 and these are hence called derived units. It is most con- 
 venient to take the unit of length as the fundamental unit. 
 
 5. The English unit of length (or distance) is the yard, 
 and is defined by Act of Parliament as the distance be- 
 tween two points on a bar of irietal at a definite temper- 
 ature. The French unit, the metres although derived or- 
 iginally from the supposed dimensions of the earth, is sim- 
 ilarly defined. The unit of length adopted by scientific 
 men is one of the submultiples of the French unit, viz., the 
 centimetre^ and its multiples and submultiples are the same 
 as the French, viz., millimetre as submultiple, and deci- 
 metre, metre, decametere, hectometre, kilometre as mul- 
 tiples. 
 
 6. Whatever unit of length be used, it is found most 
 convenient in measurements of surface to take as the unit 
 of area (or surface) the area of a square of which the side 
 is unit of length, or a multiple or submultiple thereof 
 Hence the scientific unit of area is a square cetitimetre. 
 The French unit of area, the are, is a square decametre, 
 and therefore equal to 10^ scientific units of surface. 
 
 7. Similarly the unit of volume is immediately and most 
 conveniently derived from the unit of length by defining it 
 as the volume of a cube of which the edge is unit of length 
 or a multiple or submultiple of the unit of length. Hence 
 the scientific unit of volume is a cuhio centimetre. The 
 French unit of capacity, the litre, is a cubic decimetre, and 
 therefore equal to 10^ scientific units of volume. 
 
 . I' 
 
3 
 
 by num- 
 nieasure- 
 if length, 
 ken as a 
 upon it, 
 lost cen- 
 tal unit. 
 
 he yard, 
 
 ince be- 
 
 teinper- 
 
 rived or- 
 
 i, is sini- 
 
 icientific 
 
 viz., the 
 
 the same 
 
 nd deci- 
 
 as nuil- 
 
 id most 
 tlie unit 
 the side 
 thereof- 
 iimetre, 
 iametre, 
 •e. 
 
 nd most 
 fining it 
 f length 
 Hence 
 e. The 
 tre, and 
 
 
 
 8. The unit of angle in common measurements is the 
 degree, which is the 90tli part of a right angle. It may 
 seem strange to say that the unit of angle can be derived 
 from the unit of length, but this will be understood when 
 we remember that if a circle be described with the vertex 
 of the angle as centre and with any radius, the magnitude 
 of the angle is measured by the ratio of the length of the 
 arc on which it stands to the length of the radius. We 
 may, therefore, define the unit angle as that angle which 
 is subtended by an arc of unit length at the centre of a 
 circle of unit radius. This is just the same as an angle 
 which is subtended by an arc, whose length is equal to the 
 radius, at the centre of any circle whatsoever. Such an 
 angle is the scientific unit of angle, and is called a r<tdhin. 
 
 9. As mentioned above, we learn from elementary ge- 
 ometry that, whatever units ot length and angle be adopted, 
 the following formula expresses the relation between the 
 length of any arc {a) of a circle, the length of the radius 
 (/'), and the magnitude of the angle (6) subtended at the 
 centre of the circle by the arc 
 
 a = Cr6 
 
 where C is a constant number, whose value depends upon 
 the unit of angle adopted. If we measure d in radians, 
 this reduces to the simple form 
 
 a = rd. 
 
 10. To express the value (tf a radian, in <h!</re('S. Since 
 
 the arc subtended by two rigiit angles = r/', if be the 
 
 measure of two right angles in radians, we get zr — /•//, 
 
 .'. ff = 71, i.e. two right angles := - radians, and .-. a radian 
 
 180 = 
 = -z — =57" 17' 44"\S true to the tenth part uf a second 
 
 of angle. 
 
I 
 
 11. In many dynamical investigations it is unnecessary 
 to consider in any way the dimensions of a body, or the 
 distances between the different parts of a system of bodies. 
 When this is the case, the body or system of bodies is called 
 a. particle or material particle. Thus in the explanation 
 of the seasons, or of the phases of the moon, the earth or 
 moon is a body, as we cannot neglect its dimensions, 
 whereas in the determination of the sun's positioTi in the 
 ecliptic at any time it is a particle. In considering the 
 proper motion of the solar system amongst the lixed stars, 
 the sun, and indeed the whole solar system, are merely 
 particles. The term particle is frequently defined as an 
 indefinitely small body. The terms small and large are 
 merely relative, and what is small at one time or from one 
 })oint of view is large at another or from another point of 
 view. 
 
 12. The most generally accepted theory of the ultimate 
 structure of matter at the present day is known as the 
 atomic theory. iVccordinn to this theory matter is not in- 
 finitely divisible, but consists ultimately of excessively 
 small indivisible particles. The smallest portion of any 
 body, beyond which mechanical sub-division is supposed to 
 be impossible is called a molecule. A molecule may, how- 
 ever, be chemically divided into atoms. Thus a molecule 
 of water {TI^O) may be chemically divided into three 
 atoms, two of Hydrogen, and one of Oxygen. 
 
 1 
 
 Examination I. 
 
 1. Give various examples of matter. 
 
 2. Distinguish between the terms hody.^ particle, mole- 
 cule^ atom. 
 
leceseary 
 y, or the 
 )f bodies. 
 
 I is called 
 )lanation 
 earth or 
 nensions, 
 
 II in the 
 ring the 
 ;ed stars, 
 
 merely 
 
 id as an 
 
 artre are 
 
 from one 
 
 point of 
 
 ultimate 
 n as the 
 s not in- 
 Ljossively 
 I of any 
 >posed to 
 ay, how- 
 inoleeule 
 to three 
 
 3. Define extension. 
 
 4. Distinguish between volume, hulk, and capacity. 
 
 5. What is a unit of measurement ? Give examples. 
 t>. What is a metre, a litre, a radian ? 
 
 7. Express in radians the angles of an equilateral triangle, 
 half a right angle, 30', 75', 4 right angles. 
 
 8. If the unit of length be a foot, and the unit of angle 
 a right angle, what must be the value of C in the formula 
 a = Crdl 
 
 9. Is the value of C in the above formula dependent on 
 the unit of length? Why? 
 
 le^ mole- 
 
 EXERCISE I. 
 
 Note. — The following examples in mensuration are ap- 
 pended to exercise the student in the use of the new units 
 and also of logarithmic tables to which he should early 
 accustom himself. 
 
 1. The great pyramid of Gizeh is a regular pyramid on 
 ji square base. The original length of an edge of the base 
 was 22042, and of a slant edge 23286-5 ; find (1) the area 
 of the ground on which it stands, (2) the exposed area of 
 the pyramid, (3) the volume. 
 
 2. Assuming the earth to be a sphere, and that the 
 length of an arc of a degree on a meridian is equal to 
 M119 X 10^, find (1) the length of the diameter, (2) 
 the area of the earth's surface, (3) the volume. 
 
 3. If the nature of the earth's crust be known to a depth 
 of 8 kilometres, find the total volume known, and the ratio 
 of the known to the unknown volume, supposing the earth 
 to be a sphere of 6370*9 kilometres radius. 
 
•i 1 
 
 6 
 
 11 ' 
 
 4. On the same supposition, how much of the earth's sur- 
 face could a person see who was at a height of 4 kilo- 
 metres above the sea level ? 
 
 5. If the atmosphere extend to a height of 70 kilometres, 
 what is the ratio of its volume to that of the solid and liquid 
 earth ? 
 
 6. Compare the surfaces of the torrid, temperate, and 
 frigid zones of the earth, supposing the first to extend to 
 an angular distance of 23*30' from the equator, and the 
 last to a distance of 23° 30' from each pole. Determine 
 also in square kilometres the amount of surface in each. 
 
 7. The side.-' of a triangle are 3 metres 3 centimetres, 2 
 metres 8 decimetres, and 2 metres 1 decimetre 4 centi- 
 metres ; to determine its area and greatest angle. 
 
 8. Find the slant edge, surface, and volume of a circular 
 cone, the diameter of its base, and its height being each 1 
 metre. 
 
 9. Two sectors of circles have equal areas, and the radii 
 are as 1 to 2 ; find the ratio of the angles. 
 
 10. A gravel walk of uniform breadth is made round a 
 rectangular grass-plot, the sides of which are 20 and 30 
 metres ; find the breadth of the walk, if its area be three- 
 tenths of that of the grass-plot. 
 
 1 1. The diagonals of a parallelogram are 8 and 10 metres, 
 and its area one-third of an are ; find the angle between 
 the diagonals. 
 
 12. Find the number of litres of air in a room whose di- 
 mensions are 12J m., 5*45 m., and 3*7 m. 
 
 13. Find the angle of a sector of a circle, the radius of 
 which is 20 metres, and the area a declare. 
 
 I 
 
earth's sur 
 of 4: kilo- 
 
 kilometres, 
 and liquid 
 
 erate, and 
 extend to 
 , and the 
 Determine 
 in each. 
 
 ;i metres, 2 
 e 4 centi- 
 
 • 
 
 a circular 
 ng each 1 
 
 1 the radii 
 
 3 round a 
 
 20 and 30 
 
 be three- 
 
 10 metres, 
 J between 
 
 whose di- 
 
 radius of 
 
 14. The diagonals of a parallelogram are to one another 
 as sin g to sin ~, prove that the figure is a rhombus. 
 
 15. The sides of a quadrilateral, taken in order, are 7-5, 
 5-5, 6, and 4 metres, and the angle between the first two 
 IS 74 » 40',15"; shew that the figure may be inscribed in a 
 circle and find its area. 
 
 16. The horizontal parallax of the sun {i.e. the angle 
 subtended by the earth's radius at the sun) is 8"-85, and of 
 the moon 57' 3" ; find the distances of these bodies in terms 
 of the earth's radius. 
 
 17. Find also, in terms of a great circle of the earth the 
 areas of the moon's orbit and of the ecliptic, supposincr 
 these to be circles. *^ 
 
 18. Find the circumference and area of the circle of 
 latitude passing through Kingston, Ont., latitude 44= 13'. 
 
 19. A pendulum whose length is 1^ metre swings through 
 an arc whose chord is a decimetre; find the angle and the 
 length of the arc of oscillation. ■ 
 
 20. Prove that the area of a regular polygon of n sides, 
 inscribed in a circle of radius r, is equal to 7i ~ sin ^-"^ ; 
 
 hence find the areas of an equilateral triangle, square, pent- 
 agon, hexagon, and octagon, inscribed in a circle of unit 
 radius. 
 
 4 
 
8 
 
 Chapter II. 
 
 Motion. 
 
 Velocity, 
 
 13. Motion is change of position. Although the ideas 
 conveyed by the terms matter and motion are quite differ- 
 ent, yet it is evident that all the motions vv^e are cognizant 
 of are the motions of matter directly, or are indirectly pro- 
 duced by motions of matter. Thus the motion we see when 
 a boy throws a stone is the motion of the stone directly. 
 A wave^ on the other hand, which is motion ofform^ is not 
 directly the motion of the medium through which the wave 
 is passing, but is indirectly produced by the motion of this 
 medium. 
 
 14. The opposite (or the zero) of motion is rest. All 
 the motion or rest of a bodv that we can know of is relative, 
 i.e. with respect to some other body. In infinite space ab- 
 solute motion or rest is indeterminable if indeed conceiv- 
 able. When a person is sitting at his ease in a railway 
 carriage, he is said to be at rest. But this is merely 7'e- 
 latively to the train. Relatively to the earth he is moving 
 as fast as the train is, and when we consider that the earth 
 is rotating about its axis, is further revolving around the 
 sun, and with the sun and other members of the solar sys- 
 tem careering through space, it is easily seen how complex 
 is the person's motion. The aim of the physicist is to de- 
 termine those conditions of matter and motion which, apart 
 from the world of sensation, thought, and consciousness, 
 constitute the life of the universe. 
 
 15. Velocity is rate of motion, i.e. rate of change of po- 
 sition per unit of time. By rate is meant here degree of 
 quickness. When two bodies are moving, and one moves 
 
 
he ideas 
 te differ- 
 ognizant 
 ctly pro- 
 ■>ee when 
 directly, 
 m, is not 
 the wave 
 )n of this 
 
 est. All 
 
 I relative, 
 
 space ab- 
 
 conceiv- 
 
 L railway 
 
 erely 7'e- 
 
 moving 
 
 he earth 
 
 )und the 
 
 olar sys- 
 
 coinplex 
 
 is to de- 
 
 ch, apart 
 
 iousness, 
 
 ^e of po- 
 egree of 
 e moves 
 
 over a greater distance in the same time than the other, 
 the velocity of the former is said to be the gre.iter. In any 
 motion of a body the velocity may be uniform, i.e. tlie 
 same throughout the motion, or it may be variable, i.e. 
 continuously or at intervals changing during the motion. 
 The velocities of all bodies that we see moving are really 
 variable. The motions of the hands of a chronometer, or 
 the rotations about their axes of the different members of 
 the solar system are cases of tnotion in which the velocities 
 are nearly uniform. The test of uniform velocity is that 
 equal distances are moved over in equal times, however 
 f<mall these times may he. 
 
 16. What, however, we naturally ask, are equal times f 
 We look at oar clocks or watches and say that they tell us 
 equal times. Some watches go slow, others go fast, and 
 how are we to know which go right ? It is well-known 
 that our clocks and watches are regulated by the apparent 
 motion of the sun in the sphere of the heavens. This mo- 
 tion is the resultant of two motions, viz., (1) the apparent 
 rotation of the sphere of the heavens, produced by the real 
 rotation of the earth, which takes place in a sidereal day ; 
 and (2) the apparent revolution of the sun in the ecliptic 
 in a sidereal year, produced by the real revolution of the 
 earth in its orbit around the sun. As the sidereal year is 
 however estimated in sidereal days, we find that ultimately 
 the apparent rotation of the sphere of the heavens is our 
 standard measurer of time, and we define equal times as 
 times in which the sphere of the heavens apparently ro- 
 tates through equal angles. Whether the term equality is 
 rightly applied to such times or not is a legitimate enquiry. 
 
 17. The mean or average time in which the sun appar- 
 ently rotates about the earth is called a mean solar day, and 
 
10 
 
 our unit of time, the second, is a well-known fraction of the 
 mean Bolar day. In the scientific system of units of u\e2i- 
 surement the ifi-cowr/ is, like the unit of length, one of the 
 fundamental units. 
 
 18. Tiie magnitude of a uniform velocity is measured by 
 the number of units of length passed ovtr in a unit of time. 
 The unit of velocity is derived immediately from the units of 
 length and time ; it is the velocity in which a unit of length 
 is passed over in a unit of time. Hence the scientific unit 
 of velocity is 1 centimetre per second. It will be conve- 
 nient to call this a tach. If ^ be the distance in centimetres 
 described in t seconds by a body moving uniformly with a 
 
 velocity of v tachs, then s = vt, and v =—. 
 
 t 
 
 19. When a body is moving with variable velocity it has 
 of course a definite velocity at every instant, which is 
 measured by the number of units of length which would be 
 passed over in a unit of time, if for such a period from the 
 instant in question the velocity did not change. Hence 
 we talk of a ship sailing at the rate of 12 knots an hour, 
 of a man walking at the rate of 4 miles an hour, &c., al- 
 though the velocity of the ship and of the man may not be 
 the same for any two consecutive seconds. When a body is 
 
 moving with variable velocity, the equation v = ~ gives 
 
 the mean velocity during the time ^, and, by taking t small 
 enough, we can approximate in any degree of exactitude 
 to the velocity at any instant. 
 
 20. A velocity has direction as well as magnitude, and 
 can be completely represented by a straight line, the di- 
 rection of the line representing the direction of the motion 
 (the tangent to the path of the moving particle at the 
 
n 
 
 ction of the 
 
 lits of mea- 
 
 one of the 
 
 leasured by 
 mit of time, 
 the units of 
 lit of lengtli 
 ientific unit 
 1 be conve- 
 centinietres 
 rmly with a 
 
 ocity it has 
 t, which is 
 ih would he 
 3d from the 
 Hence 
 its an hour, 
 ur, &c., al- 
 may not be 
 en a body is 
 
 gives 
 
 t 
 
 dng t small 
 exactitude 
 
 litude, and 
 ne, the di- 
 the motion 
 icle at the 
 
 8- 
 
 instant in question), and tlie length of tlie line representing 
 the magnitude of the velocity. Since a body may move 
 in two directions along a line, the one being diametrically 
 opposite to the other, it is convenient to distinguish these 
 by the signs + and — , as is cu^^totnary in the applica- 
 tions of algebra to geometry. If AB be a straight line, 
 and a velocity in the direction of AB be called + , a ve- 
 locity in the direction of BA will be called — . 
 
 Any ttla r Vtlociiy. 
 
 21. The only case of angular velocity we need at present 
 consider is that about the centre of a circle of a body mov- 
 ing in the circumference 
 
 Definition. When a body moves in a circle the rate 
 of change of the angle, which the radius through the body 
 makes with a fixed radius, is called the angular velocity of 
 the body about the centre of the circle. 
 
 If the angles described by the radius through the body 
 be equal in equal times, however small these may be, then 
 the angular velocity is uniform and is measured by the 
 angle described in a unit of time. The unit of angular ve- 
 locit}' is that in which a unit of angle is described in a unit 
 of time. In scientific measure, then, the unit of angular 
 velocity will be a radian per second. 
 
 When the body is revolving with variable velocity, the 
 angular velocity at any instant is measured by the angle 
 which would he described by the radius through the body 
 in a unit of time, if for such a period the velocity did not 
 change. 
 
 22. From the formula a — rO (art. 9) it follows at once 
 that if V represent in tachs the linear velocity of a body, 
 
 
12 
 
 moving in tlie circumference of a circle, r the radins in 
 centimetres, and m the angular velocity about the centre 
 in radians per second, 
 
 v = r(o, and to = — . 
 
 23. If the angular velocity be in the direction of the 
 hands of a watch it is represented by the — sign, and if in 
 the opposite direction by the + sign. Let it be carefully 
 observed, however, that the sign given to the angular 
 velocity of a body revolving in a circle depends upon the 
 side of the plane of motion from which the motion is ob- 
 served. Thus, if we could see the motion of the hands of 
 a watch through the back of the watch, the angular velocity 
 would be -}-. If we look northward at the rotation of the 
 sphere of the heavens it seems to be +, and if we look 
 southwards it seems to be - . 
 
 Examination II. 
 
 1. Define motion, rest, velocity. 
 
 2. What is a wave ? How is it produced ? Give examples. 
 
 3. Illustrate the meanings of the terms relative and ab- 
 solute, (1) with respect to extension, (2) with respect to 
 motion, (3) with respect to direction. 
 
 4. Define velocity, and distinguish between uniform and 
 variable velocity. 
 
 5. What is the test of uniform velocity ? 
 
 6. Define equal timeSy and the unit of time. 
 
 7. Distinguish between fundamental and derived units, 
 and give examples of each. 
 
radius in 
 the centre 
 
 on of the 
 and if in 
 1 carefully 
 3 angular 
 upon the 
 ion is ob- 
 
 hands of 
 ir velocity 
 ion of the 
 
 we look 
 
 examples. 
 
 3 and ab- 
 espect to 
 
 form and 
 
 ed units, 
 
 13 
 
 S. Name and define the scientific unit of velocity. 
 
 9. Give the relation between *, v, t in uniform motion, 
 
 10. When the motion of a body is not uniform, how is 
 the velocity measured ? 
 
 11. How may velocity be completely represented ? 
 
 12. Define angular velocity, and the unit thereof. 
 
 13. Give and prove the relation between the angular 
 and linear velocity of a body moving in the circumference 
 of a circle. 
 
 14. Distinguish between + and - angular velocity. 
 
 Exercise II. 
 
 1. A body has a velocity of 10 tachs, how long will it 
 take to pass over 600 metres ? 
 
 '2. Which is greater a velocity of 72 tachs or one of 252 
 metres per hour, and by how much ? 
 
 3. Express a velocity of 72 kilometres per hour in deci- 
 metres per minute. 
 
 4. If a line a foot long represent a velocity of 3*75 miles 
 per hour, what length of line would represent a velocity of 
 80 yards per minute ? 
 
 5. Two bodies start from the same point, the one 10 
 minutes after the other, and travel in perpendicular direc- 
 tions with velocities of 120 tachs and 100 metres per 
 minute. How far apart will they be in an hour from the 
 starting of the first ? 
 
 6. Two travellers leave the same place at the same time 
 
 
14 
 
 in directions inclined to one another at an angle of ^, and 
 
 each travels with a velocity of 160 tachs, how far apart 
 will they be in two hours? 
 
 7. A man two metres high walks in a straight line at 
 the rate of 6 kilometres an hour away from a lighted lamp 
 
 3 metres high ; find in tachs the velocity of the end of his 
 shadow, and the rate at which his shadow lengthens. 
 
 8. If 3 minutes be the unit of time, and 50 decimetres 
 the unit of length, what number measures the average rate 
 of walking of a person who goes over 40 kilometres in 12 
 hours? 
 
 9. If 7 metres per 3 minutes be the unit of velocity, and 
 
 4 decimetres the unit of length, what must be the unit of 
 time '( 
 
 10. If 3 metres per 7 minutes be the unit of velocity, 
 and 4 seconds the unit of time, what must be the unit of 
 length ? 
 
 11. A body moving uniformly in a circle describes the 
 circumference twice in 3 minutes, what is the scientific 
 measure of its angular velocity ? 
 
 12. What is the angular velocity of any body on the 
 earth's surface due to the earth's rotation ? 
 
 13. The diameter of the driving wheel of a locomotive 
 is 2 metres, what is the angular velocity of a point on the 
 circumference when the train is moving at the rate of 80 
 kilometres an hour ? 
 
 14. A body moving in the circumference of a circle of 
 radius 10 has unit angular velocity ; find the space de- 
 scribed in 10 seconds, and the time taken to complete a 
 revolution. 
 
■ 
 
 of o? a"<^ 
 far apart 
 
 ht line at 
 lited lamp 
 Bnd of his 
 lens. 
 
 lecinietres 
 erage rate 
 tres in 12 
 
 ocity, and 
 lie unit of 
 
 f velocity, 
 le unit of 
 
 cribes the 
 J scientific 
 
 iy on the 
 
 ocomotive | 
 int on the | 
 rate of 80 I 
 
 I circle of 
 space de- 
 )mplete a 
 
 16 
 
 Chapter III. 
 A ccehralion 
 
 24. Just as a body's position may be constantly changing, 
 giving rise to motion, so a body's velocity may be con- 
 stantly changing, giviiiir rise to acceleration. 
 
 Acceleration is change of velocity, and is measured by 
 the rate of this change per unit of time. A body's velocity 
 may change in magnitude only, or in direction only, or in 
 both magnitude and direction, and in any one of these 
 cases it will be found that at every instant the body has a 
 definite acceleration. 
 
 The rate at which a body's velocity changes may be slow 
 or fast. Compare the accelerations of trains in a long rail- 
 way like the Canadian Pacific, in which the stations are 
 far apart, with the acceleration of trains in large cities, run 
 for the convenience of passengers hurrying frum one part 
 of the city to another, such as the Underground Railroad 
 in London or the Elevated Railroad in New York. 
 
 25. Acceleration, as change of velocftf/, has direction as 
 well as magnitude, and the direction of a body's accelera- 
 tion at any instant may, or may not, be the same as the 
 direction of its motion. 
 
 Let us first consider acceleration, the direction of which 
 is the same as that of the body's motion, or diametrically 
 opposite thereto. The effect of such an acceleration is 
 evidently to change the magnitude of a body's velocity 
 without changing its direction. Acceleration may be uni- 
 form or variable. An acceleration is uniform when equal 
 changes of velocity take place in equal times, however 
 small these times may be, and is then measured by the 
 
16 
 
 number of units of velocity acquired in unit of time. Thus, 
 if in every unit of time the body's velocity is changed by 
 a units of velocity, then a measures the magnitude of the 
 acceleration. If the velocity is increased by 6j, the acceler- 
 ation is generally represented by -i- a ; if on the other hand 
 the velocity be diminished by « units of velocity in unit of 
 time, the acceleration is represented by - a. 
 
 26. The unit of acceleration is a change of unit of ve- 
 locity in unit of time ; hence the scientific unit of acceler- 
 ation is 1 tach per second. 
 
 Observe carefully that the unit of acceleration, by in- 
 volving the units of velocity and time, involves the unit of 
 length once and the unit of time twice. This must be par- 
 ticularly attended to if in the solution of a problem the 
 units require to be changed. Thus a tach is represented 
 by j^^ if a metre and minute be the units of length and 
 time, but with the same units the scientific unit of accel- 
 eration will be represented by ^-YtV- One of the most im- 
 ])ortant cases of the motion we are now considering is that 
 of a body moving vertically upwards or downwards in 
 vacuo. Such a body has a uniform acceleration vertically 
 downwards. This is represented by ^, and in the latitude 
 of Kingston, Ont., its value is 980*5, i.e. in every second 
 the body acquires a downward velocity of 980'5 tachs. 
 
 Tf a represent the acceleration of a body uniformly ac- 
 celerated in the direction of its motion (-}- ^y or — ly\ and 
 V be the whole change of velocity in time ^, then v - at 
 
 and a - y. 
 
 27. A body may have an acceleration constant in mag- 
 nitude and direction, but different in direction from the 
 
 it 
 ■I* 
 
me. Thus, 
 langed bj 
 Jde of the 
 le acceler- 
 ther hand | 
 in unit of I 
 
 lit of ve- 
 )f acceler- 
 
 n, by in- 
 le unit of 
 St be par- 
 blern the 
 presented 
 igth and 
 of aceeJ- 
 niost im- 
 g is that 
 i^ards in 
 ertically 
 latitude 
 second 
 chs. 
 
 nily ac- 
 
 ly\ and 
 
 V = at 
 
 in mag- 
 om the 
 
 I 
 
 17 
 
 direction of motion. The resultant motion in this case is 
 very different from that of the preceding case. Such would 
 be the motion of a body moving in vacuo in any but a ver. 
 tical direction ; it is very nearly that of a leaden bullet 
 projected in the air in any but a vertical direction, and with 
 a small velocity. The magnitude of such a body's ve- 
 locity will be constantly though not uniformly changing, 
 and the direction of motion will be constantly changing, so 
 that the path descri])ed will be a parabola with its axis in 
 the direction of acceleration. 
 
 '28. Again, a body may have an acceleration constant in 
 magnitude but not in direction. Any body revolving uni- 
 formly in a circle (which is approximately the motion of 
 the moon in its orbit) has such an acceleration. If (o be 
 the angular velocity of the body and r the radius of the 
 circle, it can be shewn that the magnitude of acceleration 
 is measured by ?'(o^, but the direction of acceleration is 
 always towards the centre of the circle and therefore 
 changing at every instant. 
 
 29. When the magnitude of a body's acceleration is var- 
 iable, the acceleration at any instant is measured by the 
 number of units of velocity by which the body's velocity 
 would he changed in a unit of time, if for such a period 
 from the instant in question the acceleration did not change. 
 
 When the acceleration is variable the formula a = .gives 
 
 the average acceleration during the time ^, and by taking t 
 small enough we may approximate as closely as we please 
 to the acceleration at the beginning of time t. 
 
 30. A cannon ball shot vertically upwards with great ve- 
 locity is an example of a body, whose acceleration is con- 
 stant in direction, but variable in magnitude on account of 
 
 I 
 
 
 
 m 
 
-1 
 
 18 
 
 the varying resistance of the air. If tlie ball be shot in any 
 but a vertical direction we have a case of motion in which 
 the acceleration is constantly changing both in magnitude 
 and direction. 
 
 31. Acceleration, like velocity, is completely represented 
 by a straight line, the direction of the line being the di- 
 rection of acceleration, and the length of the line represent- 
 ing the magnitude of the acceleration. 
 
 Examination III. 
 
 1. Define acceleration. How is it measured? 
 
 2. Define the unit of acceleration ? What fundamental 
 units does it involve ? 
 
 3. What is the acceleration of a falling body in the lat- 
 itude of Kingston, Ont. ? What of a body rising vertically 
 upwards? 
 
 4. Under what conditions is the formula v=^at true ? 
 
 5. How is acceleration measured when variable? 
 
 6. Give examples of bodies having accelerations, (1) con- 
 stant both in magnitude and direction, (2) constant in mag- 
 nitude but not in direction, (3) constant in direction but 
 not in magnitude, (4) variable both in magnitude and di- 
 rection. 
 
 7. Shew that an acceleration of a metre per minute per 
 second is equal to an acceleration of a metre per second per 
 minute. 
 
 Exercise III. 
 
 N.B. — In the following examples the acceleration is sup- 
 posed to he uniform and in the direction of motion. 
 
 4 
 
shot in any 
 
 n in which 
 
 magnitude 
 
 represented 
 }ing the di- 
 e represent- 
 
 mdamental 
 
 in the lat- 
 ff vertically 
 
 t true ? 
 le? 
 
 ns, (1) con- 
 int in inag- 
 ection but 
 de and di- 
 
 tiinute per 
 second per 
 
 ion IS sup- 
 on. 
 
 19 
 
 1. A body has an acceleration 20 ; find in decimetres per 
 minute the velocity acquired in an hour. 
 
 2. Express the acceleration of a body falling in vacuo 
 (98()'5) in units of a metre and an hour. 
 
 3. The acceleration due to the weight of a body in feet 
 and seconds is 3*2-2 ; find the same in yards and hours. 
 
 4. A body is thrown vertically upwards with a velocity 
 of r)000 tachs ; what is its velocity at the end of 4 and of 8 
 seconds, neglecting the resistance of the air ? 
 
 5. A body uniformly accelerated starts with a velocity 
 of 6 metres per minute, and in half an hour has a velocity 
 of 36 kilometres per hour ; find the acceleration in tachs 
 per second. 
 
 A 6. Compare the acceleration 2 with that in which a 
 velocity of 1 800 metres per hour is acquired in an hour. 
 
 7. Compare an acceleration 3 when a yard and minute 
 i are the units with an acceleration 1 when a foot and second 
 
 are the units. 
 
 8. If 1 tach per 10 seconds were the unit of acceleration, 
 what would be the measure of an acceleration ot 10 tachs 
 per second ? 
 
 9. If 6 kilometres per second per minute were the unit 
 of acceleration, and 1 metre the unit of length, what would 
 be the unit of time ? 
 
 10. If 1 decimetre per hour per second were the unit of 
 acceleration, and 1 metre per minute the unit of velocity, 
 what would be the units of length and time? 
 
 11. What is the diiference between an acceleration of a 
 metre per hour per second and one of a metre per minute 
 per minute ? 
 
 12. Find in tachs per second the diflference between an 
 
 ♦v;^ 
 
 f I, 
 
 «5 
 
20 
 
 acceleration of 24 metres per minute per second and one 
 of 21J kilometres per minute per hour. 
 
 13. If 216 kilometres per minute per hour be the unit of 
 acceleration, and a second be the unit of time, what must 
 be the unit of length ? 
 
 14. If the unit of velocity be 96 metres per 15 minutes 
 and 10 seconds be the unit of time, express in scientific 
 measure the unit of acceleration. 
 
 15. If the unit of velocity be 5 tachs and 3 metres be 
 the unit of length, express in kilometres per hour per hour 
 the unit of acceleration. 
 
 16. If 7 metres be the unit of length and 3 minutes the 
 unit of time, what velocity in tachs will a body acquire in 
 half an hour with an acceleration 18 ? 
 
 17. A body starts with a velocity of 120 tachs, and lias 
 an acceleration of 6 metres per minute per minute ; another 
 starts at the same time from rest with an acceleration of 36 
 kilometres per hour per hour ; when will their velocities be 
 equal 'i 
 
 18. If 5 inches represent an acceleration of 10 tachs per 
 minute, what length of line will represent an acceleration of 
 6 when a metre and minute are the units of length and time? 
 
 19. If a represent an acceleration when m seconds and 
 n feet are the units of length and time, and a when n sec- 
 onds and m feet are the units, find the ratio of — r in terms 
 
 of m and n. 
 
 20. If a and a represent the same acceleration according 
 to two sets of units, and?' be the ratio of the units of length, 
 and s the ratio of the units of time, shew that a: a : : s^ : r. 
 
 21. A body is thrown vertically upwards with a velocity 
 of 10 kilotachs ; after how many seconds will it be moving 
 downwards with a velocity of 6 kilotachs ? 
 
 .'■S& 
 ■■;!? 
 
 i 
 
 
d and one 
 
 the unit of 
 what must 
 
 15 minutes 
 1 scientific 
 
 metres be 
 r per hour 
 
 inutes the 
 acquire in 
 
 8, and hrts 
 e ; another 
 ation of 36 
 slocities be 
 
 tachs per 
 leration of 
 and time? 
 Muds and 
 len n see- 
 in terms 
 
 according 
 
 of length, 
 
 :: s^ : r. 
 
 velocity 
 e moving 
 
 21 
 
 Chapter IV. 
 
 Uniformly accelerated Motion. 
 
 32. We shall in this chapter consider more fully the 
 nature of the motion of a body which is uniformly accel- 
 erated in the direction of its motion. If u be the velocity 
 at any instant and a the acceleration, the velocity at the 
 
 I end of any time t will evidently be u -\- at; denoting this 
 by V we get the first equation of motion, 
 
 V = u -i- at. 
 
 33. To determine the apace described in the time t. Since 
 the velocity during the time t increases iiniforndy from u 
 to u + at, the average velocity is ^\u + {u-\-at)\ or u-\-^at. 
 If a body moved uniformly during the time t with this ve- 
 locity, the space described would be {u-\-^at)t or ut-\-^at^. 
 This will evidently be also the space described during the 
 time < by a body whose velocity increases uniformly from 
 a to u-\-at in that time. Hence if a body has an initial 
 velocity w, and an acceleration a in the direction of its mo- 
 tion, and Ms denote the distance described in the time t^ 
 
 s = ut -\- ^ at^. 
 
 34. The following may be considered by the student a 
 more rigorous proof of the same result : 
 
 Let the time t be divided into a7iy large number of equal 
 
 parts. If n denote the number, the duration of each little 
 
 t 
 interval will be — . The velocities at the heginning of 
 
 each of the little intervals will be 
 
 
 w 
 
 u , u-\-a 
 
 t_ 
 n 
 
 u-\-2a 
 
 n 
 
 w-f(7i— l)a 
 
 n 
 
22 
 
 The velocities at the end of each of the little intervals 
 will be 
 
 u+a — . 2C-{-2a — . u + Sa — , u-^na . 
 
 n 
 
 71 
 
 n 
 
 Suppose now that a body A moved uniformly during:: 
 each little interval with the velocity indicated above at the 
 beginning of each interval ; if s-^ be the whole distance 
 moved over during the n intervals, i.e. during the time t^ 
 
 *i =^ 1 1^+ (i^ + «— ) + (^ + 2a --)+ . . (u -fTT- la -) I 
 
 ^2 
 
 = i^^+ a^ (1 + 2 + 3+ n-\\ 
 
 t^ n{n-\) , o /. 1 \ 
 
 = ut+a^, 2 =^^ + J««^' \} - -^) 
 
 Similarly if s^ be the whole distance moved over by a 
 body B^ which moved uniformly during each little interval 
 with the velocity indicated above at the end of each in- 
 terval, 
 
 Srt-ut + \at^(l + — ) 
 
 Now if « be the whole space described during the time t 
 by the body uniformly accelerated, it is evident that s is 
 greater than s-^ and less than s^ , for the body A moved 
 during each little interval with the least velocity, which 
 the body uniformly accelerated had during that interval, 
 
 "eatest. 
 
 body 
 
 gi\ 
 
 8 
 
 >t.^ + Ja^3(l--) 
 
 <:^ut-\.^at^{\+—) 
 
ttle intervals; 
 
 .u-^na - . 
 n 
 
 rinly durin<; 
 above at the 
 ole distance 
 the time ^, 
 
 n-\a~) \ 
 n' 
 
 ) 
 
 id over by a 
 ittle interval 
 1 of each in- 
 
 g the time t 
 snt that s is 
 [y A moved 
 )city, which 
 lat interval, 
 
 I 
 
 ! 23 
 
 i 
 
 Now these two quantities between wiiich 6- lies differ 
 
 1 
 only m the sign of--. What is iit n is my 7iumber 
 
 ^ v:ha.Uoev€t\ and may be made as large as you please. But 
 
 I 1 1 
 
 by taking 7} large enough, 1 -— and 1 4- may be made 
 
 to differ from 1 by as small a fraction as you please. Hence 
 
 when n becomes 17} definitely great, the motions of A and H 
 
 do not differ from the motion of the body uniformly accel- 
 
 , crated, and the three quantities *, a'^, s^ become %a-^\at'^. 
 
 35. From the equations of uniformly accelerated motion 
 . just determined 
 
 V =u-\-at (1) 
 
 * =ut-\-\at^ (2) 
 
 \ we derive by algebraical analysis the following useful though 
 not independent equation : 
 
 ^2 _ ^2 .^2«.v (3) 
 
 Cor. 1. If the acceleration be opposite in direction to 
 that of motion, it must be represented by - «, and the equa- 
 tions become 
 
 V =u — at (4) 
 
 s =iit — \at^ (5) 
 
 ^2 _ ^3 _ 2a.<} (^Q\ 
 
 Cor. 2. If the body start from rest, n = and the equa- 
 tions become 
 
 V -at (7) 
 
 « =i«<^ (8) 
 
 t>2 - 2as (9\ 
 
 Comparing (2) or (5) with (8) we might say that ut is 
 
 1^ 
 
 I 
 
 J 
 
24 
 
 the space described in virtue of the velocity n, and lat^ 
 that described in virtue of the acceleration a. 
 
 36. As already stated in art. 26, the motion of a body 
 moving freely in a vertical direction is of the character we 
 have been considering. Strictly speaking, this applies only 
 to bodies moving in vacuo^ but unless the velocity be great 
 we may often neglect the action of the atmosphere. The 
 acceleration g of such bodies is vertically downwards, and 
 in the latitude of Kingston, Ont. is very nearly 980-5 tachs 
 per second, or 32 '17 feet per second per second. 
 
 Let us consider the motion of a body thrown vertically 
 upwards with a velocity u. 
 
 1). How long will it rise? 
 
 It rises until its velocity is zero. Hence from equation 
 
 (4) we get — u-gt, .-. t= — . 
 
 2). What is the greatest height reached f 
 
 u 
 In equation (5) putting ^ = — we get 
 
 if 
 
 
 2^- 
 
 We might get the same result more simply from equa- 
 tion (6). When the body ceases to rise, v=0 
 
 = w2-2^5, 
 
 5 = 
 
 V 
 
 3). When will the body return to the point of projection? 
 The distance described from the point of projection in 
 the required time is zero ; hence from equation (5), 
 
 (i=ut-^gt^, 
 
 2u 
 t = Oor — -. 
 9 
 
25 
 
 I, and ^aP 
 
 of a body 
 laracter we 
 -pplies only 
 ity be great 
 here. The 
 wards, and 
 980-5 tachs 
 
 vertically 
 
 m equation 
 
 from equa- 
 
 projectionf 
 qjection in 
 
 (6), 
 
 Comparing this result with 1), we see that the time taken 
 for a body to fall from the greatest height reached, back 
 again to the point of projection, is just the same as that 
 taken by the body to reach its greatest height. 
 
 4). What is the velocity of the body after returning to 
 the point of projection ? 
 
 r rom equation (4), v=ii—gy — I = — w, 
 
 that is, the velocity is the same in magnitude as that at 
 starting, but opposite in direction. Now, since any point 
 in the path might he considered a point of projection^ we 
 infer from this result that the return or downward motion 
 of the body is an exact image of the upward motion. 
 
 37. The distances described i?i successive secotids {or other 
 fqual intervals <f time) by a body^ which starts from rest 
 and 'is uniformly accelerated, are as the odd numbers. 
 
 The distances described in 1, 2, 3, . . . .{n - 1), n seconds 
 
 areia(l)2,^«(2)2, J«(3)2 ^a {n - 1^ , ^an^ ; 
 
 .'. the distances described in the 1st, 2nd, 3rd, nth 
 
 seconds are ^a, ^a,^a, ^a{2n — 1). Thus the 
 
 distance described in the/ith second, where n is any number 
 vjhatsoever, is equal to ^a{2n— i), which varies as (2n— 1) 
 the Tith odd number. 
 
 Ex. A body is thrown vertically upwards with a ve- 
 locity of 3922 tachs ; find 1) the time taken to describe 
 58'83 metres, 2) the velocity at that height, 3) the greatest 
 height reached, 4) the time of ascent, 6) the distance de- 
 scribed in the half second following the fifth second from 
 the instant of starting, 6) the distance described in 10 
 seconds. 
 
 1). Let t = time required. From equation (5), 
 
 5883 = 3922^-4(980-5^2^ ... j = 2 or 6 seconds. 
 
 m 
 
 
 a*fi 
 
26 
 
 3922 
 980-5 
 
 = 4 sec. 
 
 2). From equation (4), velocity at the end of 2 seconds 
 = 3922-2 (980-5) = 1961 tachs; velocity at the end of 6 
 seconds = 3922 -6 (980'5)= -1961 tachs. 
 
 We thus see from 1) and 2) that the body in its ascent 
 has risen 58*83 metres in 2 seconds ; that after 6 seconds it 
 is at the same height, but is then descending / that at botli 
 times the magnitude of the velocity is the same. 
 
 3). From 2), art. 36, the greatest height reached 
 
 (3922)2 
 = Q^,. > = 15688 centimetres. 
 
 4). From 1), art. 36, the time of ascent = 
 
 5). Distance described in 5^ seconds 
 
 = 3922(U)^|(980-5)X(V)' 
 Distance described in 5 seconds = 3922 X5-J(980-5)X5» 
 
 .'. the required distance 
 
 = 3922Xi-J(980-5)XV= -612|fcm. 
 The — sign tells us that the body has descended down this 
 distance in the lltli half second of its motion. 
 
 6). From equation (5) the distance required is 
 3922 X 10- 1(980-5) X 102 = _ 9805 cm. 
 
 The -sign tells us that the body is helow \\\q point of 
 projection. 
 
 Examination IV. 
 
 1. Determine the equations of motion of a body uni- 
 formly accelerated in the direction of its motion. 
 
 2. Deduce the formula v^^u'^ —2as. 
 
 M 
 
27 
 
 
 2 seconds 
 end of 6 
 
 its ascent 
 seconds it 
 at at both 
 
 (d 
 
 = 4 sec. 
 
 80-5) X5» 
 
 n. 
 
 down this 
 
 point of 
 
 ody uni- 
 
 3. A body starts from a given point with a velocity w, and 
 lias an acceleration a opposite in direction to u ; determine 
 1) after what time will the velocity be zero ? 2) After what 
 time will the body return to the point of projection? 3) 
 What is the velocity on returning to the point of projec- 
 tion ? 4) What is the greatest distance travelled over ? 
 
 4. Give the three equations of rnotion of a body let fall 
 to the ground, neglecting the resistance of the air. 
 
 5. Prove that the distances described in successive equal 
 intervals of time by a body, which starts from rest and is 
 uniformly accelerated, are as the odd numbers. 
 
 f). Trace the motion of a body projected vertically up- 
 wards, and shew that the downward return motion is an 
 exact image of the upward. 
 
 :i 
 
 Exercise IV. 
 
 ^.B. — /n thefoUoivivg examples the directions of ve- 
 locity and acceleration are the same^ and in the case of 
 bodies moving vertically the resistance of the air is neglected, 
 unless otherwise stated. 
 
 1. A stone is observed to fall to the bottom of a pre- 
 cipice in 9 seconds; what is the depth ? Given ^ = 980. 
 
 2. The height of the piers of Brooklyn Bridge is 277 ft. ; 
 how long will a stone let fall from the top take to fall into 
 the water? Given ,^ = 32 J. 
 
 3. A body is projected vertically upwards with a velocity 
 of 320 ft. per sec. 1). How long will it rise ? 2). How 
 far will it rise ? 3). When and where will its velocity be 
 150 miles per hour ? 4). How long will it take to rise 1000 
 
 
 *■.: 
 
 
 m 
 
 r, ■ si. 
 
 
 
28 
 
 ft. ? 5). What will its velocity be at that height ? 6). How 
 tar will it travel in the seventh second ? Given ^ = 32. 
 
 4. A body starts with a velocity of 1 metre per second, 
 and has an acceleration of 10 tachs per second ; what will 
 its velocity be after traversing 6| metres 'i 
 
 5. How long would a body in Kingston, Ont., which is 
 projected with a downward velocity of450tach8, take to fall 
 through 15 kilometres, if there were no atmospheric resist- 
 ance ? 
 
 6. The velocity of sound in air is uniform and at 10 'C. 
 is equal to 33833 tachs. The depth of the well in the fort- 
 ress of K(")' .gstein in Saxony is 195 metres. In what time 
 should the splash of a stone dropped into the well be heard, 
 if there were no atmospheric resistance i 
 
 7. When a bucket of water is poured into this well, the 
 splash is heard in 15 seconds; what is the average accel- 
 eration produced in the water by the resistance of the air ? 
 
 S. A body whose acceleration is 10, traverses 6 metres 
 in 10 seconds; what is the initial velocity? 
 
 9. A body moves over 34*3 metres in the fourth second 
 of its motion from rest ; find the acceleration. 
 
 10. A person, starting with a velocity of 1 metre per 
 second, and accelerating his speed uniformly, traverses 960 
 metres in a minute ; find his acceleration. 
 
 11. A body starts from a given point with a uniform 
 velocity of 9 kilometres per hour ; in an hour afterwards 
 another body starts in pursuit of the first with a velocity 
 of 2 metres per second, and an acceleration of 5 decatachs 
 per hour ; when and where will the second body overtake 
 the first ? 
 
29 
 
 ■* ■ • 
 
 12. A body projected vertically upwards in Kingston, 
 Ont., passes a point 10 metres above the point of projec- 
 tion with a velocity of 9805 tachs ; how high will it still 
 rise, and what will belts velocity on returning to the point 
 of projection ? 
 
 13. A body uniformly accelerated describes 6*5 metres 
 and 4*5 metres in the fourth and sixth seconds of its mo- 
 tion ; find the initial velocity and acceleration. 
 
 14. Two bodies uniformly accelerated in passing over the 
 same space have their velocities increased from a to J, and 
 from do d respectively ; compare their accelerations. 
 
 15. Find the acceleration when in one-tenth of a second 
 a velocity is produced, which would carry a body over 10 
 metres everv tenth of a second. 
 
 ■ 
 
 16. A particle is projected vertically upwards, and the 
 time between its leaving a point 21 feet above the point of 
 projection and returning to it again is observed to be 10 
 seconds ; find the initial velocity. Given ^=32. 
 
 17. Two bodies are let fall from the same place in King- 
 ston, Ont. at an interval of two seconds ; find their dis- 
 tance from one another at the end of five seconds from the 
 instant at which the first was allowed to fall. 
 
 18. Two bodies let fall from heights of 40 metres and 
 169 decimetres reach the ground simultaneously ; find the 
 interval between their starting. Given ^ = 980. 
 
 19. Two bodies start from rest and from the same point 
 on the circumference of a circle ; the one body moves along 
 the circumference with uniform velocity, and the other, 
 starting at the same time, moves along a diameter with 
 uniform acceleration ; they meet at the other extremity of 
 the diameter ; compare their velocities at that point. 
 
 i^ 
 
 
 
 
 14, ■ ' .. . 
 
 •li ■\:: ■ 
 
 
 M: 
 
 11 
 
30 
 
 20. A body, starting from rest with an acceleration of 20 
 tachs per second, moves over 10 metres; find the whole 
 time of motion, and the distance passed over in the last 
 second. 
 
 21. A body moves over 9 ft. whilst its velocity increases 
 uniformly from 8 to 10 ft. per second ; how much farther 
 will the body move before it acquires a velocity of 12 ft. 
 per second ? 
 
 22. The path of a body uniformly accelerated is divided 
 into a number of equal spaces. Shew that, if the times of 
 describing these spaces be in A. /*. , the mean velocities for 
 each of the spaces are in II. P. 
 
 23. A body falling freely is observed to describe 24|^ 
 metres in a certain second ; how long previously to this has 
 it been falling ? 
 
 24. A body is dropped from a height of 80 metres ; at 
 the same instant another body is started from the ground 
 upwards so as to meet the former half way ; find the initial 
 velocity of the latter body, and the velocities of the two 
 bodies when they meet. 
 
 25. A body has a vniform acceleration a. 1{ p be the 
 mean velocity, and q the change of velocity, in passing over 
 any portion s of the path, shew that jy^=:a*. 
 
 26. A body uniformly accelerated is observed to move 
 over a and h feet respectively in two consecutive seconds; 
 find the acceleration. 
 
31 
 
 CHAPTER V. 
 
 Inertia. 
 
 Mat. 
 
 ss. 
 
 88. Tvertia is the inability of a body in itself to alter its 
 own condition of motion or rest. If a body be at rest, it re- 
 mains so ; if it be in motion, it goes on moving in the same 
 direction and with the same velocity, i.e. uniformly in a 
 straight line ; and if it be rotating, it goes on rotating 
 with the same angular velocity, about the same axis, 
 which maintains a constant direction ; unless some other 
 ho(Jy interfere with it. To change the state of rest or mo- 
 tion of a body requires the presence of another body. Force 
 is the term applied to the action of a body in altering the 
 status quo of another body. 
 
 39. Inertia, although a negative property, is perhaps the 
 most obtrusive property of matter. It is lucidly illustrated 
 in railway and horse-riding accidents, in vaulting and jump- 
 ing, in shaking the dust from off a book, in the danger of 
 turning round a corner in a carriage very rapidly, in the 
 difficulty of driving over smooth ice, and in the action of 
 a fly-wheel, which is used to regulate either an irregular 
 driving power, as in a foot-lathe, or an irregular resistance, 
 as in a circular saw cutting wood. The tendency of bodies, 
 moving in circles, to fly ofl^ at every instant aloijg the tan- 
 gent, (ommonly but misleadingly called centrifugal force ^ 
 is just a case of inertia. Herein we have an explanation 
 of the spheroidal form of the earth, and of the diminuticm 
 of a body's weight, as we approach the equator. On letting 
 a bullet fall from the top of a high tower or down a deep 
 mine, it will, on account of its inertia, be found to tall 
 somewhat to the east of the point vertically below that from 
 which it fell, thus affording an ocular demonstration of the 
 
 »V 
 
 m 
 
 1:^ 
 
 w ■ 
 
 
 If 
 
32 
 
 earth's rotation from west to eaff. The rotations of the 
 earth and other members of the soUir system afford beaii- 
 tifnl examples of the inertia of rigid bodies as regards ro- 
 tation. The constancy of direction of the earth's axis^ 
 (except in so far as it is interfered with by the sun and 
 moon), furnishes us with the most important step in the 
 explanation of the changes of the seasons. The gyroscope 
 is a beautiful physical toy which illustrates the same im- 
 portant principle. In Foucault's experiment for proving 
 the earth's rotation, the same principle is assumed for an 
 oscillating pendulum as regards its plane of oscillation. 
 
 40. Newton clearly enunciated the Inertia of Matter in 
 his First Law of Motion : 
 
 Eoery body continues in its state of rest, or of unijorin. 
 motioji in a straight line, (xcept in so far as it jnay be 
 compelU'd by impret^sed force to changt that state. 
 
 Observe, however, that he takes no notice of inertia as 
 regards rotation. Here indeed there is a difficulty, for 
 evidently the individual small particles of the rotating body 
 move in circles, and must therefore be acted on by forces 
 among themselves ; else, on account of their inertia, they 
 would move uniformly in straight lines. However, when 
 once by internal forces the relative positions of the particles 
 are fixed, the body will be as inert in its rotation, as in its 
 motion of translation. 
 
 41. When the same force acts on different bodies we find 
 that the changes from the previous states ©f rest or motion 
 are different, and we express this fact by saying that the 
 bodies difter in mafis. Mass, then, is a property in which 
 bodies differ, just as they differ in colour, volume, elasticity, 
 &Q. It may not inaptly be called the dynamical measure 
 of a body^s Inertia. 
 
33 
 
 How is Mass measured ? 
 
 When the same j'orce actmg durmg the same time on 
 two bodies produces in each the same changes of velocity^ 
 the masses are defined as equal to one another / hut^ if the 
 changes of velocity he not the same, the masses are defined 
 as inversely proportional to the changes of velocity pro- 
 duced. 
 
 42. The difference in mass of diiferent bodies {e.g. of balls 
 of wood, ivory, lead, &c., of different radii) may be lucidly 
 illustrated by suspending the bodies by strings, and allow- 
 ing the same spring, bent through the same angle, to act 
 upon them in succession so as to give the bodies a hori- 
 zontal motion. It will be found that the velocities imparted 
 will be very different. 
 
 43. A unit volume of pure water at 4® C. (and under the 
 mean atmospheric pressure) is defined as having u?iit of 
 mass, which is called a gram. In the scientific system of 
 the units of measurement the gram is the third and last of 
 the fundamental units. The English standard unit of mass 
 is the pound avoirdupois, which contains 7000 grains. The 
 French standard unit of mass is the kilogramme, which is 
 the mass of a litre of pure water at 4° C. What have hith- 
 erto been called the scientific units are better called the 
 C. G. S. units, these three letters being the initial letters of 
 the fundamental units adopted by scientific writers. These 
 units, it is agreed, will be used in all international scientific 
 •juestions. With other fundamental units equally scientific 
 systems can be formed. Thus with the Ya\\^\\^\\ foot, pound, 
 and second an English scientific system of units is formed, 
 which we shall call the F. P. S. system. It will be useful 
 to the student to familiarize himself with both of these 
 
 |!v;. 
 
 A* * 
 
 V'. 
 
 I'.' ■ 
 
 '.Tl 
 
 W^''A 
 
 
 
 p*- . , 
 
84 
 
 systems of units in the solution of problems. Whe7iever 
 apecial units are not mentioned, the units of the C G. S' 
 system are to he understood. 
 
 44. Let it be observed that by means of one force we can 
 theoretically determine the masses of all bodies. 
 
 When the same force acts upon bodies of the sarne kind, 
 e.g. two pieces of iron at the same temperature, it is found 
 that the accelerations produced are inversely proportional 
 to their volumes, (take as an illustration the opening of 
 doors of the same kind of wood but of different sizes); but, 
 when the same force acts upon bodies of different material, 
 e.g. a piece of iron and a piece of wood, it is found that the 
 accelerations produced are not inversely proportional to 
 their volumes, (take as an illustration the opening of a 
 wooden and of an iron door). Hence it follows that the 
 masses of bodies of the same material (and at the same 
 temperature and pressure) are directly proportional to their 
 volumes / but not so for bodies of different material. 
 
 45. These facts lead us to the consideration of specific 
 mass, or, as it is more commonly called, density. The den- 
 sity of a body is the mass per unit oj" volume. Hence the 
 density of water at4® C. wiU be represented by 1. A unit 
 volume of gold at ® C. and under the mean atmospheric 
 pressure is found to be 19 '3 grams, of rock- crystal 2*65, of 
 cork 0-24, of mercury 13-596, of dry air 0-001293, &c. We 
 express these facts by saying that the density of gold is 19-3, 
 of rock-crystal 2-65, of cork 0-24, of mercury 13-596, of dry 
 air 0-001293, &c. 
 
 46. The densitv of water, as of all other substances, 
 varies with temperature, and under the mean atmospheric 
 pressure is a maximum at 4 ° C. Hence it is that, in 
 
35 
 
 ices, 
 leric 
 in 
 
 defining unit of mass, tlie water is taken at this temper- 
 ature. So little is the density of water chaui^ed by pressure 
 that it is hardly necessary to state tliat the water, in de- 
 fining unit of mass, is supposed to be under the mean at- 
 mospheric pressure, the chanjres of atmospheric pressure 
 making only immeasurably small changes of density. 
 
 47. The density of a body may be unifurin, i.e. every 
 part having the same density, or it may be variable. In 
 the latter case the density at any point of the body is the 
 same as that of any other body, whose density throughout 
 is uniform and the same as at the point in question. 
 
 48. We might have defined the density of a body as the 
 ratio of its mass to the mass of an eqval volume of pure 
 vmter at 4:'^ C. 
 
 In the case of a body of variable density, this definition 
 would give us the mean density of the body. It would fur- 
 ther be applicable whatever units of volume and mass be 
 taken. 
 
 49. Relation between the 7n(/ss, voUtme, and density of 
 a body, M = VD^ where M is the mass of the body in 
 grams, Y its volume in cubic centimetres, and D its den- 
 sity. 
 
 When masses are expressed in pounds avoirdupois and 
 volumes in cubic feet, then J/ = 62*4 VD gives the relation 
 between mass, volume, and density, since a cubic foot of 
 water is 624 pounds, 
 
 50. The mass of a body is sometimes defined as the 
 measure of the quantity of matter in it, or as the dynam- 
 ical measure of the quantity of matter in it. Since we 
 do not know the ultimate nature of matter this can hardly 
 be scientifically correct. We only know the properties of 
 
 y^ 
 
 V- 
 
 i 
 
36 
 
 matter, and can only measure its properties. Why then 
 should quantity of matter be measured V)y one of these 
 properties, mass, rather than by any other ? We might 
 reason thus : when the same quantity' of heat is applied to 
 bodies of the same kind, the changes of temperature pro- 
 duced are mversely proportional to their volumes / but 
 when applied to bodies of different kinds, the changes of 
 temperature are n^Hnversely proportional to their volumes. 
 We express this fact by saying that bodies differ in thermal 
 capacity y and we <'e'' .v the thermal capacities as inversely 
 proportional to ti; , -it". - :^s of temperature produced. Just 
 then as with mass Wa ni.^i.t deline the thermal capacity of 
 a body as the therm 'il measure of the quantity of matter 
 in it. We should th^n ii'id l]:^''. the thermal measure and 
 measurement by mass wore quite oifferent. 
 
 So long as we are dealing with matter of one kind there 
 are many ways in which we may measure quantity of 
 matter quite intelligibly, eg. by volume, by weighing in 
 the same place, in the case of food by the length of time it 
 will supply nourishment, in the case of fuel by the amount 
 of heat it will give out, or by the amount of oxygen gas 
 necessary for its complete combustion, tfec, and all these 
 measurements would be found to agree with one another 
 as well as with the measurement by mass. But when we 
 come to deal with bodies of different kinds, none of these 
 measurements will be found to give results consistent with 
 one another. 
 
 51. The reason, doubtless, why 7nass is stated to measure 
 the quantity of matter in a body, is that this is the only 
 property of a body which remains invariable through what- 
 soever changes the body may pass. Thus, whilst by pressure, 
 motion, heat, chemical action, or other agencies, we can 
 
37 
 
 alter the other measurable properties of a body, such as 
 its volume, elasticity, weight, thermal capacity, &c., its 
 mass, through whatever changes the body may pass, re- 
 mains unchanged. This may be clearly illustrated by many 
 experiments, e.g. by dissolving a piece of loaf sugar in tea, 
 by freezing a body of water, by mixing alcohol and water, 
 and, generally, in all chemical combinations. Hence the 
 great law which forms the foundation of chemical science, 
 the Conservation of Mas,^, which asserts that, through 
 whatever changes matter may pass, the total mass of the 
 universe remains unchanged. 
 
 Examination V. 
 
 1. Define inertia, and state the different forms thereof. 
 
 2. Give various illustrative examples of inertia. 
 
 3. What is centrifugal force ? Suggest a better name 
 for it. 
 
 4. How may the earth's rotation from west to east be 
 visibly proved ? 
 
 5. Enunciate Newton's First Law of Motion. 
 
 6. Define mass. How is it measured ? 
 
 7. Describe a simple experiment to shew difference of 
 mass in different bodies. 
 
 8. Name and define the units of mass in the C. G. S. and 
 F. P. S. svstems of measurement. 
 
 9. What relation exists between the volumes and masses 
 of bodies of the same material ? How is this proved? 
 
 10. Define density in two ways. Give the densities of 
 a few commonly found bodies. 
 
 1^ 
 
 S! 
 
 i . 
 
 I 
 
as 
 
 11. Why is water at 4® taken as the standard substance 
 in measuring mass and density? 
 
 12. Give the algebraical equation connecting the mass, 
 volume, and density of a body. In the case of a body of 
 variable density how do you express the relation ? 
 
 13. In the F. P. S. system of units what is the relation 
 between mass, volume, and density ? 
 
 14. Criticise the usual definition of mass as the measure 
 of the quantity of matter in a body. 
 
 15. How did the above definition probably arise ? 
 
 16. Enunciate the principle of the Conservation of Mass. 
 
 Exercise Y. 
 
 1. A rectangular block of limestone is 2 metres long, 
 1*5 metre broad, and 1 metre thick. If 2*7 be its density, 
 find its mass. 
 
 2. The sides of a canal shelve regularly from top to bot- 
 tom. The width of a section at the top is 10 metres, at the 
 bottom 5 metres, and the depth is 3 metres. If the canal 
 be filled with water to a depth of 2*5 metres, find the mass 
 of water per mile of length. 
 
 3. If the density of common salt is 2*3 and of sea- water 
 1*026; find the mass and volume of salt obtained in evap- 
 orating 100 litres of sea-water. 
 
 4. The density of copper is 8*8, of zinc 6*9, and of 
 brass formed from these 8*4 ; find the quantity of copper 
 in 100 grams of brass. 
 
 5. The mass of a sphere of rock-crystal is 400*5, and its 
 radius 3*3 ; find its density. 
 
t.y» 
 
 6. Find the mass of the earth, supposing it to be a sphere 
 of radius 6371 kilometres, and of mean density 5*67. 
 
 7. Equal masses of copper and tin, whose densities are 
 89 and 7*3, are melted together ; what would be the den- 
 sity of the alloy if no contraction or expansion took place ? 
 
 8. When 63 litres of sulphuric acid, whose s.g. is 1.85, 
 is mixed with 24 litres of water, the volume of the mixture 
 is 86 litres ; find the mass and density of the mixture. 
 
 9. The density of gold is 19-3, and of quartz 2*65 ; the 
 mass of a nugget of gold-quartz is 350, and its density is 
 7'4: ; find the mass of gold in it. 
 
 10. The density of sea-water is 1*026, and of salt 2*3 ; 
 100 litres of sea-water is frozen, and 200 grams of ice free 
 from salt formed therefrom ; what is the density of the 
 residue ? 
 
 11. What is the density of mercury, if 9 cubic inches 
 have a mass of 442 lbs ? 
 
 12. If 4 lbs. of silver have the same volume as 3 lbs. of 
 brass, compare the densities of silver and brass. 
 
 13. If 3 cubic inches of silver have the same mass as 4 
 cubic inches of brass, what mass of silver will have the 
 same volume as 10 lbs. of brass? 
 
 14. Two bodies whose volumes are as 3 : 4 are in ma?s 
 as 2 : 3 ; compare their densities. 
 
 
 '!'':rt| 
 
 
 
 
 
 
40 
 
 Chapter VI. 
 
 Momentum. 
 
 Force, 
 
 52. The momentum of a moving body is measured by 
 the product of the numbers whicli represent its mass and 
 velocity. 
 
 The unit of momentum is that of unit mass moving with 
 unit of velocity, and therefore in the C. G. S. system of 
 units will be the momentum of a body whose mass is 1 gram 
 moving witli a velocity of 1 tach. It thus involves each of 
 the three fundamental units once. It is obvious that with 
 the above unit the simple formula Mo = Mv expresses the 
 relation between the momentum, mass, and velocity of 
 any body. 
 
 To experience that property of a body called its momen- 
 tum.^ let a person bathe close to a waterfall, say 200 ft. 
 high, when he will feel the drops of watei\ which separate 
 from the main mass, strike his body as if they were sharp 
 stones. If he attempted to enter the main mass of falling 
 water he would be roughly thrown on the ground. 
 
 53. The rate of change of momentum per unit of time 
 will evidently be measured by the product of the numbers 
 which represent the mass and acceleration of the moving 
 body. It is called the acceleration of momentum of the 
 moving body. 
 
 54. Force is that aspect of any external influence exerted 
 on a body which is manifested by change of m.omentum. 
 Whei»ever the momentum of a body changes, a force is 
 said to act on the body. The magnitude of the force is 
 measured by the rate of change of momentum per unit of 
 
It, is well in defining force to avoid the word 
 All that we are aware of is a change of momentum 
 
 41 
 
 time, i.e. hy the acceleration of momentum^ and its direction 
 is the direction of the change of momentum. 
 
 55. This is what Newton taught in his Second Law of 
 Motion : 
 
 Change of momentum is proportional to the impressed 
 force and takes place in the direction of the straight line 
 in which the force acts. 
 
 By impressed force Newton meant external to the body 
 concerned 
 cause. 
 
 and the word force is conveniently used as a measure of 
 the rate of this change. Under energy^ one of the most 
 important properties of matter, the student will learn that 
 force may be defined as the rate of expenditure of energy 
 per unit of length. 
 
 56. The unit of force is that force which produces unit 
 acceleration of momentum, /. «., in C. G. S. measure, the 
 force which, acting upon a body whose mass is 1 gram, 
 produces in 1 second a velocity of 1 tach. Such a force is 
 called a dyne, and in terms of this unit we have the simple 
 relation F=Ma, where Jt^\9, a force in dynes, Jf the mass 
 in grams of the body on which the force acts, and a the ac- 
 celeration produced in tachs per second. 
 
 57. When the change of momentum produced by a force 
 takes place in an immeasurably short time {e. g. when a 
 cricket ball is struck by a bat), it is practically impossible 
 to measure the force in dynes. Such a force is called an 
 impulsive force, and is measured by the whole change (f 
 momentum produced, without any account being taken of 
 time. 
 
 58. Since force has direction as well as magnitude, it is, 
 
 ■W'--'^n 
 
 A '' 
 
42 
 
 like velocity, acceleration, or monientum, completely re- 
 presented by a straight line; the direction of the line being 
 the direction of the force, and the length of the line repre- 
 senting the magnitude of the force. 
 
 59. What does the second law of motion really teach us ? 
 1). It defines the measurement of mass and force. Just as 
 it is theoretically possible to measure all masses hv means 
 of one force, so is it possible to measure the magnitudes of 
 all forces theoretically by one mass. When different forces 
 act upon the same body, the magnitudes of the forces are 
 by definition directly proportional to the accelerations pro- 
 duced. It is indeed evident that the mass of any body is 
 measured in grams by the reciprocal of the acceleration in 
 tachs per second produced, when a dyne acts upon it ; and 
 any force is measured in dynes by the acceleration in tachs 
 per second produced, when it acts upon a body whose mass 
 is a gram. 2). It enunciates the important experimental 
 fact, that with whatever force different masses be measured, 
 and with whatever mass different forces be measured, the 
 measurements will always be alike. 3). It asserts that the 
 effect of a force depends in no way upon the motion of the 
 body, and that, when more than one force is acting on the 
 body, each force produces its effect quite independently of 
 the others. 
 
 60. Different names are given to different aspects of force, 
 such as pressure^ tension^ attraction^ weighty repulsion^ re- 
 sistance^ friction., &c. 
 
 Pressvre is applied to a force which calls up the idea of 
 pushing^ i.e. a force acting between two bodies, already 
 close together, in consequence of which they tend to ap- 
 proach still nearer to one another. 
 
 
43 
 
 
 4 
 
 Tension calls up the idea o\ pulling. It is a force acting 
 between two bodies close together, in consequence of which 
 they tend to move away from one anotlier. Hence we 
 speak of the tension of a stretched rope. Both pressure 
 and tension are applied to the elastic force of a gas ; press- 
 ure, when attention is drawn to the gas pushing against 
 the sides of the containing vessel ; and tension, when we 
 think of the particles of gas tending to separate from one 
 another, so as to occupy, if possible, a greater space. 
 
 Attraction is a term applied to forces exerted betweeii 
 bodies, when there i*^ no sensible material medium through 
 which the force is exerted, and in consequence of which 
 the bodies approach one another. The force ])etween two 
 unlike magnetic poles is a familiar case of attraction. 
 
 Weighty a well known form of attraction, is applied to 
 the force exerted between the eartli and any body on its 
 surface. Forces are often conveniently measured by the 
 weights of bodies of standard mass. Thus, when a force 
 of/) grams is spoken of, a force equal to the weight of a 
 body whose mass is p grams is meant. It would be better 
 to speak of a force of/? grams-weight. 
 
 Repulsion is a term generally applied to forces between 
 bodies, when there is no sensible material medium through 
 which the force is exerted, and in consequence of which 
 the bodies recede from one another. The force between 
 two like magnetic poles is a fatniiiar example of a force of 
 repulsion. 
 
 Resistance is a term frequently applied to any force op- 
 posing i\\Q motion of a body, t*. <?. producing a negative 
 acceleration. One of the most familiar and important of 
 such resistances is the ubiquitous force oi friction^ a term 
 applied to that force which is called into play, when one 
 
 
 li-.':;> 
 
 ^:ji-^l 
 
44 
 
 body moves or tends to move over the surface of another 
 body. It ia principally the force of friction which a locomo- 
 tive is constantly working against in pulling a train along. 
 The resistance which bodies experience in falling through 
 the air is largely the force of friction between the bodies 
 and the aerial particles they rub against. 
 
 61. By many writers the science of force is called Me- 
 chiiiics^ and is divided into the two branches Statics and 
 Dynamics. There has, however, been a much better nom- 
 enclature adopted lately by the best modern writers on 
 Natural Philosophy. By them the science of force is called 
 Di/7iamics, and they divide it into Statics and Kinetics. 
 
 Statics treixts of equilihrium or the balancing of forces. 
 It is chiefly concerned in determining the relations which 
 must exist amongst a set of forces which keep a body at 
 rest. 
 
 Kinetics tvQiits of change of momentum. The investiga- 
 tion of the motiotis of the Solar System is the grandest 
 problem in Kinetics, and is commonly known as Physical 
 Astronomy. 
 
 Kinematics is the science of motion^ when studied with- 
 out any reference to mass. It forms an appropriate intro- 
 duction to Kinetics. Chapters II., III., IV. belong to 
 Kinematics. 
 
 Mechanics is the science which treats of the construction 
 and uses of machines. 
 
 Examination VI. 
 
 1. How is the momentum of a moving body measured '( 
 
 2. Define the unit of momentum, and give the relation 
 between the momentum, mass, and velocity of a body. 
 
 or 
 m 
 
to 
 
 45 
 
 3. Define and give the measure of acceleration of mo- 
 mentum. 
 
 4. Define force and its measure in magnitude and di- 
 rection. 
 
 5. Enunciate Newton's Second Law of Motion. 
 
 6. Name and define the unit of force in the C. G. S. sys- 
 tem, and give the relation between force, mass, and accel- 
 eration. 
 
 7. What is an impulsive force? How is it measured? 
 
 8. How may a force be completely represented ? 
 
 9. State in full what the Second Law of Motion teaches 
 us. 
 
 10. Define the terms pressure, tension, attraction, weight, 
 repulsion, resistance, friction. 
 
 11. What is meant by a force of 10 pounds, or a force of 
 10 kilograms? Give better expressions for these. 
 
 12. Define the terms Dynamics, Statics, Kinetics, Kin- 
 ematics, Mechanics. 
 
 13. How can the property of matter called momentum be 
 illustrated ? 
 
 
 Exercise VL 
 
 1. Akilodyne acts upon a body whose mass is 50 grams; 
 find the velocity and distance passed over at the end of 10 
 seconds. 
 
 2. A body, whose mass is 5, has an acceleration 2. At 
 one instant the velocity is 10; what is the momentum a 
 minute afterwards ? 
 
 V ' 
 
46 
 
 3. Find the acceleration produced by a megadyne act- 
 ing on a solid sphere whose diameter is a metre and 
 density 10. 
 
 4. A force of 50 kilodynes acts upon a body which ac- 
 quires in 10 seconds a velocity of a kilotach ; find the mass 
 of the body. 
 
 5. The mean radius of the earth is 20902070 feet, its 
 mean density 5*67, its mean distance from the sun 92| 
 million miles, and the time of its revolution around the 
 sun SG6^ days ; compare its momentum with that of a train 
 of 1000 tons mass, rushing along at 60 miles an hour. 
 
 6. The distance of Jupiter from the sun is 5'2 times that 
 of the earth, its period 4332^ d^-js, its mass 310 times that 
 of the earth ; compare the momenta of Jupiter and the 
 earth. 
 
 7. A body of 10 grams mass has a uniform acceleration 
 of 1 metre per minute per minute ; what force is acting 
 upon it? 
 
 8. A body acted on by a uniform force is found to be 
 moving, at the end of the first minute from rest, with a 
 velocity which would carry it through 20 kilometres in the 
 next hour ; compare the force with the weight of the body, 
 which would give it an acceleration g - 980*5. 
 
 9. "What is the change of momentum in a minute of a 
 body whose mass is 10 and acceleration 10 ? 
 
 10. Compare the momentum of a man, whose mass is 
 140 lbs. and latitude that of Kingston, Ont. (44® 13'), 
 arising from the earth's rotation, with that of a steamer of 
 10000 tons mass ppJling at the rate of 15 miles an hour. 
 Radius of the earth = 20902070 ft. 
 
f y 
 
 47 
 
 be 
 
 11. A body, acted on by a force of 100 kilodynes, has its 
 velocity increased from 6 to 8 kilometres per hour in pass- 
 ing over 84 metres ; find the mass of the body. 
 
 12. A body of 1 kilogram mass is acted on by a uniform 
 force in the direction of its motion, and is found to pass 
 over 9*05 and 8'05 metres in the 10th and 20th seconds 
 of its motion ; find the force acting on it, and its initial 
 velocity. 
 
 13. Two bodies, acted on by equal forces, describe the 
 same distance from rest, the one in half the time the other 
 does ; compare their final velocities and momenta. 
 
 14. Two bodies of equal mass, uniformly accelerated 
 from rest, describe the same distance;, the one in half the 
 time the other does; compare the forces acting on the 
 bodies. 
 
 15. Two balls, one of silver and the other of ivory, whose 
 diameters are as 1 to 2, are acted upon by the same impul- 
 sive force ; the velocities produced are as 11 to 15; com- 
 pare the densities of silver and ivory. 
 
 16. If a ship be sailing with a uniform velocity, what 
 relation must exist between the driving power and the re- 
 sistances of the air and water ? 
 
 17. The density of lead is 11*4 and of cork 0-24. Two 
 balls of these substances, whose diameters are as 1 to 10 
 are acted upon the same force during the same time; com- 
 pare their momenta and velocities. 
 
 !r of 
 our. 
 
48 
 
 Chapter VII. 
 Weight, 
 
 62. Weight is the force which acts between the earth and 
 every body on its surface, in virtue of which any body, unless 
 it be supported, falls to the ground. It is also called the 
 force of gravity. All bodies at the sanie place are found 
 to fall invariably in the same direction relatively to the 
 surface of the earth, and bodies falling in contiguous places 
 fall in parallel straight lines. The direction in which a 
 body falls at any place is called the vertical direction at 
 that place, and is easily found by means of a plumb-line. 
 Any direction at right angles to the vertical is called hor- 
 izontal. The surface of any liquid, at rest relatively to the 
 earth, is a horizontal plane, except at the edges of the vessel 
 containing it. 
 
 63. When bodies fall in vacuo under the action of their 
 weights, the accelerations are found to be the same for all 
 at the same place. Hence we deduce the very important 
 fact : The weights of bodies at the same place are directly 
 proportional to their masses. 
 
 Let PT, w be the weights in dynes of two bodies at the 
 same place ; J!/, m the masses in grams ; g the acceleration 
 in tachs per second of each body, when falling under the 
 action of its weight, then 
 
 W=Mg^ w — mg^ .'. W : w : : M : m. 
 
 64. The following extract from Lucretius, translated in 
 Young's Lectures on Natural Philosophy, shews that the 
 fact that all bodies would fall equally fast in vacuo at the 
 same place, was believed in, if not proved, nearly 2000 
 years ago : 
 
40 
 
 In water or in air when weights descend, 
 
 The heavier weiglits more swiftly downwards tend ; 
 
 The limpid waves, the gales that gently play, 
 
 Yield to the weightier mass a readier way ; 
 
 Bnt if the weights in empty space should fall, 
 
 One common swiftness we should find in all. 
 
 65. Weight is measured like any other force in dynes. 
 Thus the weight of a hody whose mass is 1 gram is g dynes 
 (in the latitude of Kingston, Ont., 980'.5 dynes). Forces are 
 often conveniently measured in terms of the weights of 
 bodies of known mass. Thus we read of a force of a 
 kilogram weight or a force of 10 pounds-weight, and these 
 expressions are generally abbreviated into a force of a kilo- 
 gram or a force of 10 pounds. The measure of a force in 
 terms of weight is called its gravitation measure, that in 
 dynes being called in contradistinction its absolute measure. 
 Since g the acceleration due to the force of gravity varies 
 with latitude, it is evident that the gravitation measure of 
 a force has not a definite value, unless the place, at which 
 the body of definite mass is weighed, be given. The dyne 
 on the other hand is independent of time and place ^ and is 
 hence called an absolute unit. 
 
 ^Q. In the F. P. S. system the unit of force is that force 
 which, acting on a body whose mass is 1 pound for a second, 
 generates a velocity of a foot per second, and is known as 
 fipoundal. Evidently a pound-weight is eqnal to g pound- 
 als, g being now measured in units of a foot and second (32| 
 very nearly in the latitude of Kingston, Ont.) 
 
 67. Let us now measure in absolute units the mean 
 pressure of the atmosphere. This, like a. y other fluid 
 pressure, is measured in terms of the force applied per unit 
 
50 
 
 of area. In absolute measure the unit of fluid pressure^ or 
 o^ pressure- inteiitiity generally, is a pressure of unit of force 
 per unit of area, and will therefore be in C. G. S. units a 
 dyne per square centimetre. In the absence of a better 
 name this unit may be called a prem. The mean pressure 
 of the atmosphere (as indicated by the barometer) is the 
 same as what would be produced by the weight of a ver- 
 tical column of mercury 76 cm. long at * in the latitude 
 of Paris. The pressure per sq. cm,, is therefore equal to 
 the weight of 76 cub. cm. of mercury, i.e, (since the den- 
 sity of mercury at 0°= 13-596) 76x13-596 or 1033-3 
 grams-wt. per sq. cm., i.e. (since g at Paris = 980-94) 
 76 X 13-596 X 980-94 or 1-0136 X lO^ prems. 
 
 68. The simple relation between the weights of bodies 
 at the same place and their masses gives the best practical 
 method of measuring the masses of bodies, as is done in a 
 common balance. Observe that in a common balance^ bv 
 comparing the weights of bodies with those of standard 
 masses, we really measure 7nass / whereas in a spring bal- 
 ance we directly measure weight. 
 
 The law which explains to us the measurement of weight 
 by means of a spring balance is known as Ilooke's law : 
 The extension., compression^ or distortion of a solid body., 
 within the limits of elasticity., is directly proportional to 
 the force which produces it. 
 
 69. The direct proportionality between the masses of 
 bodies and their weights at the same place is the probable 
 cause of mass and weight being constantly confounded. 
 The following illustrations in which these two properties of 
 matter are contrasted, will assist the student to apprehend 
 their difference : 
 
51 
 
 1. a). The rmiss of a body i? tlie same at wliatever part 
 of the earth's surface it be. 
 
 b). The weight changes with change of place, and is dif- 
 ferent at the Equator, at either Pole, or at the summit of 
 the Rocky Mountains, from what it is in the class-roonj. 
 
 2. a). The opening of a room door is essentially a ques- 
 tion oi mass y and, however heavy the door may be, if the 
 hinges are truly vertical and well oiled, a small child may 
 open it, though slowly. 
 
 h). If the same door formed the lid of a box, and swung 
 on horizontal hinges well oiled, the child could not open it 
 unless he had strength enough to exert muscular force equal 
 to at least half the weight of the door. 
 
 In either case the child lias to overcome the force of 
 friction, wdiich, though greater in the first than in the sec- 
 ond case, is in either case small. 
 
 3. a). In moving a cart along a level road the horse has 
 to exert a greater force at starting than afterwards, because 
 he has to exert force to give the mass a given velocity, i.e. 
 to produce momentuin. After having started he has only 
 to balance the force of friction. 
 
 V). When, however, he comes to a hill he has again to 
 put forth his strength, for now he has, in addition to the 
 force of friction, to overcome part of the vmghf of the cart. 
 
 4. a). The action of a fly-wheel depends entirely upon 
 its mass. 
 
 h. The action of a larsje steel hammer, worked by steam, 
 depends essentially upon its weight. 
 
 5. a). In athletic sports the " long jump " is essentially a 
 question of mass. 
 
52 
 
 h). In the " high jump " roeight in addition has to be 
 considered. Hence the actual distance of the high jump is 
 never so great as that of the long jutnp. 
 
 6. a). In an undershot water wheel the miller depends 
 upon the momentum (and hence also the wass) of the run- 
 ning water to drive the wheel. 
 
 h). In an overshot water wheel he depends upon the 
 imight of the water which enters the buckets of the wheel. 
 
 70. How is g the acceleration due to the force of gravity 
 at any place measured ? The most accurate method of 
 finding this important physical constant is by means of 
 ])endulum experiments. There is, however, one method of 
 finding a very accurate value of ^, which at this stage the 
 student can understand. This is by means of the well- 
 known physical instrument called Attwood's machine. The 
 essential part of the apparatus is a grooved wheel which 
 turns upon an axle, each end of which rests on two other 
 wheels called i\\Q friction wheels^ so that the friction on the 
 axle of the first wheel is reduced to a minimum ; over this 
 wheel passes a fine thread connecting two bodies of difiier- 
 ent weights. If m, and m! be their masses, and m be the 
 greater, the bodies will move on account of the greater 
 weight of m with an acceleration equal to (m - m) g 
 -r- {m -f m') if we neglect friction and the motion of the 
 wheels. This acceleration can evidently be made as small 
 as the experimenter pleases, by making the difference be- 
 tween 7ti and m' small enough. By a clock and suitable 
 adjuncts the acceleration of the moving bodies can be very 
 accurately measured, and therefore g determined. 
 
 The following values of ^ at the sea-level have been deter- 
 mined by experiment and calculation : 
 
53 
 
 Latitude. Value of g. 
 
 Equator 0® 0' 978*10 
 
 Washington 38^54:' 980-08 
 
 Kingston, Ont 44^13' 980-54 
 
 Paris 48°50' 980-94 
 
 Greenwich 51 ° 29' 98 1 -17 
 
 Berlin 52^30' 981-25 
 
 Edinburgh 55 = 57' 981*54 
 
 Pole 90= 0' 98311 
 
 We thus see that the maximum variation is about ^ p.c. 
 of the mean value. 
 
 deter- 
 
 EXAMINATION VII. 
 
 1. Define weight. Bv what other name is it known ? 
 
 2. How is the direction of weight practically found ? 
 
 3. Define the terms vertical and horizontal, and give an 
 illustration of each. 
 
 4. Prove that the weights of bodies at the same place are 
 directly proportional to their masses. 
 
 5. Explain what is meant by an absolute unit of force, 
 and express, in absolute units, forces of a pound-weight and 
 of a kilogram-weight. 
 
 6. Name and define the absolute unit of pressure-inten- 
 sity ; express, both in gravitation and absolute measure, 
 the mean pressure of the atmosphere. 
 
 7. How is mass practically measured ? 
 
 8. Give illustrations of weight and mass, which contrast 
 with one another, to shew the difference between these two 
 properties of matter. 
 
 ?• '- 
 
54 
 
 9. How is the value of g experimentally determined ? 
 
 10. Describe the essential parts of Attwood's machine. 
 
 11. Give the values of ^ at the Equator, Kingston, Ont.. 
 Paris, and the North Pole, true to a decitach per second. 
 
 12. Enunciate Hooke's Law, and apply it to the spring 
 balance. 
 
 13. If a merchant buys goods in London by means of a 
 spring balance, and with the same balance sells in King- 
 ston, Ont., will he gain or loose in the transaction ? Why ''i 
 
 14. Shew that a poundal is nearly equal to the weight of 
 a body whose mass is half an ounce. 
 
 Exercise VII. 
 
 ^.B. — Take g in the following examples equal to 980'5 
 or 32J. In examples on Attwoocfs machine friction and 
 the motion of the wheels are to he neglected. 
 
 1. A body whose mass is 10 grams is falling in vacuo ; 
 what is the force acting on it, and its momentum at the 
 end of 10 seconds ? 
 
 2. A force of 60 grams-weight acts upon a body which 
 acquires in 10 seconds a velocity of 39*22 tachs ; find the 
 mass of the body. 
 
 3. A force produces in a sphere of radius 10 and density 
 10 an acceleration 100 ; find what weight the force could 
 balance. 
 
 4. A body of 3 kilograms mass pulls by its weight an- 
 other body of 2 kilograms mass along a smooth horizontal 
 plane ; find the momentum at the end of 5 seconds, and 
 the distance passed over. 
 
 5. In Attwood's machine, if 10 kilograms be the mass of 
 
 
 :'!■„,! I 
 
55 
 
 one body, and 15 kilograms that ot'tlie other; find the ac- 
 celeration of momentum, and the velocity at the end of two 
 seconds. 
 
 6. A force of 10 pounds-weight acts upon a mass of 2 
 pounds ; what is the velocity after traversing a kilometre ? 
 
 7. A mass of 10 pounds is acted on by a uniform force, 
 and in 4 seconds passes over 200 feet ; express in gravita- 
 tion measure the force acting. 
 
 8. A body of 6 lbs. mass pulls by its weight another body 
 of 4 lbs. mass along a smooth horizontal table; find the 
 time taken to move through 965 feet, and the distance 
 described in the last second. 
 
 9. In Attwood's machine one mass is known to be 10 
 lbs., and the distance described in 2 sec. is found to be ir» 
 ft. 1 in. ; find the other mass. 
 
 10. A mass 5 has an acceleration 2 ; at one instant the 
 velocity is 10 ; what is the momentum a minute after- 
 wards ? Express in gravitation measure the force acting 
 on the body. 
 
 11. Find the unit of time, if a centimetre be the unit of 
 length, a gram the unit of mass, and a gram-weight the 
 unit of force. 
 
 12. Find the unit of length, if a second be the unit of time, 
 a gram the unit of mass, and a gram-weight the unit of force. 
 
 13. Find the unit of mass, if a second be the unit of 
 time, a centimetre the unit of length, and a gram-weight 
 the unit of force. 
 
 14. A sphere of rock-crystal of density 2*68 has a diameter 
 6*5; find its volume, mass, and weight in Kingston, Ont. 
 
 15. The density of sea-water is 1*028 ; find the pressure 
 in the ocean at the depth of a kilometre. 
 
56 
 
 16. Find in poniidals the tensions of the three parts of a 
 ^5tring, which supports at different heights bodies of 12, 6, 
 and 4 lbs. mass respectively. 
 
 17. Oxygen combines chemically with hydrogen to form 
 steam in the proportion of 8 parts by mass of oxygen to 1 
 of hydrogen. If the gases be weighed by means of a spring 
 balance graduated at Edinburgh, how many milligrams- 
 weiglit of oxygen at the Equator will combine with 100 
 juilligrams-woight of hydrogen at Edinburgh to form steam ? 
 
 18. Determine the mass of steam so formed, and its 
 weight, as indicated on the above spring balance, at King- 
 ston, Ont. 
 
 19. Answer the above (18 and 19) when tlie gases are 
 weighed in a common balance, and explain your answers. 
 
 '20. If a mass of a kilogram be placed on a horizontal 
 plane, which is made to descend vertically with an accel- 
 eration of 100 ; find in gravitation measure the pressure on 
 the plane. 
 
 21. If a mass of 10 lbs. be placed on a horizontal plane, 
 which is made to ascend vertically with an acceleration of 
 20 feet per second per second ; find in poundals the pres- 
 sure on the plane. 
 
 22. If the velocity of each of the bodies in Attwood's 
 machine be 20 feet per second, when they are at the same 
 height above the ground, and if at that instant the string 
 be cut, find how far apart the bodies will be in 5 secont^ 
 
 23. Two bodies of 4 and 5 kilograms together pull ol 
 of 6 kilograms over a smooth peg by means of a connecting 
 string ; after descending through 10 metres, the 5 kilograms 
 mass is detached without interrupting the motion ; find 
 through what distance the remaining 4 kilograms will 
 descend. 
 
 I 
 
 tt 
 
) 
 
 will 
 
 57 
 
 Chapter VIII. 
 
 Archimedes' Principle. 
 
 71. Since the weights of bodies at the same place are 
 directly proportional to their masses, and since different 
 bodies differ in their specific masses or densities, therefore 
 they will also have different specific weights^ or, as they are 
 more commonly called, specific gravities. 
 
 The specific griwity of a body is the ratio of its weight 
 to the weight of an equal volume of pure water at 4®C 
 (its maximum density point) at the same place. 
 
 The specific gravity of water at 4 ° C as well as its den- 
 sity will thus be represented by unity, and it is evident that 
 the numbers which represent the density and specific grav- 
 ity of the same body are the same. Let J/, W, />, 6", re- 
 present the mass, weight, density, and specific gravity of a 
 body, and tw, w^ the mass and weight of an equal volume of 
 water at 4 ® C, then M \ m = />, and W \ w = S^ by de- 
 finition, but (art. 63) J/ : m : : W : w^ .'. D=S. 
 
 In the case of a body whose specific gravity is not uni- 
 form throughout, the above definition gives the mean spe- 
 cific qravity of the body. 
 
 2. The most convenient methods of determining the 
 S| ific gravities of liquid and solid bodies depends upon 
 the Princi|^ie of Archimedes : 
 
 Every hody immersed hi a fluid is sahjected to a vertical 
 upward p '^sure equal to the weight of the fluid displaced. 
 
 The tr i of this principle is at once seen when we think 
 that, if li body were replaced with a portion of fluid of 
 
58 
 
 the same kind without any other change, the weight of the 
 fluid would be supported. Its truth is setisiblt/ felt in 
 bathing on a shinirly beach, when it is found that, the 
 deeper one enters the water, tlie less are the soles of the 
 feet hurt by the pressure of the stones on them. It can be 
 proved directly by immersing in a liquid, a body, whose 
 volume can be measured exactly (such as a cube, cylinder, 
 or sphere), observing the apparent loss of weight of the 
 body, and then weighing the amount of liquid displaced. 
 In the case of a floating body, the volume of the part of 
 the body immersed is to be calculated, and then the weight 
 of this volume of fluid will be found to be equal to the en- 
 tire weight of the floating body. Convenient experiments 
 to show these facts are given in books on Experimental 
 Physics. 
 
 73. The occasion which l,ed Archimedes "-o the discovery 
 of this principle was the giving to him by King Hiero of 
 Syracuse the problem : — to discover the amount of alloy 
 which, the king suspected, had been fraudulently put int(» 
 a crown, wliich he had ordered to be made of pure gold. 
 It is said that Archimedes saw the solution of the problem 
 one day on entering the bath, and no doubt it was by his 
 observation of the buoyancy of the water. It may have 
 been, however, by his noticing that the volume of the water 
 which he displaced would just be equal, by the principle 
 of impenetrability (art. 2), to the volume of the immersed 
 part of his body. Indeed, one of the most important ap- 
 plications of the principle of impenetrability is to determine 
 exactly the volume of any irregularly shaped body, by im- 
 mersing it in a liquid contained in a measuring glass, and 
 noting the change of level which takes place. 
 
 As is pointed out by Prof. Tait in his "Properties of 
 
59 
 
 ap- 
 Uine 
 
 im- 
 laiul 
 
 Is of 
 
 Matter," it was fortunate for Archimedes that the densities 
 of the alloys of gold are not very different from the means 
 of the densities of the component metals, else the fraud- 
 ulent goldsmith might have escaped. The discovery of a 
 most important hydrostatic principle was however far more 
 important than the solution of King Hiero's problem. Let 
 uj* consider how it enables us to determine the specific grav- 
 ities of liquid and solid bodies. 
 
 74. L Liquids^ to an approximation of the Jirst fie- 
 (jvee : 
 
 a). By means of a balance^ either common or spring. 
 
 Weigh a body which is not attacked either by water or 
 the liquid, e.g. a piece of quartz, or a platinum ball, firstly 
 in air, secondly in water, thirdly in the liquid whose spe- 
 cific gravity is required : 
 
 Let 
 
 u'l = 
 
 weight of th 
 
 e body 
 
 in 
 
 air, 
 
 
 w,^ = 
 
 • • • • 
 
 • • • 
 
 . 
 
 water, 
 
 
 7V^ = 
 
 • • • • 
 
 • • • 
 
 • 
 
 the liquid. 
 
 
 then 
 
 s. g. of the 
 
 liquid 
 
 — 
 
 Wy - w,^ 
 
 h). By means of hydrometers or areometers. 
 
 These are instruments, essentially hollow closed tubes, 
 weighted below, for determining specific gravities by ob- 
 serving how far they sink in water and other liquids, or by 
 observing what weight will make them sink to a certain 
 ilepth. The latter are called hydronjeters of constant im- 
 mersion, the former hydrometers oi' variable immersion. 
 
 I. By means of an hydrometer of constant immersion, 
 e.g. Nicholson's. 
 
60 
 
 Let Wi = weight of the hydrometer in air, 
 
 w„ = weight required to sink the hydrometer to the 
 
 marked depth in water, 
 w^ = ditto, ditto, ditto, in the liquid, 
 
 then 8. s. of the liquid = —^—, — - 
 
 2. By means of hydrometers of variable immersion. 
 
 These are called sallmetera or alcoholimeters^ according 
 as they are used for liquids of greator or less specific grav- 
 ity than that of water. Both kinds have scales attached to 
 them which tell either the specific gravity directly for any 
 immersion, or the volume immersed, in which case the 
 specific gravity must be calculated. A thermometer is fre- 
 quently attached to tell the temperature of the liquid. 
 
 g). By means of a specific gravity bottle. 
 
 Let w?i = weight of the s. g. bottle empty, 
 
 full of water, 
 
 full of the liquid. 
 
 w?2 = 
 
 Wg = 
 
 then s. g. of the liquid = —^^ ^ 
 
 w. 
 
 w, 
 
 75. IL Solids^ to an approximation of the first de- 
 gree : 
 
 a). By means of a balance, either common or spring. 
 
 Let w^ — weight of the body in the air. 
 
 w^ = 
 
 water. 
 
 then 8. g. of the body = 
 
 «'i 
 
 w. 
 
 w, 
 
 b). By means of an hydrometer of constant immersion. 
 
61 
 
 Let Wj = weight of the body in air, 
 
 w^ = weight required to make the Iiydrometer alone 
 
 sink to the marked depth in water, 
 10^ = weight required to make the hydrometer, with 
 the body attached to the lower part of it, sink 
 to the marked depth in water, 
 
 then s. g. of the body = '^ 
 
 tor evidently W w denote the weight of the hydrometer in 
 air, then w + w^ will be the weight of water displaced by 
 the hydrometer, and w + w^-^-w^ the weight of water dis- 
 placed by the hydrometer and body together ; therefore the 
 difference ^v^-Y^v^ -ic^ will be the weight of water dis- 
 placed by the body alone : w^ can easily be determined 
 by the hydrometer, although it is simpler to measure it by 
 means of a common balance. 
 
 c). By means of a specific gravity bottle. 
 
 This method is particularly convenient for finding the 
 specific gravities of powders. 
 
 Let w^ = weight of the powder. 
 
 ^'2 = weight of the specific gravity bottle, full of 
 water, 
 
 w^ = weight of the specific gravity bottle, after the 
 powder has been inserted, and the bottle 
 thereafter filled up with water, 
 
 then 8. g. of the powder = —^ 
 
 ^/). When a solid body is soluble in water, we find its 
 specific weight relatively to a liquid in which it is insoluble 
 and multiplying this by the specific gravity of the liquid', 
 we get the specific gravity of the body. As an example let 
 us take common salt, and adopt method a). 
 
 Ill' 
 
 ,t.. 
 
 I- 
 
 
62 
 
 Let vji = weight of the salt in air, 
 
 ?/;2= weight in kerosene or turpentine of a vessel to 
 
 hold the salt, 
 'Mjgrr weight in kerosene or turpentine of the vessel 
 
 containing the salt, 
 s =the s. g. of kerosene or turpentine, 
 
 then s. fir. of the salt = 
 
 ?^i s 
 
 w^ -\-Wc^ —w^ 
 
 Examination VIII. 
 
 1. Define the specific gravity of a body. 
 
 2. Prove that the numbers which measure the density 
 and specific gravity of any body are the same- 
 
 3. Give the specific gravities of a few well-known sub- 
 stances. 
 
 4. Enunciate and prove Archimedes' principle. 
 
 5. Describe several illustrative experiments which prove 
 the same principle. 
 
 6. Explain why a boat built entirely of iron can float in 
 water. What is the s. g. of iron ? 
 
 7. Give the history of the discovery by Archimedes of 
 his Principle. 
 
 8. Is the density of an alloy ai\\ ays equal to the mean of 
 the densities of the component metals ? Give examples. 
 
 9. What are the three practical methods of determining 
 the specific gravities of solid and liquid bodies ? 
 
 10. Give formulae for all the methods in the case of both 
 solid and liquid bodies. 
 
63 
 
 11. What is an hydrometer? Give the names of the 
 vh'fFerent kinds. 
 
 12. When a solid body is sohible in water, how is its 
 specific gravity found ? 
 
 13. How would you find the specific gravity of (1) sul- 
 phuric acid, (2) nitric acid, (3) bichromate of potash (4) 
 sand, (5) a piece of cork ? 
 
 14. Given a common balance with a hook to weigh 
 l)odies in water, a piece of cork, and a piece of lead suf- 
 ficient to sink the cork in water ; shew (giving foi-mulae) 
 how to determine the specific gravity of the cork. 
 
 15. How can the volume of an irregularly shaped stone 
 he accurately determined i 
 
 ■f^. 
 
 Exercise VIIT. 
 
 1. A piece of limestone balances 20-21 grams in air, and 
 12-82 in water; find its specific gravity. 
 
 2. An egg whose volume is 48 and s. g. 1-015 is put into 
 salt water whose s. g. is 1-026 ; find what volume of the 
 egg will be above the surface of the water. 
 
 3.^ An empty bottle hermetically sealed, whose mass is 
 33, 18 attached to a heavy body of mass 203, and the whole 
 is found to balance 10 grams in water. The lieavv bodv 
 alone balances 178 grams in water; find the s. g." of the 
 heavy body and of the empty bottle. 
 
 4. When 1 lb. of cork is attached to 21 lbs. of silver, the 
 whole is found to balance 16 lbs. in water. If the s ..• (»f 
 silver be 10^ ; find that of cork. 
 
64 
 
 5. A person whose mass is 140 lbs. enters the sea to 
 bathe. If the s. g. of sea-water be 1"028, and of the human 
 body 0*9, find the pressure on his feet when three-fourths 
 of his body is immersed. 
 
 6. A ball of platinum whose mass is a kilogram, when in 
 water, balances 955 grams ; what will it balance when in 
 mercury (s. g. 13'6)? 
 
 7. A piece of iron, whose s. g. is 7*5, floats in mercury 
 whose s. g. is 13*r) ; find what part of the iron is above the 
 surface of the mercury. 
 
 8. The s. g. of ice is 0*92, and of sea-water 1*028; find 
 what fraction of an ice-berg is below the surface of the sea. 
 
 9. A vessel, containing water, balances 2034 grams ; a 
 body of mass 1000, whose s. g. is 8*4, is held in the water ; 
 find what will now be the apparent weight of the vessel 
 and water. 
 
 10. A piece of cork, whose s. g. is J, has mass 534; find 
 the pressure necessary to keep the cork under sea-water 
 whose s. g. is 1*028. 
 
 11. A kilogram of lead, whose s. g. is 11*35, is suspended 
 in water by a string ; find the tension of the string. 
 
 12. A vessel partially filled with mercury (s. g. 13*6) 
 balances 72^ kilograms ; a kilogram of iron (s. g. 7*5) is 
 held completely immersed in the mercury ; what will now 
 be the apparent weight of the vessel and mercury ? 
 
 13. A block of pine, the volume of which is 4 litres, 340 
 cub. cm., fioats in water with a volume of 2 litres, 240 cub. 
 cm. above the surface ; find the s. g. of the pine. 
 
 14. Find what force would be necessary to immerse a 
 kilogram of oak (s. g. 097) in mercurj (s. g. 13*6). 
 
65 
 
 3-6) 
 
 )i8 
 
 low 
 
 1 5. A body of mass 58 grams floats in water with two- 
 thirds of its bulk submerged ; find its volume. 
 
 16. A certain body A is observed to float in water with 
 half its volume submerged, and when attached to another 
 body B of twice its own volume, the combined mass is just 
 submerged ; find the specific gravities of -4 and JB. 
 
 17. A piece of platinum of mass 15 lbs. is attached to a 
 piece of iron of mass 10 lbs., and the whole is found to 
 balance 1 lb. in mercury (s. g. 13'6); the platinum by itself 
 balances 6 lbs. in mercury; find (I) the s. g. of the plat- 
 inum, (2) the s. g. of the iron, (3) the weight in water of 
 the iron. 
 
 18. Eight cub. ft. of maple (s. g. 0*75) floats in sea water 
 (s. g. 1*026) ; find what volume of water it displaces. 
 
 19. Two bodies, which balance 50 and 60 grams in air, 
 balance each 30 grams in alcohol (s. g. 08) ; compare their 
 volumes and densities, and find the s. g. of the first. 
 
 20. A man whose mass is 63*5 kilograms can just float in 
 fresh water ; find the maximum weight he could bear up 
 clear of the water, when floating in the sea (s. g. 1'028). 
 
 21. How much lead (s. g. 11*35) will a kilogram of cork 
 (s. g. 0*25) keep from sinking in salt-water (s. g. 1*028) ? 
 
 22. A piece of hard wood of mass 7*6 grams is at- 
 tached to the lower part of Nicholson's hydrometer, and it 
 is then found that the buoyancy of the hydrometer in salt 
 water is just the same as before the wood was attached, 
 viz., 12*6 grams-weight. If 1*03 be the s. g. of the salt 
 water, find that of the wood. 
 
 • t^'i^ 
 
 ke a 
 
 VMW ■ 
 
66 
 
 Chapter IX. 
 
 Weight of Gases, 
 
 76. We are aware from the effects of wind in driving 
 windmills, ships, &c., that air has mass. That it has weight 
 like solid and liquid bodies can easily be proved by the 
 following experiments : 
 
 Exp. 1. Weigh a globe with a tightly fitting stop-cock, 
 firstly full of air, secondly after the air has been extracted 
 from it by an air-pump. 
 
 Exp. 2. Boil water in a flask until all the air is driven 
 out, cork it up tightly, weigh when cool, admit air and 
 weigh again. 
 
 Exp. 3. Instead of extracting air from the globe in exp. 
 1, compress air into it, when it will be found to become 
 heavier. 
 
 Exp. 4. Weigh the globe in exp. 1 when filled, firstly 
 with air, secondly with hydrogen, thirdly with carbonic 
 acid. 
 
 The second and third experiments are due to Galileo ; 
 from the third the specific gravit}' of air may be roughly 
 measured by collecting the compressed air in a pneumatic 
 trough. The fourth experiment proves that gases, like 
 liquid and solid bodies, differ in specific gravity. 
 
 77. The fact that gases have weight, and even flame, 
 which is essentially incandescent gas, was known to the 
 Epicureans, if we take Lucretius as their mouth piece. In 
 his great poem, "De rerum natura, " written about 56 
 B.C., he says : 
 
67 
 
 In 
 56 
 
 See with what force yon river's crystal stream 
 
 Resists the weight of many a massy beam : 
 
 To sink the wood, tlie more we vainly toil, 
 
 The higher it rebounds with swift recoil : 
 
 Yet that the beam would of itself ascend, 
 
 No man will rashly venture to contend : 
 
 Thus too the flame has weight, though highly rare, 
 
 Nor mounts but when compelled by heavier air. 
 
 78. Before considering how the specific weights of gases 
 have been determined, it will be necessary to know how 
 the density of a gas depends upon its pressure or tension- 
 The physical law which tells us this is known as Boyle's 
 law, which may be enunciated in either of the following 
 ways : 
 
 In a gas^ far removed from its 'point of condensation^ 
 the volume at a given temperature is inversely proportional 
 to the pressure. P V= G, where 6* is a constant so long as 
 we keep to the same gaseous body. Or thus : 
 
 In a gas, far removed from its point of condensation , the 
 density at a given temperature is directly proportional to 
 the pressure. D oc P. 
 
 The truth of Boyle's law depends of course entirely upon 
 experimental evidence. Dry air is an example of a gas far 
 removed from its point of condensation, and for dry air at 
 all ordinary pressures and temperatures the law may be 
 said to be exact. In the case of solid and liquid bodies it 
 is not necessary to consider the pressure to which they are 
 subjected, for the small changes of density, arising from the 
 changes of pressure to which bodies are in general exposed, 
 would be immeasurable. The case of gases is very different. 
 
 79. It was the great French physicist, Regnault, who 
 
68 
 
 first overcame the experimental difficulties necessary to an 
 exact determination of the specific weights of gases. The 
 secret of his success depended upon counterbalancing the 
 globe containing the gas he was weighing, with another 
 globe of equal volume and weight as nearly as possible, so 
 that it was unnecessary for him to make any corrections for 
 barometric, thermometric, or hygrometric changes in the 
 state of the atmosphere during the time of experimentation. 
 Having several times exhausted one of these globes and 
 filled it with a dried gas until he was satisfied that the globe 
 was thoroughly dry, he put the globe in a mixture of ice 
 and water (0®C), and filled it once again with the dried 
 gas at the pressure of the atmosphere, say P cm. of mer- 
 cury at 0°C. He then partially exhausted the globe, to 
 pressure jp say, keeping it at the same temperature ® C, 
 and noted the change of weight. This change of weight 
 (denote it by w) will, by Boyle's law, be the weight of the 
 gas atO" which would fill the globe at pressure P—p', 
 therefore the weight of gas at ° required to fill the globe 
 at the mean atmospheric pressure will, by the same law, be 
 1/7 X 76 -^■ (P - p). 
 
 In this way Regnault found the weights of equal vol- 
 umes of dry air and other gases. It remained for him to 
 determine the specific weight of dry air with respect to the 
 standard substance, pure water at 4 ® C. If w denote the dif- 
 ference of weight between the globe when filled with water 
 at ° , and when filled with dry air at ° and pressure P, 
 then w -\- w y. P -^ {P -p) will denote the weight of the 
 water which the globe would hold at ® . If therefore s 
 denote the s. g. of pure water at ° , the s. g. of dry air at 
 ® and under the mean atmospheric pressure will be 
 
 IQws , , P ^ 
 
69 
 
 so 
 
 
 If it be necessary to take into consideration tliebnoyancy 
 of the air in determining w and w\ tlie methods indicated 
 in the next cliapter will exphiiii how tliis can be done. 
 The following table gives the results of some of Regnault's 
 experiments : 
 
 Mass of 1 litre at 0° and 76 cm. Specific gravity. 
 
 Air (dry) 1-293187 0-0012932 
 
 Oxygen 1-429802 0-0014298 
 
 Nitrogen 1-256167 000125<;2 
 
 Carbonic Acid .... 1-977414 0-0019774 
 
 Hydrogen 0-089578 0-0000896 - 
 
 80. On account of the small densities of gases it is gen- 
 erally more convenient to measure their specific weights 
 with respect to dry air or hydrogen as a standard, than 
 with respect to water at 4 ° C. 
 
 The specific weight of a gas with respect to dry air (or 
 hydrogen), is defined as the ratio of the weight of any vol- 
 ume of the gas at 0°C and under the mean atmospheric 
 pressure, to the weight at the same place of an equal volume 
 of dry air (or hydrogen) at the same temperature and 
 pressure. 
 
 Tlie following table gives the specific weights of the above 
 gases with respect to dry air and hydrogen : 
 
 Hydrogen 00693 1-000 
 
 Nitrogen 0-9714 14-023 
 
 Air (dry) 1-0000 14-436 
 
 Oxygen 1-1056 15-9r,2 
 
 Carbonic Acid 1*5291 22-075 
 
 81. The numbers in the last column and similar results 
 have enabled chemists to establish a most important law 
 
70 
 
 relating to the molecular volumes of gaseous bodies. It 
 may be expressed thus : 
 
 The speoijlc gravities of gases' ^ far removed from their 
 points of condensation, are, at the same pressure and tern.- 
 perature, directly proportional to their molecular weights. 
 
 In the following table for gases the specific gravities 
 have been calculated from the molecular weights, with the 
 exception of those of nitrogen, air, and oxygen, which are 
 Regnault's experimental njeasurenients. 
 
 By means of a good air-pump hydrogen can be rarified 
 to a density 10* times less than what it has under the 
 mean atmospheric pressure. We thus see, by comparing 
 the density of platinum with that of rarified hydrogen, 
 what a very great range of density there is, even in sub- 
 stances which can be easily obtained. It will be seen from 
 the following table that the density' of a solid substance 
 depends to a certain extent on the way in which it has been 
 prepared. Different specimens of the same material may 
 be found to have diflferent densities arising from the pres- 
 ence of impurities. Even in the case of natural bodies like 
 sapphires or diamonds, diflferent specimens from diff'erent 
 places are found to vary in density. In mixtures the den- 
 sity is not always the mean of the densities of the compo- 
 nent parts. Thus bell metal has a greater density than the 
 mean of the densities of the component metals ; so with a 
 mixture of alcohol and water. In the case of woods differ- 
 ent parts of the same tree may vary in density, as well as 
 specimens from diff'erent trees of the same species. Liquids 
 can be obtained more easily in a state of purity, but in such 
 liquids as blood, milk, or sea-water, slight differences of 
 density may be found in different specimens. 
 
71 
 
 Table of Densities and Specific Gravities. 
 
 /. Solids at O^C, 
 
 Platinum, stamped 2210 
 
 rolled 2207 
 
 " cast 20-86 
 
 Gold, coin 19-36 
 
 " cast 19-26 
 
 Lead, cast 11-35 
 
 Silver, cast 10-47 
 
 Copper, hammered 8 88 
 
 cast 8-79 
 
 Bronze,] 
 Brass, 
 
 Steel 7-82 
 
 Iron, wrought 779 
 
 " cast 7-21 
 
 Tin. cast 729 
 
 Zinc, cast 700 
 
 Aluminium 267 
 
 Magnesium 174 
 
 f average 8-40 
 
 Sapphire 401 
 
 Diamond 3 52 
 
 Glass 2-5 to 3-3 
 
 Kingston Limestone 2 70 
 
 Kock -crystal (Quartz) 266 
 
 Ivory 1 92 
 
 Anthracite 180 
 
 Ebony, American . . 133 
 
 Mahogany, Spanish 106 
 
 Box, French 103 
 
 Oak, English 097 
 
 Apple 0-79 
 
 Maple 75 
 
 Walnut 068 
 
 Elm 0-70 
 
 Willow 0-58 
 
 Poplar 0-38 
 
 Cork 0-24 
 
 //. Liquids at 0° O, 
 
 Mercury 13-596 
 
 Sulphuric Acid 184 
 
 Human Blood 1.05 
 
 Milk, of cow 1-03 
 
 Sea-water 1027 
 
 Water at 4° 1000000 
 
 " at0° 999873 
 
 Olive Oil . . 0-915 
 
 Alcohol 0-80 
 
 Sulphuric Ether 0-72 
 
 ///. Gases at 0^ C and 70 cm. pressure. 
 
 Hydrogen 00000896 
 
 Ammonia 0-0007619 
 
 Aqueous Vapour 00008044 
 
 Carbonic Oxide 00012510 
 
 Nitrogen 00012562 
 
 Air (dry) 000.2932 
 
 Oxygen 00014298 
 
 Sulphuretted Hydrogen 00015219 
 
 Carbonic Acid 00019658 
 
 Chlorine 00031684 
 
72 
 
 Examination IX. 
 
 1. How do we become aware that air possesses the 
 property of mass ? 
 
 2. Describe three experiments wliicli prove that air has 
 weight. 
 
 3. How is it proved that gases differ like solid and liquid 
 bodies in specific gravity ? 
 
 4. Enunciate Boyle's law in two ways, and shew that 
 the one follows from the other. 
 
 5. What was the secret of Regnault's success in weighing 
 gases ? 
 
 t). Describe fully Regnault's method of determining the 
 specific gravity of dry air. 
 
 7. Define the specific weight of a gas with respect to dry 
 air, and also with respect to hydrogen. 
 
 8. What is the specific gravity of dry air, 1) with respect 
 to water at 4 *^ , 2) with respect to hydrogen ^ 
 
 9. Enunciate the law which connects the specific gravity 
 of a gas with its molecular weight. 
 
 10. What is the range of density as found by experiment ? 
 
 EXEIICISE IX. 
 
 1. Determine the mass and weight of 10 litres of oxygen 
 at 0® and at a pressure of 74 cm. of Hg. at 0° in the lat- 
 itude of Kingston, Ont. 
 
 2. Determine the pressure in prems under which chlorine 
 has a density 3 with respect to air. 
 
73 
 
 CllAl'TER X. 
 
 Exact Specific WiMjhts. 
 
 hlorine 
 
 82. Since the principle of Archimedes evidently {ip])lies 
 to gases as well as to liipiids, all bodies in the atmosphere 
 are subjected to a vertically upward pressure equal to the 
 weight of the air displaced by them. This maybe illus- 
 trated experimentally by the baroscope and Ixdlonns. In 
 determining the specific weights of solid and liquid bodies 
 to an approximation of the first degree, we neglected the 
 buoyancy of the surrounding atmosphere. Let us now de- 
 termine these specific weights to an approximation of the 
 second deicrce. This is done by takini^ into consideration 
 the buoyancy of the air, but reckoning its specific gravity 
 as 1*2932 X K)^^, without noting what may be its barom- 
 etric, thermometric, and hygrometric stjites. The follow- 
 ing problem will illustrate the process: 
 
 Given w^, tv^, if-^', the number of grams which balance 
 a solid body in air, water, and another licpiid respectively; 
 to determine the specifics gravities of the solid body and 
 liquid to an iqtproximation of the second degree. 
 
 Let /i denote 0-0012032grajn-weight. The approximate 
 volume of the solid body will i)e {i/\ — W2) cub. cm,, 
 and therefore the weight of air (lisj)laced by the solid body 
 will be {u\—u\) A^ gram-weight approximately. 
 
 Let s denote the s. g, of the standard masses against 
 which the body is weighed. This should bo determitied 
 bv tlie maker of the standard masses. Then ?r,-:-,s* is the 
 volume of the standard masses which balance the body in 
 air, and therefore (wj^.v) A* the ai)proximate weight of 
 
74 
 
 air, in grams-wciglit, displaced by them. Therefore the 
 approximate weiglit of the solid body in vacuo 
 
 iv 
 
 — w^- ;-7?+ (ijo-^ — w^) li grams- weight = W^ 
 s 
 
 The approximate weight of the solid body in water 
 
 = w^— — R grams-weight = \\\ 
 
 The approximate weight of the solid body in the liquid 
 w. 
 
 = w. 
 
 ^ R irrams-weiirht = Wo 
 
 s 
 
 w. 
 
 Then tiie specific gravity of the solid body =-t^>— "117^ 
 
 1 M 
 
 and liquid body = ii;- _ IF 
 
 each to an api)roxiination of the second degree. 
 
 83. To get the specific weight of a body to a degree of 
 approximation of the third degree, we require to calculate 
 the density of air at the |)ressure, temperature, and hygro- 
 nietric state in which it is at the time the weighings are 
 performed, as well as to allow for the temperature of the 
 water in which the body is weighed. We have already 
 learned from Boyle's law (art. 78) how the density of a gas 
 depends uj)on its pressure. The law of change of density of 
 a gas, arising from change of temperature, was first dis- 
 covered by Charles. It may be enunciated thus : 
 
 The dilatation of a gas^ far removed from its point of 
 condensation^ and at a constant pressure^ is directly pro- 
 portional to the increase of temperature ,' and the coeficient 
 of dilatation is the same for all gases. 
 
 If the dilatation be reckoned from ° (?, the coefficient of 
 dilatation is very approximately 0'<i030()5or ^^,s. Hence 
 
75 
 
 gas 
 
 of 
 pro- 
 
 ient 
 
 
 if Ff be tlie volume at temperature t' , and Vq the volume 
 at 0®, we may express the law algebraically thus : 
 
 Vt = Fo (1-M)-003C65 t), or Vt = V^ (1+273^ 
 
 If now temperature be reckoned from the zero of the air 
 tliermorneter, i.e. from --273 °C, Charles' law may be ex- 
 pressed thus : 
 
 The volume of a gas ^ far removed from, its point of eon - 
 densatioii, is, at a constant pressure.^ directly proportional 
 to its tan pe rat are reckoned from the zero of the air ther- 
 mometer. Or thus : 
 
 The density of a gas, far removed frotn its point of con- 
 de7isation, is, at a constant pressure, inversely proportional 
 1') its temperature reckoned from the zero of the air ther- 
 mometer. Or thus : 
 
 The pressure (f a gas, far removed from its point <f 
 condensation, is, at a constant volume, directly propor- 
 tional to its temperature reckoned from the zero of the air 
 thermometer. 
 
 The equation PY— C'J' evidently expresses both Boyle's 
 
 and Charles' laws. The quantity C is constant, so long as 
 
 we keep to the same gaseous body, however the pressure, 
 
 volume, or temperature be changed: T must he reckoned 
 
 from the zero <f the air thermometer. 
 
 84. To allow for the hygrometric state of the air, we re- 
 quire first to know the law of Dalton relating to the i>re8- 
 sure of a mixture of gases. It may be enunciated thus : 
 
 When two or more gases, which do not act chemically on 
 one another, are enclosed m a vessel, the 7'esulta?it pressure 
 of the pressures of the gases wl 
 
 in the vessel. 
 
 placed singly 
 
76 
 
 The plij'sical principle underlying Boyle's and Dalton's 
 laws has been beaatifully expressed by Rankine thus : When 
 one, or more gases, which do not act Ghemiaally on one 
 another, is confined in a vensel, each portion of gas, how- 
 ever small, exerts its pressure quite in dependent I ij of the 
 presence of the rest of the gas in the vessel. Dalton's law 
 tells us that the pressure of moist air is just the sum of the 
 pressures of the dry air and aqueous vapour mixed with it. 
 
 85. From the classical experiments of Regnault the pres- 
 sure of the aqueous vapour in the atmosphere can be 
 determined, so soon as the dew-point is known. The dew- 
 point is the teuiperature at which the atmosphere at any 
 place would be saturated with the aqueous vapour which 
 it contains. It is found experimentally by means of an 
 hygrometer. 
 
 The following is a part of Regnault's table of the max- 
 iuium pressures (or pressures of saturation) of aqueous 
 vapour at dilierent temperatures. It gives the pressure of 
 the aqueous vapour in the atmosphere, in centimetres of 
 mercury at 0" at the latitude of Paris, for dew-points from 
 0° to 20 = C. 
 
 reinp, 
 
 Pressure. 
 
 Temp. 
 
 Pre.s8ure. j 
 
 Temp. Pressure. 
 
 Temp. 
 
 Pressure. 
 
 0° 
 
 0-4600 
 
 5° 
 
 0-6534 ! 
 
 10 = 
 
 0-9165 
 
 15 = 
 
 1-2699 
 
 1° 
 
 0-4940 
 
 6° 
 
 0-6998 
 
 11 = 
 
 0-9792 
 
 16" 
 
 1-3536 
 
 2° 
 
 0-5302 
 
 7° 
 
 0-7492 1 
 
 12 = 
 
 1-0457 
 
 17 = 
 
 1-4421 
 
 3° 
 
 0-5687 
 
 8° 
 
 0-8017 i 
 
 13 = 
 
 1-1162 
 
 18 = 
 
 1-5357 
 
 i° 
 
 6097 
 
 9» 
 
 08574 ^ 
 
 14 = 
 
 1-19(»8 
 
 19 = 
 
 1-6346 
 
 5 = 
 
 0-6534 
 
 10=* 
 
 09165 
 
 ;i5° 
 
 1-2699 
 
 20 = 
 
 1-7391 
 
 Observe, that whilst the i)ressure of a gas far removed 
 from its point of condensation dei»ends upon its tempera- 
 ture and volume, the pressure of the same gas id its j>oint 
 of condensation (or, in contact with its own liquid) (lej^ends 
 
t421 
 
 >357 
 
 •)3+r> 
 
 f 391 
 
 loved 
 pera- 
 
 upon its femperature alone. The following example will 
 illustrate how the density of the atmospheric air can be cal- 
 culated when its barometric, thermometric and hygrometric 
 states are known. 
 
 £b. The reading of the barometer is 76 "i-, the temper- 
 ature 20®, the dew-point 8°, and the latitude 44' 13' ; to 
 determine the density of the air, given the coefficient of 
 dilatation of the barometer scale to be O'OOOOiS, and the 
 mean coefficient of dilatation of mercury between 0° and 
 20° to be 000017951. 
 
 The barometric pressure in centimetres of mercury atO° 
 in the latitude of Paris will be 
 
 76-4 (l-f'><>X< ••00001 8) "080-54 
 = ~T + -20X "^000 17951 ^ 980-94 ="^'1230. 
 
 This pressure is due, according to Dalton's law, partly 
 to dry air, and partly to the aqueous vapour in the air. 
 According to Regnault's tables the })ressure of the aqueous 
 vapour for the dew-point 8' is 0"8017. Therefore the 
 pressure of the dry air in the atmosphere is 75-3213. Hence, 
 applying Boyle's and Charles' laws, the density of the dry 
 air in the atmosphere 
 
 75-3213 273 
 
 = 1-2932 X 10-3 X 
 
 - X 
 
 = 1-1942 X 10-3, 
 
 76 -^ 293 
 
 the density of the aqueous vapour in the atmosphere 
 
 0-8017 273 
 = 8-044X10-4 X ,^^. -X.^j).^ =--7-9X 10 «, 
 
 .'. the density of the atmospheric air = 1-2021 X 10-3. 
 
 86. To detpriiihie the f<pecific gravity of a solid or liquid 
 hody to an approximation of the third degree. 
 
 Ex. Given ?<>|, w^, w^, the number of grams which bal- 
 ance a solid body in air, distilled water, and another lic^uid 
 
78 
 
 respectively ; tlie reading of the barometer 76'4, the tem- 
 perature 20°, tlie dew-j)oiiit S°, and the hititude 44° 1^^ ; 
 the 3oetticient of dilatation of tlie barometer scale 000018, 
 the mean coefficient of dilatation of mercury between 0° 
 and 2i>°, according to Regnanlt, 0-00017951, and the den- 
 sity of distilled water at 20°, according to Despretz, 
 0"Si98213; to determine the specific gravities of the solid 
 body and liquid to an approximation of the third degree. 
 
 Find, as in art. 82, IFj the approximate weight of the 
 solid body in vacuo, S the s. g. of the solid body to an ap- 
 proximation of the second degree, and, as in last article, 
 li the density of the atmospheric air. Denote by s the s. g. 
 of the standard masses against which the body is weighed, 
 and by *S^ the s. g. of distilled water at 20°. 
 
 The weight of the solid body in vacuo (in grams-weiglit) 
 very nearly = tt\ -f ( vv — -^) H = W^ 
 
 O o 
 
 The weight of the body in distilled water at 20° 
 
 w 
 
 = 90. 
 
 - — ' i? = IF, 
 
 The weight of the body in the liqiid at 20 
 
 w< 
 
 = w^ - " A* = W^ 
 
 Then s. g. of the solid body at 20 ° = 
 
 
 w,-w,^ 
 
 W, - w. 
 
 s 
 
 and s. g. of the liquid at 20° = 
 
 to an approximation of the third degree. 
 
 If the coefficients of dilatation of the solid body and 
 liquid be known, the specific gravities at any other tem- 
 perature m;iy be determined. By taking the specific gravity 
 
 of the solid body just determined in place of aS^, and Wi in 
 
79 
 
 in place of TFj, and re])eatin<i; the method above, we could 
 find the specific gravities to an a])proximati(>n of the fourtli 
 de<;ree, and so on to hii^her degrees. This would however 
 be useless, as the errors of experimentation would certainly 
 be greater than any errors, from the exact values, of the 
 specific gravities to an approximation of the third degree. 
 
 Examination X. 
 
 1. How can it be proved experimentally that Archimedes' 
 principle applies to gases? 
 
 2. Given the apparent weights of a solid body in air, 
 water, and another liquid, to determine the specific gravities 
 of the solid body and liquid to an approximation of the 
 second degree. 
 
 3. Enunciate in four ways the law of Charles, and de- 
 duce each one from the others. 
 
 4. Enunciate Dalton's law relating to the pressure of a 
 mixture of gases. Give Rankine's statement of the physical 
 principle underlying Boyle's and Dalton's laws. 
 
 5. What are the various corrections to be made in de- 
 termining the specific gravity of a l)ody to an approxima- 
 tion of the third degree? What are the physical instruments 
 used for this purpose ? 
 
 6. Define the dev-point. What does it tell us ? 
 
 7. Write down an algebraical equation which expresses 
 the f^iets contained in Boyle's and Charles' laws. 
 
 Exercise X. 
 
 1. The reading of the barometer in a room is 77*34, the 
 thermometer 15°, the dew-point 10°, the latitude 44° 13' ; 
 
80 
 
 the coefficient of dilatation of tlie barometer scale is 
 0*000018, and tlie mean coefficient of dilatation of mercury 
 between 0— and 15°, according to Regnault, « 1-0001794:; 
 the room is 12*5 m. long, 5-45 m. broad, and 3'7 m. high : 
 find the volume, mass, and weight of the air in the room. 
 
 2. A piece of cork balances 50 grams in air ; when at- 
 tached to the bottom of Nicholson's hydrometer, it is found 
 that 175 grams-weight are required to sink the hydrometer 
 to the marked depth in distilled water, whilst only 25 
 grams-weight are required to sink the hydrometer alone ; the 
 reading of the barometer is 78 cm., of the thermometer 14°, 
 the dew-point 12° ; the latitude, that of Edinburirh ; the 
 specific gravity of the standard masses is 8'4, the coefficient 
 of linear dilatation of the barometer scale 0m>00018, the mean 
 coefficient of dilatation of mercury between 0° and 14°, 
 according to Regnault, 0'00017937, and the density of dis- 
 tilled water at 14°, according to Despretz, 0-90928") ; to 
 determine the specific gravity of cork to approximations of 
 the first, second and third degrees. 
 
 3. A lump of gold balances 437"008 grams in air, 414*357 
 in distilled water, and 420*699 in sulphuric ether ; the 
 reading of the barometer is 77*3 cm., of the thermometer 
 9°, the dew-point 4° ; the latitude, that of Greetiwich; 
 the 8. g. of the standard masses against which the body is 
 weighed is 8*4, the coefficient of linear dilatation of the bar- 
 ometer scale 0*000018, the mean coefficient of dilatation of 
 mercury between 0° and 9°, according to Regnault, 
 0*0001792, and the density of distilled water at 9°, accord- 
 ing to Despretz, 0*999812 ; to determine the specific grav- 
 ities of gold and sulphuric ether to approximations of the 
 first, second, and third degrees. 
 
81 
 
 18 
 
 Chapter XI. 
 
 Ei\i^r(jy. 
 
 Work. 
 
 87. Evergy is the power to overcome resistance through 
 space. Worl' is the cxpenditnre of energy, or is the trans- 
 ference of energy from one body to another. Work is phys- 
 ically manifested either by accelerating the motions of ma- 
 terial bodies, or by changing the configuration of a material 
 system against resistance. Thus when a man raises a body 
 vertically upwards hv does work against the body's weight, 
 and by the work done produces a change of configuration 
 of the material system consisting of the body and the earth. 
 Again, when he throws a cricket ball, he does work in 
 giving the ball motion. 
 
 We have firstly defined energy, then work. The order 
 might have been reversed, thus : work is the production of 
 motion against resistance ; energy^ the power of doing work. 
 
 88. The property of mass is essential to any body possess- 
 ing energy. If further the body be in motion, it has energy 
 in virtue of this motion. Thus, the energy of a moving 
 cannon ball is due entirely to its mass and velocity, and it 
 is this energy which enables it to tear down a rampart 
 against the resisting molecular forces. Similarly, in virtue 
 of its mass and velocity, a running stream can drive a 
 water-wheel and thus grind onr corn. These are examples 
 of visible kinetic energy. 
 
 89. A body may further possess energy in virtue of its 
 position with respect to other bodies, which, along with it, 
 form a material system. There are forces which are con- 
 stantly acting between ever}^ pair of particles of a material 
 system. The force of gravitation, the molecular forces 
 
82 
 
 (cohesion, elasticity, crystalline force, &c.), the atomic force 
 or cbeinical affinity, are different aspects of force which is 
 found by experience to act between every pair of particles 
 in the universe. When work is done a^iinst such force 
 upon a body which forms a j)art of a material system, so as 
 to alter the configuration of that system, the l)()dy in virtue 
 of its new position has energy which it did not previously 
 possess. Thus a head of water has energy in virtue of its 
 position with respect to the earth. The wound up spring 
 of a clock can keep it going for a week or longer. Com- 
 pressed, air, such as is used for the conveyance of letters in 
 Paris and elsewhere, is a store of energy in virtue of the 
 configuration of the aerial particles. These are examples ot 
 visible potential energy. 
 
 A material system, such e.g. as the solar system, possesses 
 visible energy ; firstly, on account of the motions of its 
 component parts ; this is its kinetic energy ; secondly, on 
 account of its configuration, i.e. the relative positions of its 
 component parts ; this is its potential energy. An oscill- 
 ating pendulum, or a vibrating spiral spring, is a beautiful 
 and simple example of a body whose visible energy is con- 
 stantly passing from the one form into the other. At the 
 extremities of the line of vibration, the energy is wholly 
 potential; at the middle point, it is wholly kinetic; and at 
 intermediate positions, it is partly kinetic and partly po- 
 tential. In an undershot water-wheel the miller depends 
 upon the kinetic energy of the water to grind his corn ; in 
 an overshot water-wheel he depends upon the potential 
 energy of the water to drive the wheel. 
 
 90. Work is measured by the force overcome and the 
 distance through which it is overcome conjointly. Thus in 
 measuring the work done in raising bricks to the top of a 
 
83 
 
 po- 
 inds 
 in 
 itial 
 
 ^ 
 
 house, the builder multiplies the wei<i;ht of the bricks by 
 the vertical height tlirough which they are raised. To 
 raise double the number of bricks through double the 
 height will evidently require four times as much work. 
 The unit of work is that in which unit of force is overcome 
 through unit of distance. In the C. G. S. system the unit 
 of work is the work of overcoming a dyne through a centi- 
 metre, and is called an ery. The equation E= A evidently 
 gives the relation between the work done in ergs, the force 
 overcome in dynes, and the distance in centimetres through 
 which the force is overcome. 
 
 91. To determine the kinetic energy of a body v:ho8e 
 mass is m and velocity v. 
 
 Let the bodv move against a uniform resistance /'. This 
 will give the body an acceleration /-i-m, opposite in direc- 
 tion to the body's motion. If s be the whole distance 
 through which the body can act against this resistance, so 
 that after passing through the distance s the velocity is 
 zero, by equation (6) art. 35, 
 
 / 
 
 m 
 
 .'. fs = ^?nr'-i 
 
 hut fs is evidently the total work done by the body against 
 the resistance /, and is all that it can do, since, after doing 
 this work, the velocity is zero. Hence |^///v^ measures 
 the body's kinetic energy, or amount of work the body can 
 do, in virtue of having mass vi and velocity r. If 7n be 
 measured in grams, and v in tachs, ^mr"^ is the measure 
 of the body's kinetic energy in ergs. Since the kinetic 
 energy of a body varies as the square of its velocity, it is 
 evident that it has no direction ; in this respect it is well 
 to note, energy differs from nnomentum and force. 
 
 92. As an illustration of the preceding nrticle let us 
 

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 84 
 
 (jonsider the case of a body of weight i&and mass w, thrown 
 vertically upwards in vacno with velocity n. In virtue of 
 its kinetic energy it raises itself against its own weight. 
 If h be the greatest height reached, the work done is w/t. 
 Now w = w<f (art. 63), and /t = w'-7-'2y (art. ?^^), therefore 
 wh — ^mu^^ wliich, as might be expected, is the same result 
 as we got in last article. 
 
 Is the energy of the body in its ('/4>vutt;d position de- 
 stroyed? No, it is merely in a latent form ; for, without 
 imparting any more energy to the body, we can get out of 
 it, in virtue of its new position^ the same amount of work 
 as it was capable of doing at starting. This will be at once 
 understood when we remember, that by letting the body 
 fall to its point of starting, it acquires the same velocity 
 which it had at starting (art. 36), and has therefore again 
 the original kinetic energy imparted to it. In its elevated 
 potsitiun the energy of the body is called potential. Such 
 is the energy of a head of water used to drive machinery, 
 or of the elevated massive bodies whose energy is used to 
 <lrive piles into the ground. 
 
 We see from the above that the potential energy of an 
 elevated body is measured by wh, where w is the body's 
 weight, and h its height above the ground. If w be mea- 
 sured in dynes and /t in centimetres, then wh measures the 
 potential energy in ergs. 
 
 Cor. From the above and art. 35, equations (3) and (6), 
 it is easily seen, that in any intermediate position of the 
 body between the ground and height A, the energy of the 
 body is partly kinetic and partly potential, and that the 
 total energy is constant and equal to wh or ^mn^. 
 
 93. Just as it is convenient in many practical questions 
 to hare a gravitation as well as an absolute unit of force. 
 
85 
 
 BtionR 
 force, 
 
 80 in the practical estimate of work it is often convenient 
 to use a gravitation unit. Such a unit is the kUogrammetre^ 
 or work done in raising a body of 1 kilogram mass ver- 
 tically upwards against its weight through the height of I 
 metre. Evidently 1 kilogrammetre = 1000 g X 100 ergs, 
 ( = 98054000 ergs in the latitude of Kingston, Ont). 
 
 94. The F. P. S. unit of work is that required to overcome 
 a poundal through the distance of a foot, and may be called 
 2ifoot-pimihdal. English engineers use as a gravitation unit 
 of work Afoot-jM.und^ i.e. the work done in raising a body of 
 1 pound mass vertically upwards through the distance of 1 
 foot. The foot-pound is evidently equal to y or nearly 32J 
 foot- poun dais.. 
 
 If m be the mass of a body in pounds and v its velocity 
 in feet per second, then ^mv^ measures its kinetic energy 
 in foot-poundals (art. 91), and therefore J mv^-^g^ its en- 
 ergy in foot-pounds. Similarly, if a body, whose mass ism 
 pounds and weight w poundals, be at a height of A feet above 
 the earth's surface, it has potential energy measured by wk 
 foot-poundals, i.e. mgh foot-poundals, or nih foot-pounds. 
 
 95. The unit rate of worMmj in the C. G. S. system is 1 
 erg per second. If II denote the rate of working in ergs 
 per second,/' the resistance in dynes, and v the velocity in 
 tachs of the body moved against this resistance, then II 
 =-fv. This formula suggests the name ilijntacli for the unit 
 rate of working. Watt's hwae-poicer is a convenient grav 
 itation unit adopted by English engineers, and is equal to 
 33000 foot-pounds per minute, or nearly 7*46 X 10® dyn- 
 tachs. The Yxewi^ force-de-cheval is a similar gravitation 
 unit equal to 75 kilogram metres per second, or nearly 
 7*36 X 10* dyntachs. Each of these was supposed to re- 
 
I ♦''! 'f %^' 
 
 
 86 
 
 present the rate at which a good horse works, but is now 
 allowed to be too high. 
 
 96. The examples of energy we have hitherto taken as 
 illustrations are energies of systems, the motions and con- 
 figurations of whose parts are plainly visible to us. Our 
 grandest sources of energy are, however, derived from sys- 
 tems, the motions and configurations of whose parts are 
 invisible to us. Whence the energy which enables the 
 labourer to dig the ground, the student to pursue his 
 studies, or the horse to draw his load ? These are examples 
 o^ vital cneryy which the man and horse derive from the 
 food they eat and drink, and the air they breathe. The 
 energy of gunpowder, of steam, and of a voltaic battery are 
 other examples of what is called moleeular enery^y. 
 
 Food and fuel are our principal immediate sources of ener- 
 gy. Thus coal and the oxygen of the air form a system which, 
 before combustion, in virtue of the separation of the atoms 
 of coal and the atoms of oxygen, pos^sesses, potential energy of 
 atomic separation. During combustion the energy becomes 
 kinetic, and may be communicated to the water in a boiler 
 so as to heat the water and form steam, and through this 
 be used to drive an engine, and by means of the engine do 
 all sorts of mechanical work. Similarly, food and air form 
 a great store of potential molecular energy, which is trans- 
 formed during digestion into the vital energy by means of 
 which we do our daily work. 
 
 Heat, light, and electricity, in their physical aspects, are 
 well defined as forms of molecular energy. Sound forms a 
 sort of connecting link between visible and invisible or 
 molecular energy. The Tra/nafonnation of Energy is the 
 enunciation of the fact : 
 
S7 
 
 Any ono form f^f ''i^'ryy rnay he tran^fnvmed^ directly or 
 Indirectly, into an exact enH.'nudent of any otfier form. 
 
 It is principally plainly vlsihh' t'nenjy that is studied in 
 the science of dyn amies. 
 
 97. AinonfTSt the most important of the recent advances 
 in Physical Science is the measurement of the different 
 forms of molecular enerjj;y in dynami(;al units. Thus the 
 energy of a unU of heat., (the heat required to raise the 
 temperature of 1 gram of pure water from 4'C to 5®C), 
 is determined experimentally to be equal to 42 million ergs 
 nearly. 
 
 From such measurements the very important general- 
 ization, known as the Conservation of Enmfy^ has been de- 
 duced. This principle asserts : 
 
 Through whatever forma eneryy may pans^ it can i tot A^ 
 cJianged in quantity, and hence tfte tot<d energy in the uni- 
 verse remahis constant. 
 
 As the Conservation of Mass forms the foundation of 
 the science of modern chemistry, the Conservation of En- 
 ergy may be said to form the foundation of the science of 
 modern physics. 
 
 Although the total energy in the universe remains con- 
 stant, it is gradually being transformed into lower forms so 
 as to be less useful to man. This is the principle enun- 
 ciated by Sir W. Thomson, and known as the DeAjradatiim 
 of Energy : 
 
 The energy of the universe i^gradnally being transformed . 
 into a form in which it cannot he made ase of hy man, v/*., 
 that of uniformly diffused heat. 
 
 98. Perhaps the principle force through which energy is 
 
1 1 "■}./>' 
 
 88 
 
 being constantly dissipated, or degraded into the useless 
 form of diffused heat, is friction (art. 60). The direction of 
 this force is always diametrically opposite to the direction 
 of motion, or to that in which motion would take place 
 under the influence of the other actiuir forces. When the 
 surfaces between which friction is called into play are plane, 
 the laws of friction, as determined by experiment, are well 
 defined and may be enunciated thus : 
 
 1 . Thefrwtwn per unit of area is directly proportional 
 to thr. normal premun' iwr unit of area^ so long as the 
 surfaces In contact are similar aufl of the satne materials. 
 
 f-^fir. 
 
 2. When there Is sliding motion^ the friction Is indepen- 
 dent of the velocity. 
 
 The first of these laws is generally divided into two, 
 which may be enunciated thus: 
 
 ( 1 ) For similar surfaces qftfm sam*'. tnaterlals^ the total fric- 
 tion varies di/rectly as the total normal pressure. F-=. [iR. 
 
 (2) The total friction is independent (f the areas of the 
 surfaces in contact. 
 
 The maximum amount of friction is called into play 
 when motion i^just about to take place, being then goner- 
 ally greater than when motion actually takes place. The 
 constant /i, which measures the ratio of the friction to the 
 normal pressure, is then called the coefficient <f friction for 
 the two surfaces in question. Rankine has shown that the 
 value of /i lies between 0*2 and 0*5 for wood on wood, 0*2 
 and 0*6 for wood on metals, 0'3 and 0*7 for metals on stone, 
 and 0'15 and 0*25 for metals on metals. 
 
 On account of the dissipation of energy through friction 
 
89 
 
 and other causes, a machine does not do as much useful 
 work as the equivalent of the energy imparted to it. The 
 ratio of the useful work done to the energy supplied is 
 called the efficiency or modulus of the machine. The duty 
 of a steam-engine is the amount of useful work performed 
 per unit r jass of fuel consumed. 
 
 Examination XI. 
 
 1. Define energy and work. How is work physically 
 manifested ? Give examples. 
 
 2. What other property of matter is invariahly associated 
 with the possession of energy? 
 
 3. Define kinetic energy, and give examples of it. 
 
 4. Define potential energy, and give examples of it. 
 
 5. How is work measured ? Give examples. 
 
 6. Name and define the unit of energy and work. 
 
 7. Determine the kinetic energy of a body whose mass 
 is m and velocity v. 
 
 8. Prove that the potential energy of a body whose 
 mass is m, and height above the earth's surface A, is mgh. 
 
 9. Prove that when a body is moving vertically, under 
 no other force than its weight, its total e gy is indepen- 
 dent of its position. 
 
 10. Define a kilograrametre and foot-pound, and deter- 
 mine their values in absolute measure. 
 
 11. If a body of m pounds be moving with a velocity of 
 t) ft. per sec. ; find its kinetic energy in foot-pounds. 
 
IP 
 
 l.i 
 
 If. 
 
 90 
 
 12. Name and define the unit rates of working in abso- 
 lute and gravitation measures, according to both the C. G. S. 
 and F. P. S. systems. 
 
 13. Give various examples of molecular energy, both 
 kinetic and potential. 
 
 14. What are our principal immediate sources of energy ? 
 Explain your answer. 
 
 15. Define the unit of heat, and give its ineasuretnent in 
 dynamical units. 
 
 16. Enunciate the principles known as the Transforma- 
 tion, Conservation, and Degradation of Energy. 
 
 17. Enunciate the laws of friction for plane surfaces, and 
 define the coefficient of friction. 
 
 18. Define the modulus of a machine, and the dut}' of a 
 steam-engine. 
 
 Exercise XI. 
 
 1. How much work must be done to pump 1000 cub. ft. 
 of water from a mine 150 fathoms deep? 
 
 2. In pile-driving 30 men raised a rammer of 500 kilo- 
 grams through a height of 40 metres 12 times in an hour ; 
 find the rate of working per man ? 
 
 3. How many ergs are stored up in a mill-pond near 
 Kingston, Ont., which is 40 m. long, 20 m. broad, and 1 
 metre deep, and has a fall of 5 metres ? 
 
 4. A ball of 40 lbs. is moving at the rate of 300 miles 
 per hour ; find its kinetic energy in ft.-lbs. 
 
 5. A machine (modulus J) for raising coals is worked by 
 two horses ; how much coal will be raised in a day of 8 
 working hours from a pit 90 metres deep ? 
 
91 
 
 kilo- 
 lour ; 
 
 niilep 
 
 6. An engine is found to raise 5 tons of material per 
 hour from a mine 110 fathoms deep ; find the II. P. of the 
 engine, supposing J of its energy to be lost in unavoidable 
 resistances. 
 
 7. A railway train of 60 tons in passing over a certain 
 mile hj\s its velocity increased froin 40 to 50 miles per hour. 
 If tlie average friction be 10 Ibs.-wt. per ton, find the work 
 done by the engine in passing over the mile, and the kinetic 
 energy of the train at the end of the mile. 
 
 8. What must be the horse-power of an engine whose 
 modulus is J, working 8 hours per day, which supplies 3000 
 families with 100 gallons of water each per day, the mean 
 height to which the water is raised being 60 feet ? 
 
 9. How many bricks will a labourer raise to the mean 
 height of 20 ft., working 8 hours per day ; given that the 
 volume of 17 bricks is a cubic ft. and the mass 125 lbs., and 
 that the average rate of doing such work is 1200 ft.-lbs. 
 per minute? 
 
 10. If a load be 10 bricks (Ex. 9), and the man's own 
 mass 140 lbs., what is the rate at which he expends his vital 
 energy when working ? 
 
 11. What would be the cost per ton to raise coals from 
 a pit 25 fathoms deep, allowing $3 per day for a horse and 
 driver, and that the horse performs 22000 effective units of 
 work per minute, working 8 hours per day ? 
 
 12. At what rate will a train of 80 tons be drawn by a 
 locomotive-engine of 70 H.P., the frictional resistance being 
 10 Ibs.-wt. per ton ; and how far, after steam is shut off, 
 will it go before being brought to rest ? 
 
 13. If 8 Ibs.-wt. per ton (Ex. 12) be the average frictional 
 
rnr^ 
 
 92 
 
 reeistance until full speed is attained, how long will it take 
 for the train to attain its maximum speed after starting; 
 and how far will it have travelled in this time ? 
 
 14. In what tinu will a locomotive of 100 force-de-cheval , 
 drawing a train of 100 tonnes, complete a journey of 100 
 kilometres, supposing that the frictional resistance until 
 full speed is attained, and after steam is shut off until it 
 stops, be on the average 3 kilograms-weight per tonne, and 
 after full speed is attained, 4 kilograms- weight per tonne. 
 (1 tonne = 10* grams). 
 
 15. According to Navier it requires 43333 kilogram- 
 metres to saw tlirougli a square metre of green oak ; how 
 many ares of oak planking will an engine of 5 force-de-cheval 
 cut in a day of 8 hours ? 
 
 16. If 5 H. P. be applied to a machine which lifts 3000 
 cub. ft. of water per hour to the height of 40 feet, what is 
 the modulus of the machine? 
 
 17. The section of a mill stream is 4 ft. by 2 ft., the mean 
 velocity 20 ft. per min., and the height of the fall 15 ft. ; 
 what will be the H. P. of the water-wheel whose modulus 
 is 0*7, and how many cub. ft. of water will the wheel pump 
 per day to the height of 100 ft. ? 
 
 18. The mean annual rainfall over the earth's surface is 
 estimated at about 5 feet, and 1000 feet the mean height 
 from which rain falls ; calculate the sun's horse-power used 
 in merely raising the water in the process of evaporation. 
 (Earth's mean radius = 20902070 feet). 
 
 19. Determine the H. P. of the river Niagara which has 
 a total descent of 334 feet, and discharges about 4 X lO** tons 
 of water per hour. 
 
93 
 
 20. A ball of 10 kilograms is fired from the mouth of a 
 cannon 3 metres long with the velocity of 15 kilotachs ; find 
 the mean pressure of the gaseous products on the ball. 
 
 21. There were 4000 cub. ft. of water in a mine of depth 
 60 fathoms, when an engine of 70 H. P.began to work the 
 pump ; the engine worked for 5 hours before the mine was 
 cleared of the water ; if the modulus of the engine were J, 
 find the rate at which water was entering the mine. 
 
 22. A steam-engine raises 70 cub. ft. of water per min. 
 from a depth of 800 ft. ; how many tons of coal are burned 
 per day, supposing the duty of the engine to be ^ million 
 ft. -lbs. per lb. of coal. 
 
 23. Find in dyntachs the rate at which a fire-engine 
 works, which discharges 10 kilograms of water per second 
 with a velocity of 1500 tachs. 
 
 24. A railway carriage of two tons mass is started on a 
 level railroad with a velocity of 8 ft. per sec, and moves 
 over 400 ft. before it stops ; determine the coefficient of fric- 
 tional resistance. 
 
 25. A cistern is 10 ft. long, 7 ft. broad, and 8 ft. deep. 
 The height of the bottom of the cistern from the water in 
 the well is 56 ft. If a man can work with a pump at the rate 
 of 2600 ft. -lbs. per minute, and the modulus of the pump 
 is 0*66, how long will he take to fill the cistern ? 
 
 26. A spring tide raises the level of the river Thames, 
 between London and Battersea bridges, on an average 16 
 feet. If 5 miles be the distance between the bridges and 
 900 feet the mean breadth of the river, find the potential 
 energy of the spring tide when full. 
 
94 
 
 Ghaptbb XII. 
 
 
 Action and Reaction. 
 
 99. It has been pointed out, that whenever a force acts, 
 there are always two bodies concerned. We generally speak 
 of one of the two as receiving a change of momentum, and 
 of the other as being concerned in the production of this 
 change. Newton clearly pointed out in his third law of 
 motion that the action was a mutual one ; that change of 
 momentum was received by both bodies, equal in magni- 
 tude but diametrically opposite in direction : 
 
 To every action there is always an equal and c&ntrary 
 reaction / or, the mutual addons of any two bodies are al 
 ways equal (in magnitude) and opposite in direction. 
 
 The word force is properly used when we consider the 
 effect of the action between any two bodies in changing the 
 momentum of one of them only. Stress is a term applied 
 to the mutual action between any two bodies, when there 
 is special reference to the dual character of that action, as 
 enunciated by Newton. This third law tells us that all 
 dynamical actions between bodies are of the nature of 
 stresses. When a body falls to the ground under the action 
 of its weight, the earth rises to meet it with an equal mo- 
 mentum. Since, however, the mass of the earth is so very 
 much greater than that of any body on its surface, the mo- 
 tion of the earth is so small that it may be neglected. 
 When two like magnetic poles, free to move, are brought 
 near one another, it will be found that the repulsion is 
 mutual. When the loadstone attracts a piece of iron, the 
 iron attracts the loadstone with an exactly equal force. 
 When the table is pressed by the hand, we feel that the 
 hand is likewise pressed by the table. When a horse draws a 
 
96 
 
 canal boat by means of a stretched rope, the horse is drawn 
 backwards as much as the boat is drawn forwards. This 
 may easily be proved by cutting the rope, when immediately 
 the horse falls forwards. This is further easily seen, when 
 we reflect that relatively to the boat the horse does not 
 move at all. According to Archimedes' principle, any body 
 immersed in a fluid is subjected to a vertically upward 
 pressure equal to the weight of the fluid displaced. The 
 fluid, on the other hand, is subjected to a vertically down- 
 ward pressure equal to the weight of the fluid displaced. 
 This can be easily shewn experimentally by balancing a 
 vessel filled with water in a common balance, and immer- 
 sing in the water a body held by a cord. The equilibrium 
 will be immediately destroyed, and the force necessary to re- 
 store equilibrium will be found to be equal to the weight of 
 the water displaced. When two railroad trains or other 
 bodies collide, the change of momentum in the one is just 
 equal, and opposite in direction, to the change of momentum 
 in the other, whatever be the original direction or rate of 
 motion of either train. 
 
 100. Since momentum has direction as well as magni- 
 tude, it at once follows from the above law, that the total 
 momentum of two bodies is not altered by their mutual ac- 
 tion. From this the important principle called the Con- 
 servation of Momentum is at once deduced : 
 
 The total momentum of any hotly ^ or systevi of hodiea^ 
 rannot be altered hy the mutual artnms of Ux several j>arf.s. 
 
 As an illustration of this principle let us consider the A*^»/' 
 of a gun. Here we have a system consisting of 3 bodies, the 
 gun, the gas formed from the gunpowder, and the ball ; it 
 will be at once seen that the backward momentum of the gun 
 is just the equivalent of the forward momentum]of the ball. 
 
mm 
 
 mmm 
 
 ' 
 
 96 
 
 The total mometitwm of the universe, wf a conniiint quantity 
 is an immediate deduction from the same principle. 
 
 101. The Conservation of Momenturti teaches us that 
 (jhange of momentum in a body or system of bodies must 
 be produced by forces external to the body or system. Let 
 any forces act upon a body of mass m and produce in it an 
 acceleration a, then ma is the measure of the single force 
 which would produce the same dynamical effect on the 
 body. If, after Newton, we call a force measured by -ma 
 the resistance to acceleration^ which the body offers in virtue 
 of its mass and inertia, then D* AUmher€s Principle at once 
 follows as a corollary to Newton's third law : 
 
 The external forces acting upon a body {or system of 
 hodi€s\ togetJier with the resistance {or resistances) to accel- 
 eration, form a system rf forces in equilihiium. 
 
 This principle evidently amounts to saying that the 
 molecular or internal forces acting within a body or system 
 of bodies are themselves a system of forces in equilibrium. 
 
 102. Newton published his axioms or laws of motion in 
 his celebrated work " Principia Philosophiae Naturalis ". 
 In the scholium appended to his third law he points out that 
 an additional meaning may be attached to the words ac- 
 tion and reaction besides that of force. Tait and Thomson 
 in their treatise on Natural Philosophy have recently 
 brought this passage to light and thus translated it : 
 
 If the action of an argent he measured by the product of 
 its force into its mloeity ; a,nd if, si^nilarly, the reaction 
 of the resistance be measured by the velocities of its several 
 parts into their several forces, whether these arise from 
 friction, cohesion, weight, or acceleration f' action and re- 
 action, in all combinations of machines, will be equal and 
 opposite. 
 
 :m 
 
97 
 
 As pointed out by Tait and Thomson, this remarkable 
 passage contains in it nothing less than the foundation of 
 that great modern generalization, the Conservation of En- 
 ergy. 
 
 103. Ti'^o heavy bodies are connecled hy an inextensible 
 string which passes over a fixed smooth peg, {or p^dly, as 
 in Ait/wood's machine) ; required to determine the tension 
 of tJie string. 
 
 Let T denote the tension of the string, m and ni the 
 masses of the bodies, m being the greater. Since the ten- 
 sion of the string is the same throughout by Newton's third 
 law, the acceleration of the heavier body will be {mg - 7^ 
 -^wi downwards, and of the lighter body {T—m'g)-^7r^ 
 upwards ; since these must be equal, 
 
 T T _ 2mm' 
 
 m 
 
 = m'-^' 
 
 T= 
 
 Cor. The acceleration = 
 
 mg-T 
 
 m + m 
 T 
 
 9 
 
 m 
 
 or 
 
 rng 
 
 m.-m 
 
 m 
 
 m-\-7n ^ 
 
 as already proved (art. 70). If m = m', the tension of the 
 string is wg, and there is no acceleration, so that the bodies 
 must either be at rest or moving with uniform velocity. 
 
 The above completes the solution of the problem of Att- 
 wood's machine (art. 70), when the weight of the string, 
 the pully's mass, and the friction may be neglected. 
 
 104. As an additional illustration of Newton's third law 
 let us consider one of the very simplest cases of impulsive 
 force (art. 57), viz., the direct impact of two spheres of uni- 
 form density without rotation. If the centres of two 
 spheres move in the straight line joining them, and one im- 
 pinges on the other, the impact is called direct ; otherwise, 
 the impact is called oblique. 
 

 98 
 
 Denote by Wj, 7/?j, the masses of the spheres, and by 
 ?^i, ^8» their velocities before impact. If tlie direction of 
 u^ be called + , -j/g will be 4- or — according as m^ is or- 
 iginally moving in the same or opposite direction to niy^. 
 The action which takes place during impact may be ex- 
 plained thus: 
 
 a). Alterations of form and volume take place by work 
 being done against the molecular forces, until the relative 
 velocity of the two bodies is destroyed. If l{ denote the 
 total force called into play during this first stage of the im- 
 pact, and V denote the common velocity, we get from New- 
 ton's second and third laws 
 
 whence v = ; — — .... 
 
 m^ -f-z/ij 
 
 • • • • 
 
 and R = 
 
 
 O'l-'^a) 
 
 {a) 
 (1) 
 
 (2) 
 
 '\ ~ "'2 
 
 These equations contain the complete solution of the prob- 
 lem, if the bodies do not separate again after impact. This 
 will be the case, when the force of adhesion between the 
 bodies counterbalances the force of elasticity^ which tends 
 to separate them. 
 
 h). If the bodies be sufficiently elastic, they have the com- 
 mon velocity v only for an instant, for an amount oi potential 
 energy of molecular separation has been stored up in con- 
 sequence of the change of configuration of the material 
 particles of each sphere, and, in the transformation of this 
 energy into the kinetic form, the molecular forces continue 
 acting, until the original forms and volumes are as much 
 as possible restored. During this second stage of the im- 
 pact it is evident that the bodies receive accelerations of 
 
(a) 
 (1) 
 
 (2) 
 
 99 
 
 momentum in the same directions as during the first stage 
 and if ^ denote the total force called into play, 
 
 Ii' = m^{v~v\)=.m,{v,~v) ^^) 
 
 Now it has been proved by experiment, that if the im- 
 pact do not make any sensible permanent alteration of form 
 the relative velocity of the bodies after impact bears a conl 
 stant ratio to the relative velocity before impact, i.e. v^-v 
 ---e{u,-u^\ where ^i8 a proper fraction, whose value 
 depends only upon the material natures of the spheres 
 From this and equations (a) and (/>) we deduce by alge- 
 braical analysis ^=t^^. Also, 
 
 ^1=^1- 
 
 m. 
 
 ^2='^2 + 
 
 nil 
 
 nil -f ^2 
 
 
 E-\-R' = 
 
 m^m.^jX+e) 
 
 m. 
 
 T^^ (^i-^a) 
 
 (3) 
 (4) 
 (5) 
 
 The value of e was found by Newton to be f for balls of 
 compressed wool and steel, f for balls of ivory, and \t for 
 balls of glass. It is called by most writers the coefficknt of 
 elasticity, a name strongly objected to by Tait and Thom- 
 son, who call it the coefficient of restitution. When ^= 1 
 the bodies are called perfectly elastic, a condition never 
 perfectly realized; when ^ = 0, the bodies are called in- 
 elastic. 
 
 Cor. 1. If mj, = Qo, and ./^ =0, the case is that of a 
 sphere impinging normally on a fixed plane. The equa- 
 tions (3), (4), (5), become then 
 
 Cor.2. If/^^i=m2,and^ = l,theni;i = f,,,andi', = wi, 
 i,e, the bodies interchange velocities. This may be shown 
 
m^ 
 
 100 
 
 experimentally to be nearly the case for balls of ivory or 
 glass. Also, if ^2=0, and m^^em^, theniJi=0, and 
 
 105. The following results are at once deduced from the 
 preceding investigation : 
 
 1 . Whether the bodies he elastic or not, the total momen- 
 turn, is not affected hy the impact. 
 
 This is just a particular case of art. 100. 
 
 2. The total visible kineJc energy after impact is less 
 than before impact. 
 
 7/7 772* 
 
 and ^rn^v^^ -'r^m^v^^ = 
 
 imiWi^-f^y/iat^^a-J^^^— j-^^(l_e«)(w, -^2)2. 
 
 What becomes of the visible kinetic energy lost f It is 
 transformed into the molecular kinetic energy of heat, so 
 that the bodies after impact are warmer than before impact. 
 
 Examination XII. 
 
 1. Enunciate Newton's third law of motion, and give 
 numerous illustrations of it. 
 
 2. What is the diflFerence between the meanings of the 
 terms force and stress ? 
 
 3. Enunciate and prove the Conservation of Momentum. 
 
 4. Explain the kick of a gun after it is fired. 
 
 5. Enunciate and explain D'Alembert's principle. 
 
101 
 
 6. How can it be said that Newton in his third law laid 
 the foundation of the science of energy ? 
 
 7. A string passing over a smooth peg connects two 
 heavy bodies ; determine its tension, 1) when the bodies 
 have different weights, 2) when the weights are the same. 
 
 8. Two spheres of uniform density without rotation im- 
 pinge directly; describe the nature of the impact, and de- 
 termine the equations of motion. 
 
 9. What is denoted by e in the theory of impact? How 
 is it experimentally determined? Give its value for cer- 
 tain substances. 
 
 10. How do we deduce the equations of impact of a 
 sphere on a fixed plane. Give the equations. 
 
 11. Determine under what conditions will two spheres, 
 impinging directly, interchange velocities ? 
 
 12. Prove that the momentum of a system of uniform 
 spheres without rotation is not altered by direct impacts of 
 its component parts. 
 
 13. Determine the loss of visible kinetic energy when 
 two spheres impinge directly. What becomes of it ? 
 
 Exercise XII. 
 
 1. A boulder of 2 tonnes mass is rolled from the summit of 
 El Capitan in the Yosemite valley, a rock rising vertically 
 through the height of 3000 feet; find the velocity of, and 
 distance travelled by, the earth when the boulder strikes 
 the ground, in virtue of their mutual attraction. (See 6, 
 Ex. V). 
 
 2. An 81 ton gun is charged with a shot of 100 lbs. 
 mass ; if the ball leave the gun with a velocity of 200 ft. 
 per sec. ; find the velocity of recoil of the gun. 
 

 i. ■■■ 
 
 102 
 
 3. Find the tensions of the strings in 8, 9, 23, of Ex. VII. 
 
 4. Find tlie pressures on the water in 2, 4, 5, of Ex. VIII. 
 
 5. Prove that in Attwood's !nac5hine, if the total mass of 
 the moving bodies be constant, the greater the tension of 
 the string is, the less is the acceleration. 
 
 6. A body of 5 kilograms, moving with a velocity of ii 
 kilotachs, impinges on a body of 3 kilograms moving with 
 a velocity of 1 kilotach ; If e = J, find the velocities after 
 impact. 
 
 7. Two bodies of unequal masses, moving in opposite 
 directions with momenta equal in magnitude, meet; shew 
 that the momenta are equal in magnitude after impact. 
 
 8. Two wooden balls whose masses are 12 and 16 ozs. 
 are made to impinge directly on one another. One of the 
 balls is furnished with a spike to prevent the rebo md. If 
 the velocity of the lighter be 12 ft. per sec, what must 
 that of the larger be that the motion may be destroyed by 
 the impact. 
 
 9. The result of an impact between two bodies moving 
 with equal velocities in opposite directions, is that one of 
 them turns back with its original velocity, and the other 
 follows it with half that velocity ; find e and the ratio of 
 the masses. 
 
 10. A bomb-shell moving with a velocity of 50 ft. per 
 see. bursts into two parts whose masses are 70 and 40 lbs. 
 After bursting, the larger part turns back with a velocity of 
 10 ft. per sec. ; find ths velocity of the smaller part. 
 
 11. A and B are two uniform spheres of the same ma- 
 terial and of given masses. If A impinges directly upon a 
 
after 
 
 103 
 
 third sphere C at rest, and then C on B at rest, find tlie 
 mass of O in order that the vehKtity of B may he tl«e great- 
 est possible for a given initial vrdocity of .4. 
 
 12. Find the necessary and sufficient condition that one 
 body moves after direct impact with the original velocity 
 of the other. 
 
 13. Two bodies, wluse masses are as 10 to 1, start from 
 rest and move under the action of their mutual attraction. 
 At one instant the velocity of the greater mass is 100 ; find 
 the velocity of the smaller mass at the same instant. 
 
 14. A number (n) of balls whose masses form a G. P. 
 whose common ratio is 1 -^ e, hang close to one another 
 with their centres in one line. The first is then made to 
 strike the second with a given velocity, in consequence of 
 which the second strikes the third, the third the fourth, &c. ; 
 find the resultant motions of the balls. 
 
 15. A strikes B, which is at rest, and after impact re- 
 bounds with a velocity equal to that of B ; shew that ^'s 
 mass is at least 3 times ^^s mass. 
 
 16. When two uniform spheres impinge directly, find 
 the changes of visible kinetic energy during each of the 
 stages of impact. 
 
 17. If the snm of the masses of the impinging bodies be 
 constant, find when the loss of visible kinetic energy will 
 be a maximum ; what will the loss then be ? 
 
 18. Two cannon balls of masses 20 and 50 lbs. meet one 
 another with velocities of 40 and 20 ft. per sec. respectively; 
 find their velocities and momenta after impact, (^ = ^), and 
 the visible kinetic energy lost by the impact. 
 
^p 
 
 I? ' 
 
 104 
 
 
 ''*>. 
 
 Chapter XIII. 
 
 Dimenmonal Equations, 
 
 106. In the previous pages the student has been intro- 
 duced to two distinct scientific systems of units, called the 
 C. G. S. and F. P. S. systems respectively. In both sys- 
 tems three independent or fundamental units are chosen, 
 and from these all others are derived. It is not necessary 
 that any three special units must be taken as the funda- 
 mental ones. The three, however, which are most easily 
 fixed upon : and with standards of which, comparisons are 
 most easily and directly made, at all times and at all places : 
 and in relation to which the derived units are most easily 
 defined, and are of the simplest dhiensi/yns /in virtue of the 
 established rel.ations between the different units : are the 
 units of lentrth (or distance), mass, and time. 
 
 The relations between the derived and fundamental units, 
 already expressed by algebraical formulae, leads us to the 
 the consideration of dimensiondl equations, or such as ex- 
 press how the derived units depend upon the fundamental 
 units. 
 
 Whatever units of length, time, and velocity be used, 
 F QC Z -T- 7", where V measures the velocity of a body 
 moving uniformly, and L is the distance passed over by the 
 body in the time T. Now if we take the unit of velocity 
 as that in which unit of length is passed over in unit of 
 time, the relation is expressed thus, Y - L -~ T. Hence 
 if v, Z, t, denote the units of velocity, length, and time in a 
 scientific system, v — I ~-t. This is called a dimensional 
 equation. It tells us that the unit of velocity involves the 
 unit of length to the first power direetly, and the unit of 
 time to the first power invet'sdy. 
 
105 
 
 Similarly, if w, a,/, w, h, denote respectivolj the units 
 of mass, acceleration, force, work, nnd rnte of workini,', in 
 a scientific system, 
 
 I ml »»/2 „.. y^^2 
 
 V 
 
 
 /^ = 
 
 I/; 
 
 «5 " 7» 
 
 The last equation tells us that the unit of rate of work- 
 ing involves the unit of mass to the first power, and the 
 unit of length to the second power, (Ihectly^ and the unit 
 of time to the third power inverstUj. 
 
 arc 
 
 / 
 
 If^ denote the unit of angle, ^= ',. =-.— /o •. Hw. 
 
 ^ ' radius / ~ ' '•'• ^'^^ 
 
 unit of angle is independent of the fundamental units. 
 If s, p, denote the units of surface and pressure-intensity, 
 
 / 
 
 m, 
 
 U b, (/, denote the units of bulk and density, 
 h =/3^and./=-y=-^3- 
 
 In whatsoever way the dimensions of a derived unit be de 
 duced, they must of necessity always be the same. Thus 
 it has been proved that the visible kinetic energy of a body 
 whose mass is M and velocity Fis ^ MV\ therefore the 
 dimensions of energy or work must be 7nv^, i.e. nil^ -h j!2, 
 as above proved. Again, the unit of force is that which' 
 generates unit of momentum {mv) in unit of time ; there- 
 fore the dimensions of force are mv-^t, i.e. ml-f^^ as above 
 proved. 
 
 107. An important use of dimensional equations is to 
 facilitate the calculation of the numerical relations between 
 the derived units of different systems, when the numerical 
 
 
 

 pi . 
 
 I' I 
 
 km 
 
 I .i•i'^;'■- 
 
 ■' ■'f 
 
 t 
 
 106 
 
 relations between the fundamental units are known. Thus 
 if ^', m', t' denote the fundamental units in the F. P. S. sys- 
 tem, and p the derived unit of pressure-intensity, 
 
 , m' m p m I 
 
 P = xfi^ P ='lt^^ •'• p = m'T- ' t 
 
 = 453-593 X 0-0328087 = 14-8818 (see tables art. 108), *>. 
 1 poundal per square foot = 1488 18 prems. 
 
 Ex. Find the fundamental units in a scientific system in 
 which a mile per hour is the unit of velocity, a pound- 
 weight the unit of force, and a foot-pound the unit of work. 
 
 Let Z, Jf, T, denote the fundamental units : 
 
 L 
 
 5280 r 22 V 
 
 
 T - 
 
 3600* ^' ~ 15 • «' •••• 
 
 • •• •••• \^ J 
 
 ML 
 
 m'V 193 m'l' 
 
 3-^ X ^^2 — g • ^'3 • • • 
 
 (2) 
 
 ML'' 
 
 193 m7'2 
 
 
 Tt - 
 
 /% • tf a •• «••• ■••• 
 
 (3) 
 
 .'. L 
 
 15 15 
 -i" - I foot, T- 22< - 22 
 
 second, and 
 
 108. The following tables give the numerical relations 
 between the C. G. S., F. P. S., and a few other frequently 
 occurring units. The numbers in the tables of length and 
 mass give the results of the most accurate observations made 
 in the comparisons of the French and English standards of 
 measurement. Those in the other tables are calculated 
 from the dimensional equations of the units, as explained 
 in last article. Each number is true to the last decimal 
 place given, and the mantissae of the logarithms of the true 
 ratios are added. 
 
 ■; ^^yjiil 
 
1«»7 
 
 /. Length or Distmu'f. 
 
 (1) 
 (2) 
 (3) 
 
 1 inch 
 
 1 foot 
 
 1 yard 
 
 1 mile (statute) 
 
 1 decimetre 
 1 metre 
 1 kilometre 
 
 1 square inch 
 1 " foot 
 1 acre 
 1 square mih' 
 
 
 Mantissae. 
 
 2-53998 centimetres 
 
 4048298 
 
 30-4797 " 
 
 4840111 
 
 91-4392 
 
 9611324 
 
 1»;0933 
 
 2066451 
 
 3-93704 inches 
 
 5951702 
 
 3-28087 feet 
 
 5159889 
 
 0-62138 mile 
 
 7933549 
 
 //. Area or Surface. 
 
 - 6-45148 square centime's 8096597 
 
 = 929-014 '' " 9680222 
 
 = 40-4678 ares 6071101 
 
 = 2-58994 square kilometres 4132901 
 
 1 square decimetre 
 
 1 are 
 
 1 square kilometre 
 
 1 
 
 (( 
 
 u 
 
 = 15-5003 square inches 1903403 
 
 = 1076-41 " feet 0319778 
 
 = 247-110 acres 3928899 
 
 = 0-38611 square mile 5867099 
 
 ///. Volume^ Bulk, or Capacity. 
 
 1 cubic inch 
 1 " foot 
 
 1 gallon 
 
 1 litre 
 
 1 decalitre 
 1 « 
 
 = 16-3866 cubic centimetres 2144895 
 
 = 28-3161 litres 4520332 
 
 = 4-54102 " 6571531 
 
 = 61-0254 cubic inches 7855105 
 
 = 0-35316 cubic foot 5479668 
 
 - 2-20215 gallons 3428469 
 
ml 
 
 
 >^ "' 
 
 
 1 ■'■• ■ 
 
 • 
 
 
 1 degree, or 1 
 
 
 1 right angle 
 
 .^ ■■. ' 
 
 1 radian 
 
 1 ' ' ' 
 
 T 
 
 W'il ' ' ■ 
 
 I -^;r 
 
 
 n^ 
 
 • «•'.* 
 
 
 Tt' 
 
 1 grain 
 
 1 ounce avoir, 
 1 pound " 
 1 ton 
 
 1 gram 
 1 kilogram 
 1 tonne 
 
 108 
 / V. A nyfe, 
 
 = 0-0174532925 radian 
 = 1-5707963268 " 
 = 57-295779513 degrees 
 
 = 31415926536 
 
 = 0-3183098862 
 
 = 9-8696044011 
 
 = 0-1013211836 
 
 V. Ma/i8, 
 
 = 0-064799 gram 
 
 = 28-34954 " 
 
 = 453-5927 " 
 
 = 1016048 " 
 
 = 15-43235 grains 
 
 = 2'204621 lbs. avoirdupoif 
 
 = 0-984206 ton 
 
 V/. Density. 
 
 Mantisrtae. 
 2418774 
 1961199 
 7581226 
 
 4971499 
 5028501 
 9942997 
 0057003 
 
 8115679 
 4525461 
 6566661 
 0069141 
 
 1884321 
 3433339 
 9930859 
 
 1 lb. avoir, per cub. ft. = 0-01602 gram per cub. cm. 
 1 gram per cub. cm. = 62-4262 lbs. av. per cub. ft. 
 
 VII. Thne. 
 
 1 day (mean solar) = 86400 seconds 
 
 1 sidereal day = 86164-1 " 
 
 1 mean sidereal month = 2360591-5 " 
 
 1 " " " = 27-321661 days 
 
 1 mean synodic " = 29*530589 " 
 
 1 sidereal year =31558149*6 seconds 
 
 " " = ( 365*2564 days 
 
 1 mean tropical year = 365*2422 
 
 i( 
 
 2046328 
 7953672 
 
 9365137 
 9353264 
 3730210 
 4365071 
 4702721 
 4991116 
 5625978 
 5625809 
 
109 
 
 Note. A solar day is the time in which tlie snn ap- 
 parently revolves around the earth. A sidereal da^, is the 
 time of the apparent rotation of the sphere of the lieavens 
 A. sidereal month is the time in which the moon makes a 
 complete revolution in the sphere of the heavens amon-^st 
 the fixed stars. A synodic month is the time between two 
 consecutive full moons. A sidereal year is the time in 
 which the sun apparently makes a complete revolution in 
 the sphere of the heavens amon^rgt the fixed stars. A 
 tropical year is the time between two consecutive appear- 
 ances of the sun on the vernal equinox, one of the points in 
 which the equinoctial cuts the ecliptic; it governs the return 
 of the seasons, and varies slowly through a maximum range 
 of about a minute on each side of the mean value The 
 student will do well to satisfy himself that a real positive 
 ( + ) rotation of the earth would produce an apparent my. 
 atwe ( - ) rotation of the sphere of the heavens, and apos- 
 itive{ + ) revolution of the earth around the sun would pro- 
 duce an aip^ixrent positive { + ) revolution of the sun around 
 the earth. It follows from this that the number of sidereal 
 days in a sidereal year exceeds the number of mean solar 
 days by unity ; whence the relation between these days. 
 
 VI/I. Velocity. 
 
 Mantissae. 
 = 30-4797 tachs 4840111 
 
 = 44-7036 " 6503425 
 
 = 22-^15o^l•46ft. per sec. 1663314 
 = 3-28087 ft. per sec. 51598S9 
 
 1 kilometre per hour = 250-J-9 or 27*7 tachs 4436975 
 
 /JT. Momentum. 
 1 lb. 1 ft. per sec. = 13825-4 gramtach 1406772 
 
 1 gramtach = 607578 grains 1 in. per sec. 7836022 
 
 1 foot per second 
 1 mile per hour 
 1 " t( 
 
 1 hectotach 
 

 Ktf. 
 
 
 W''' 
 
 w 
 
 110 
 
 X. Force {tahing g = 980*54). 
 
 '■■ i" "' t 
 
 
 
 
 Mantissae. 
 
 'j ■'i- .■■ 
 
 1 poundal 
 
 = 
 
 13825-4 dynes 
 
 1406772 
 
 iki 
 
 1 megadyne 
 
 = 
 
 72-3307 poundals 
 
 8593228 
 
 ;' '^■''■■. ", 
 
 1 kilodyne 
 
 z=z 
 
 15-7386 grains- weight 
 
 1969668 
 
 
 1 
 
 ^ 
 
 10 1985 gram-weight 
 
 0085347 
 
 
 1 grain-weight 
 
 = 
 
 63-5380 dynes 
 
 8030332 
 
 ' ► ■■- 
 
 1 pound-weight 
 
 = 
 
 444766 " 
 
 6481314 
 
 XI. Pressure-intensity. 
 
 1 poundal per sq. ft. =14-8818 prerns 1726550 
 
 1 decaprem .-=(>67196 poundals per sq. ft. 8273450 
 
 1 Ib.-wt. per sq. in. = 70-3083 grams-wt. " cm. 8470064 
 
 1 ton-wt. per sq. ft. = 15*5555 Ibs.-wt per sq. in. 1918855 
 
 1 mean atmosphere =1*01360 megaprem 0058672 
 
 ( = 76 em. of mercury = 1033"0 gram-wt. per sq.cm.0 142248 
 at 0® at the latitude = 146967 Ibs.-wt. per sq. in. 1672184 
 of Paris) =094478 ton-wt. per sq. ft. 9753329 
 
 X/I. Work and Energy. 
 
 1 foot-poundal 
 1 million ergs 
 1 foot-pound 
 
 1 
 
 1 
 1 
 
 9 
 
 kilogrammetre 
 
 = 421394 ergs 
 
 = 2-37308 foot-poundals 
 
 = 0*13825 kilogrammetre 
 
 = 7*23307 foot-pounds 
 
 Watts' horse-power = 745599 megadyntaphs 
 force-de-cheval = 7354*05 " 
 
 = 980*54 tachs per sec. at Kingston, Ont. 
 = 980-61 tachs per sec. at lat. 45°, and the 
 
 mean value over the earth's surface. 9914963 
 
 = 980-94 tachs per sec. at Paris. 9916424 
 
 = 32^ ft. per sec. per sec. at Kingston, Ont., 
 
 and the mean value over the earth's surface 5074061 
 = 32*2 ft. per sec. per sec. the mean value in 
 
 the British Isles. 5078559 
 
 6246883 
 3753117 
 1406772 
 8593228 
 
 8725052 
 8665266 
 
 9914653 
 
' 
 
 111 
 
 Examination XIII. 
 
 1. What determines the choice of fundamental units? 
 
 2. Why is the French metliod of forming multiples and 
 submultiples of standard units the best? 
 
 3. Define a dimensional equation. Write down the dimen- 
 sional equations of ancrular velocity, momentum, energy, 
 angle, pressure-intensity, and density. 
 
 4. Determine the ratios of the units of acceleration, 
 angular velocity, density, and the gravitation units of the 
 rates of doing work, in the F. P. S. and C. G. S. systems. 
 
 6. Define the following terms : mean solar day, sidereal 
 day, sidereal month, synodic month, sidereal year, tropical 
 year, equinox, equinoctial, ecliptic. 
 
 6. How is the ratio of the sidereal day to the mean solar 
 day determined ? Calculate the ratio. 
 
 7. Check all the ratios given in tables II., III., VI., and 
 VIII. to XII. of art. 108. 
 
 Miscellaneous Examples. 
 
 1. In a scientific system, the unit of velocity is 1 kilo- 
 metre per hour, the unit of acceleration is g (981), and the 
 unit of force is the weight of a kilogram ; find the funda- 
 mental units, and the substance of unit density, in terms of 
 the C. G. S. units. 
 
 2. A saw-mill was driven by an engine of 3 H. P., and 
 in 10 minutes 12 square feet of green oak were sawn by the 
 mill; find the modulus of the mill. (See 15, Ex. XI). 
 
 3. Name the absolute and gravitation units of weight, 
 according to both the C. G. S. and F. P. S. systems. 
 
w 
 
 
 w 
 
 rt/'' 
 
 mm 
 
 f?.^«!'- 
 
 112 
 
 4. Two bodies of 9 and 5 grams draw by their weiglit a 
 body of 7 grams over a smooth pully. After moving for 
 2 seconds, the 9 grams are removed without disturbing the 
 motion ; find how long the 7 grams will continue to rise. 
 
 5. If a metre be the unit of length, 10""* of a day the 
 unit of time, and a kilogram the unit of mass ; find in 
 dyntachs the derived unit rate of working, and in j3rems 
 the unit of pressure-intensity, 
 
 6. A solid body balances 108 grams in air, 71 in water, 
 and 76 in alcohol ; find to an approximation of the second 
 degree the specific gravities of the solid body and alcohol, 
 the specific gravity of the standard masses being 8'4. 
 
 7. Oxygen at 0° and 76 cm. pressure has density 11056 
 with respect to air; find its density at 100° and 70 cm., 
 1) with respect to air at 0° and 76 cm., 2) with respect to 
 air at 100° and 70 cm. 
 
 8. A flask of 2 litres capacity was found to balance 1*6 
 grains more, when filled with carbonic acid, than when filled 
 with air at 0°; find the presure of the atmosphere, given 
 the specific gravity of air and of carbonic acid (p. 71), 
 
 9. Define equal timeSy and state in terms cf your defi- 
 nition the great physical truth contained in Newton's First 
 Law of Motion. 
 
 10. Given the densities of air and oxygen (p. 71), find at 
 what pressure the density of oxygen at - 30 °C will be 1*6 
 with respect to air. 
 
 11. A cube of metal has an edge of 1 metre, and density 
 10 ; find its volume, mass, and weight at Washington. 
 
 12. Find in dynes the apparent weight of a body of 240 
 grams mass and of volume 1 litre, when weighed in air 
 at - 30° C and 80 cm. pressure at Kingston, Out. 
 
113 
 
 240 
 air 
 
 13. When a body is moving in a curved path with a 
 variable velocity and variable acceleration, what is meant 
 by its direction of motion, velocity, and acceleration at any 
 instant ? 
 
 14. The weight of a body at Paris is 98094 dynes, what 
 would be the mass of that body at the surface of the sun ? 
 Would the weight there be the same as at Paris ? 
 
 15. A sunken vessel, whose bulk is a megalitre and m".s8 
 10* kilograms, is to be raised by attaching water-tight 
 barrels to it. If the mass of each barrel be 30 kilograms, 
 and the volume a kilolitre, find how many will be required. 
 
 16. If a mercurial barometer of 1 sq. in. section stand at 
 30 inches, what will be the height of a sulphuric acid bar- 
 ometer of section 1 -i- 1'84 sq. in. ? Given the densities of 
 mercury and sulphuric acid (p. 71). 
 
 17. Find the height of the water barometer under the 
 mean atmospheric pressure, when the temperature is 15 ® C ; 
 given the tables at pp. 71 and 76. 
 
 18. If the temperature be 15® at Edinburgh, and the 
 coefficients of dilatation of the barometer scale and mercury 
 be 1-8 X 10-» and 1-794 X 10"* ; find the barometric read 
 ing when the pressure of the atmosphere is a megaprem. 
 
 19. What effect has the dilatation of the glass tube on 
 the height of the barometric column ? Explain your answer. 
 
 20. If a force-de-cheval be the unit rate of working, g the 
 unit of acceleration, and the weight of a kilogram the 
 unit of force ; find the units of pressure-intensity and mo- 
 mentum. 
 
 21. A shaft a metres deep is full of water ; find the depth 
 of the surface of the water when J of the work required to 
 empty the shaft has been done. 
 
h '■ "-9 
 
 y- '■■: 
 
 ' *■■ ft! 
 
 114 
 
 ik 
 
 
 
 y- ■ ''' 
 
 22. The mass of a specific gravity bottle is 20*5 when 
 empty, 70*5 when filled with water, 63 when filled with 
 turpentine; when 10 grams of salt are put into it, and it 
 is thereafter filled up with turpentine, the mass is 69 6 ; 
 find the s. g. of the turpentine, and of the salt to an ap- 
 proximation of the first degree. 
 
 23. A number n of balls A, B, C\ formed of the 
 
 same substance, are placed in a straight line. ^, whoso 
 mass is m, is then projected with a given velocity uso as to 
 impinge on B / then B impinges on (7; and so on ; find 
 
 the masses of By C, so that each ball may be at rest 
 
 after impinging on the next, and find also the velocity of 
 the last ball. 
 
 24. A uniform force acting on a body for one-tenth of a 
 second produces a velocity of a mile per minute ; compare 
 the force with the weight of the body at Greenwich. 
 
 25. One end of a string is fastened to a body of 10 kilo- 
 grams ; the string passes over a fixed pully, then under a 
 movable pully, and has its other end attached to a fixed 
 hook ; 7f kilograms are attached to the movable pully 
 whose mass is 250 grams ; if the three parts of the string be 
 parallel, and friction and the masses of the string and fixed 
 pully may be neglected, find the accelerations of the masses 
 and the tension ot the string. 
 
 26. A ball is let fall from a height of 100 metres, and 
 strikes a horizontal surface (e = |) ; find how high the ball 
 will rise again, neglecting the resistance of the air. 
 
 27. A body of 100 kilograms pulls by its weight 200 
 kilograms along a rough horizontal plane ; if the coefficient 
 of friction be 0*2, find the velocity after moving through 
 a distance of a hectometre. 
 
 f'.' 
 
115 
 
 28. Two bodies of different volumes have the same weight 
 in water. Will heir weights be the same in air and in 
 mercury ? If not, how will they differ ? 
 
 29. A stream is a feet broad, l feet deep, and flows at 
 the rate of c feet per hour ; there is a fall of d feet ; the 
 water turns a machine of which the efficiency is e ; it re- 
 quires / foot-pounds per minute for 1 hour to grind a 
 bushel of corn ; determine how much corn the machine 
 will grind in 1 hour. 
 
 30. Find what will the volume of a litre of air at 0= and 
 under the mean atmospheric pressure become, when at the 
 bottom of the deepest known part of the ocean (s. g. 1-027) 
 viz., 5 miles, and what will be its mass at the same place ? 
 
 31. Find the visible energy of the boulder in 1, Ex. 
 XII. What becomes of it when the boulder strikes the 
 ground ? 
 
 32. Find what must be the area of a cake of ice (s. g. 
 0-91674), 18 inches thick, sufficient to bear the aggregate 
 weight of three school boys whose aggregate mass is 280 
 lbs. ; 1) in fresh water, 2) in sea-water, (see tables, art. 81 
 and 108), 
 
 33. Three bodies P, Q, /?, of masses 30, 1.5, 10 kilo- 
 grams respectively, are connected by strings AB and BC. 
 whose lengths are 5 m. and 70 cm. Q, R, BC, and half of 
 AB lie on the edge of a table vertically under a peg, over 
 which the other half of AB is placed holding P. If P be 
 now allowed to fall freely, find the motions of P, Q, and 
 H, the tensions of the strings after both become stretched, 
 and the measures of the impulsive tensions which set Q 
 and B in motion. Friction and the masses of the strings 
 may be neglected, and ff = 980. 
 
m 
 
 i 
 
 ft ;■■• ■« 
 
 m 
 
 
 m 
 
 :¥h 
 
 116 
 
 Answers to the Exercises. 
 
 When no unit is (appended to an answer, the units of the 
 C, G. 8. or F. P. S. system are to he understood. When 
 the correct a/nswer cannot he eoiypressed exactly in the usual 
 notation, or only hy a large numher of decimal places, the 
 answer given is true to the last figure. 
 
 Exercise I. 
 
 1. 
 2. 
 
 4. 
 
 485-85 ares; 904*31 ares; 2-80195 X 10» litres. 
 12741 kilom. ; 5-1002 X lO^^ ares; 10831 X 10i« 
 cub. kilom. 3. 4075 X 10^ cub. kilom. ; 1 : 266. 
 1-60018 X 10» ares. 5. 1:30. 6. If tbe earth's sur- 
 face be unity, 0-3987491 : 0518311 : 0-0829399; 
 torrid zone 2-0338 X 10® sq. kilom. ; temperate 
 zones 2-6436 X 10® sq. kilom. ; frigid zones 
 4-2303 X 107 gq. kiiom. 7. 28845 ; 74® 19' 14"-6. 
 8. 111-8; 25416; 261-8 litres. 9. 4:1. 10. 168-6. 
 11. 56*26' 34". 12. 252062-5. 13. l-v-20,or2®5r53"-2. 
 31-464 sq. m. 16. 23307 ; 602585. 17. 3631-1; 
 5-43205x108. 18. 28689-5; 6-54991x107. 
 3° 49' 13" -5; 10-002. 20. 1*299038; 2; 2-377641; 
 2-598076; 2-828427. 
 
 15. 
 
 19. 
 
 Exercise II. 
 
 ■■'■ \. i* 
 
 1. 
 
 4,. 
 
 10^ 
 
 -\i 
 
 o o 
 
 ;«in. 2. 72 tachs; 65 tachs. 3. 12000. 
 . 5. 6607-75 m. 6. 11952 m. 7. 500; 333*3. 
 9 lOfsec. 10. 2fcm. 11. ;r -^ 45. 
 
 12. n -^ ^o'^OO. 13. 22-2. 14. Im. ;2 7rsec. 
 
 
117 
 
 Exercise III. 
 
 1. 432000. 2. 127072800. 3. 139104000. 4. 2078 
 
 upwards; 1844 downwards. 5. J}. 6. 144:1. 
 7. 1 : 400. 8. 100. 9. j\ sec. 10. 10 m. ; 10 min. 
 
 11. No differenee. 12. 30. 13. 1 m. 14. 1*06. 
 
 15. 10-8. 16.700. 17. 18 min. 18. 5 in. 19. ^s. ^s_ 
 21. 15-3. 
 
 Exercise IV. 
 
 1. 396-9 m. 2. 4-15 sec. 3. 10 sec. ; 1(;00 ft. ; 3Jsec., 
 or 16| sec, 843 ft. 9 in. high ; 3-9 sec; 195*96; 
 112 ft. 4. 112-25. 5. 54-8 sec. 6. 6-9 sec 
 
 7. 793. 8. 10. 9. 980. 10. 50. 11. 4 h. 18 min. 
 59-8 sec ; 47849*6 m. 12. 490*25 m. ; 9904*5. 
 
 13. 1000,-100. 14. h^-a^^td^-eK 15. 105. 
 
 16. 16415. 17. 78*44 m. 18. 1 sec 19. t: : 4. 
 
 20. 10 sec; 1-9 m. 21. lift. 23. 2 sec (take ^ = 980). 
 24. 2800 ; 2800, 0. 26. h-a. 
 
 Exercise V. 
 
 1. 8100. 2. 28498552 kilogrs. (see p. 107). 3. 4*6 
 kilogrs. ; 2 litres. 4. 82-7 grs. 5. 2-66. 
 
 6. 6-1418 X 1021 tonnes. 7. 8*02. 8. 140*55 kilogrs. ; 
 1-634. 9. 260-42. 10. 1-02605. 11. 13-6. 
 
 12. 4 : 3. 13. 13-3 lbs. 14. 8 : 9. 
 
 Exercise YI. 
 
 1. 200; 10 m. 2. 650. 3. 0-191. 4. 500. 
 5. 6-6584x1021: 1. 6. 135*9:1. 7. 0-27. 
 
 8. 1 : 105-894. 9. 6000. 10. 1 : 3231. 11. 77*76 
 kilogrs. 12. 10 kilodynes ; 1 kilotach. 13. 2:1; 
 1:2. 14. 4 : 1. 15. 60 : 11. 17. 1:1; 400 : 19. 
 
m 
 
 ;^5 
 
 \>i : 
 
 Iff*::,'. 
 
 I. .*■ t. 
 
 hi."M 
 
 118 
 Exercise VII. 
 
 1. 9805; 98050. 2. 12-5 kilogrs. 3. 4-2721 kilogrs. 
 4. 14707500; 7353-75. 5. 4902500 ; 392-2. 6. 31313 
 
 tachs. 7. 7-772 Ibs.-wt. 8. 10 sec; 183-35 ft. 
 9. 6 lbs. or 161 lbs. 10. 
 
 650 ; 10-2 milligrams-wt. 
 
 13. 980-5 grs. 
 15 
 
 100-795 mega- 
 
 11. 0-032 sec. 12. 980-5 cm. 
 14. 143-793; 385-37; 377852. 
 
 preins + pressure of the atmosphere. 16, 707*6; 
 
 321-6; 128-6. 17. 7972. 18. 9(»0 milligrs. ; 899-08. 
 
 20. 898-01 grams-wt. 21, 521-6. 22 200 ft. 23. 10 m. 
 
 Exercise VIII. 
 
 
 
 
 
 
 1. 2-735. 2. 0-515. 3. 8-12; 0-164. 4. ^. 5. 20-06 
 
 Ibs.-wt. 6. 388 grs. 7. 61 ^ 136. 8. 0-895. 
 9. 2153 grs.-wt. 10. 1661-808 grs.-wt. 11. 894112 
 
 dynes. 12. 74-313 kilogrs.-wt. 13. 15-r-31. 
 14. 13-0206 kilogrs.-wt. 15. 87 cub. cm. 16. |;f. 
 
 17. 22-6; 9-06; 8-9 lbs. 18. 5-848 cub. ft. 19. 2:3; 
 5:4; 2. 20. 1778 grs.-wt. 21. 3421*9 grs. 
 22. 1-03. 
 
 Exercise IX. 
 
 1. 13*9218 grs. ; 13650-8 dynes. 2. 1-24112 X 10«. 
 
 Exercise X. 
 
 1. 252062*5 litres ; 312143; 3*06069 X 108. 2. 0-25; 
 0-250969; 0-250755. 3. 19-2931,0-720012; 19-2695, 
 0-720374; 19-2663,0-720367. 
 
 Exercise XI. 
 
 1. 5*616 X 10' ft.-lbs. 2. 0-2 X ^ X 10«. 
 
 3. 3*92216X101*. 4. 120373. 5. 32 tonnes. 6.4.6. 
 
119 
 
 ft. 
 
 15. 
 18. 
 21. 
 23. 
 26. 
 
 7. 7212535 ft.-lbs. ; M 2348 X 10' ft.-lbs. 8. 142, 
 (1 gal. of water=10 lbs). 9. 3916-8. 10. For 3920 
 bricks, 3487-65 ft.-lbs. per min. 11. 9-54 cts. 
 12. 32j| miles per hour ; 8064 ft. 13. 4 rain. 39-3 sec. ; 
 1 mile 480 yds. 14. 1 hr. 37 min. 23-3 sec. 
 
 2-49233. 16. 0-756. 17. 3-177 effective H. P. ; 24192 
 9-87 X 1010. 19. 1.5114 >< j^^, ^^^ g.^^ ^ ^^^^ 
 
 55-22 cub. ft. per min. 22. 8 tons 2207-7 lbs 
 1-125X1010. 24. 12-4825. 25. 20^ l>rs 
 1-66795 X lO'i ft.-lbs. 
 
 Exercise XII. 
 
 1. 1-375 75 X 10-10 f.^ per year; 2-9776 X lO-n cm. 
 
 2. 6-6 ft. per min. 3. 2-4 Ibs.-wt. ; 7-5 Ibs.-wt., or 
 
 12-5 Ibs.-wt. ; tension of the string connecting the 
 
 4 and 6 kilogrs., at first 7-2 kilogrs.-wt., afterwards, 
 4-8 kilogrs.-wt. ; tension of string connecting the 4 and 
 
 5 kilogrs., 4 kilogrs.-wt. 4. 48-72; 6 Ibs.-wt.; 
 119-93 Ibs.-wt. 6. 1750; 3083-3. 8 -9 9 i- 
 1:4. 10. 155. 11. (7= ^ ^aB). 
 
 12. Ratio of masses e: 1. 13. 1000. 14. They are all 
 left at rest except the last, which moves away with a 
 velocity ^"-1 times that of the original velocity of the 
 first ball. 16. 7n,m,{u,-u^)^ ~ 2 (m, -f m,); 
 ^^m,m,{n,-n,)^-^2{m^+m^). 17. mieum.,=m,; 
 
 im^il-e^Xu^-u^)^. 18. -26-6, 6-6; -533.3, 333-3 - 
 652-677 ft.-lbs. 
 
 Miscellaneous Examples. 
 
 1. 10«-(362x981)cm.; 1 kilogr. ; 103-^(36x981) sec. ; 
 substance whose density relatively to water is 
 
 4 sec. 
 
 36« X 9813 -J- 1016. 2. 0-353. 4. 
 
120 
 
 I 
 
 >.*A 
 
 .1 "i" 
 
 M ' 
 
 . •"3 
 
 
 
 t >\ 
 
 1 
 
 5. 10»s-^8648; 10» ^ 864^. 6. 2'91C44; 086504. 
 7. 0-982333; 1- 1056. 8. 90-4 cm. 10. 91-778 cm. 
 11. 106 . i()7 . 9-8008 X 10». 12. 233830. 
 14. 100 gre. ; no. 15. (bulk, halfa megalitre), 516. 
 16. 18 ft. 5-674 in. 17. 10 m. 16 cm. 18. 75*12. 
 g^ -~ 10* X 75* prems; 7*5 megagramtach. 
 a- 
 
 20. 
 21. 
 
 2. 22. 0-85 ; 236. 23. A -^ e, A ^ e^. 
 
 6"-* u. 
 
 24. 1 : 131-69. 25. ^-t-2; ^^4; 
 
 5 kilogrs.-wt. 26. 44-4m. 27. 1980-44 tachs. 
 29. Q2-4:alcde -^ 60/ 30. 1*25 cub. cm. ; 1-2932. 
 31. 1-79 X 1014 ergs. 32. 35-914 sq. ft. ; 27-12 sq. ft. 
 
 33. Q starts with vel. 1400 ^ 3 , ^ with 420 ; 27*27 and 
 
 10-90 kilogrs.-wt. ; impulsive tension of AB when Q 
 was set in motion, 7 megagramtachs ; when Ji was 
 set in motion, impulsive tension of AB was 2-8 mega- 
 gramtachsV and of ^6^4-2 megagramtachs. 
 
 * 
 
 s 
 
 U' 
 
 T 
 
 
 COPRIGENDA. 
 
 n 
 
 2ir 
 
 p. 7, Ex. 14, after -w- , insert and one of the angles -«- , 
 
 (3922)' 
 p. 26, 3)., instead of the second li/ne^ ^^^^~ 2v98(V5 ~ ^^^ 
 
 p. 47, Ex. 15,/o7' 11 to 15, read 22 to 15. 
 p. 110, line li,for 1033-0 read 1033-30. 
 p. 113, Ex. 15, after bulk is, insert half. 
 
6504. 
 
 . ft. 
 
 and 
 3n Q 
 
 was 
 lega- 
 
 ^ 
 
 H4