I H JOSEPH A, HOFMANN, SIXTEEN YEARS WITH THE LATE FIRM OF A. ROMAN & CO. BOOKSELLER AND STATIONEF Special filgcnt fat j. cfa. Sippincott (Eo's ^u&ficatioHd, 208 MONTGOMERY ST. BET. BUSH AND PINE, Piatt's Hall Block, SAN FRANCISCO, C/ Printing and Bookbinding done on the most reasonable Terms. Yearly Subscriptions received for LIPPINCOTT'S MAGAZINE and other Popular Periodicals. UNIVERSITY OF CALIFORNIA LIBRARY OF THE Accessions Afo...^.-2'..o... Book No.. ..... NATUKAL PHILOSOPHY BY ISAAC SHARPLESS, Sc.D., PROFESSOR OF MATHEMATICS AND ASTRONOMY IN HAVERFORD COLLEGE, AND GEO. MORRIS PHILIPS, PH.D., PRINCIPAL OF STATE NORMAL SCHOOL, WEST CHESTER PA. **-*? JTf o* PHILADELPHIA: J. B. LIPPINCOTT COMPANY. JOSEPH A. HOFMANN, Bookseller & Stationer, 208 MONTCOMERY ST.. San Francisco, Cal. , Copyright, 1883, by J. B. LIPPINCOTT & Co. PREFACE. THIS Treatise on Natural Philosophy differs from others in the large number of practical experiments and exercises which it contains. The authors believe that students of science should be, as far as possible, investigators, and, to encourage the spirit of research, they have given sugges- tions tending to lead them on in this way. The experi- ments can nearly .all be performed with very simple and inexpensive materials, such as any school or home can furnish. More elaborate instruments are described for the benefit of classes which have access to them. The book can also be used by classes which have not time to perform the experiments. Yet it is strongly recommended that as many as possible be tried. Two sizes of type are used through the book. The matter printed in large type will form a complete ele- mentary course, and the whole book a more exhaustive one. Those who take the former are advised to include as many as convenient of the experiments, exercises, and questions. The large number given will allow the teacher to make selections suited to the ability of the class. The use of technical terms, except where they seemed necessary to the better comprehension of the subject, has been avoided. It has been recognized that the majority 673234 3 PREFACE. of students of natural philosophy have no use for these terms. What they want is a practical knowledge of the subject and the cultivation of scientific habits of mind. The methods of the leading scientific men of the present time have been incorporated, and their instruments de- scribed and figured. In any treatise on the subject which embraces an account of these methods, the doctrine of the conservation of energy must have a prominent place. The great advances in practical science within the last few years, especially in sound, electricity, and meteorology, have also been utilized so far as they seem to bear on the principles. The work has been greatly benefited by the criticisms and suggestions of C. Canby Balderston, of Westtown School, Pennsylvania. The chapters on Magnetism and Electricity were written by him. CONTENTS. PREFACE 3 CHAPTER I. Matter 7 II. Motion and Force 19 Gravity and Stability 35 Falling Bodies 40 The Pendulum 44 Machines ........ 47 III. Liquids 63 Hydrostatics . . . . . . .63 Specific Gravity 78 Hydraulics 83 Water-Machines 87 IV. Gases 95 The Atmosphere 100 Pneumatic Machines ..... 103 V. Sound 120 Cause and Phenomena ..... 120 Musical Sound 129 Musical Instruments 133 Music 150 VI. Light 162 Keflection 169 Refraction 177 Dispersion 188 Polarization ....... 202 Optical Instruments 205 VII. Heat 213 Conduction 234 Convection 236 Steam-Engine 237 VIII. Magnetism 244 1* 5 CONTENTS. PAGE CHAPTER IX. Electricity 255 Fractional Electricity 255 Current Electricity 277 Electro-Magnetism . . . . .289 Magneto-Electricity 304 Kadiant Matter 313 X. Meteorology 319 The Atmosphere 322 APPENDIX I. The Metric System 343 II. Table of Specific Gravities . . . .345 INDEX 346 NATURAL PHILOSOPHY. CHAPTEE I: , ,, v , >B /, ' i j.. ,'-.-:' MATTER. 1. What is Matter? All the bodies which occupy space, the stars and the planets, rocks, water, and air, and everything we can see or feel, are embraced under the term matter. We can crumble a rock or divide a quantity of water into smaller portions. These can again be subdivided, and all the fragments will resemble the original in their prop- erties. There is a practical limit to this subdivision, arising from the imperfection of our senses or our tools, but we may suppose it carried on till the very smallest possible fragments remain which possess the properties of the substance. 2. Molecules. To these fragments we give the name molecules. They are definite quantities of matter, which have size and weight. Hence a molecule is the smallest portion of any substance in which its properties reside* All matter is made up of molecules. We know that molecules must be extremely small. Sixteen ounces of gold, which in the form of a cube would not measure an inch and a quarter on a side, can be spread out so that it would gild silver wire sufficient to reach around the earth. Its thickness must then be at least 1 The properties of matter are those qualities which are peculiar to it, which belong to it and to nothing else. 7 8 NATURAL PHILOSOPHY. one molecule, and is doubtless many. In odors, which produce sensation by invisible particles, the molecules scatter about through the atmosphere for years without apparently diminishing the size of the substance from which they are separated. Microscopists have found ani- mals so minute that four million of them would not be so large as & slti.glcr grin of sand, yet each has its organs and its circulating fluids. ., 3. Siz of a Mpkenle.- The methods of attaining an idea of the actual dize of a molecule "are too abstruse for explanation here, but the figures, derived from experiments of different kinds, point to UTo'.TJ'oir.irUTF f an i nc h as the mean diameter. This is too minute a quantity for comprehension, and may be better understood by the illustration of Sir William Thomson : "If we conceive a sphere of water of the size of a pea to be magnified to the size of the earth, each molecule being magnified to the same extent, the magnified structure would be coarser-grained than a heap of small lead shot, but less coarse-grained than a heap of cricket-balls." The molecules of hydrogen gas are about 7,-jnri.TnnF of an i nc h apart, so that the spaces between are much greater than the molecules themselves. 4. Atoms. When the division is carried any further than molecules, a form of matter with new properties is produced. It is not possible to divide a molecule by me- chanical means, but heat or chemical agents can separate it into two or more portions. Each of these is called an atom. An atom cannot be further divided by any means known to us. Hence an atom is the smallest possible portion of matter. Experiment i. Put a piece of marble or chalk (not a crayon) into a vessel, and pour on it some good vinegar. Bubbles of gas will arise through the water. A molecule of marble is composed of a number of atoms of dif- ferent substances. The acid in the vinegar causes a division of the molecule, forming new substances. One of these substances (carbonic acid) is a gas, which passes off into the air. The others remain in the vessel. 5. Constitution of Molecules. The molecules of some MATTER. 9 substances are made up of two or more similar atoms. A molecule of hydrogen gas contains two atoms exactly alike. On the other hand, a molecule of common salt contains one atom of sodium and one of chlorine, which are widely different from each other and from salt. In their ordinary state, sodium is a soft inflammable solid, and chlorine a greenish gas. A molecule of sugar is composed of forty- five atoms of three different kinds, carbon, which we can see as charcoal, and hydrogen and oxygen, which are color- less invisible gases. Experiment 2. In a vessel heat a small portion of sugar over a fire. A black substance will remain. In this case heat effected a separation of the atoms of the molecules ; the gases passed off into the air, and the solid carbon remained. 6. Elements. If the molecules of a substance are com- posed of one kind of atoms only, it is said to be an element. Sixty-five elements have been discovered on the earth. Iron, copper, carbon, are elements. "Water and air are not. 7. Matter Indestructible. If the escaping gases and the carbon of the last experiment could be weighed, the sum of the weights would be found to be just equal to the weight of the original sugar. Hence we arrive at an important property of matter, it is indestructible. There are many cases of the apparent destruction of matter in combustion and chemical action, but all that is done is to change its form. The molecules are divided, and the atoms form new combinations, some or all of which are invisible. In all the various changes continually going on, in our furnaces and laboratories, and in nature, not a new atom is ever created. According to the best of our knowledge, the amount of matter in the universe has re- mained unchanged since the original creation. 8. Matter Porous. The molecules of matter do not fit 10 NATURAL PHILOSOPHY. closely together. Hence open spaces, or pores, are left between them. We then arrive at a property of matter which is believed to be universal, it is porous. Experiment 3. Fill a tumbler with cotton-wool, pressing it down so firmly that the vessel will hold no more. Now remove the cotton and fill the vessel with alcohol. With care, the cotton may all be replaced without spilling the alcohol. The cotton has gone into the ?ores of the alcohol, and the alcohol into the pores of the cotton, t is impossible to conceive that the molecules of both substances occupy the same space. 9. Matter can be Expanded and Compressed. As a result of the porosity of matter, it is possible to expand or to com- press it. The molecules are not changed in form or size, but they are further separated in expansion, and crowded together in contraction, so that the substance becomes more porous in one case and less so in the other. Heat in general separates the molecules from one another. A ball that will just go through a ring when cold will not do so when heated. The mercury in a thermometer-tube rises in hot weather because the heat separates the molecules and there is no chance for expansion in any other direction. The ends of the rails of a railroad-track which touch each other in summer are separated in winter. A nail can be driven into wood because it causes a compression of the molecules around to make a place for it. Experiment 4. On a cork floating on water place a shav- ing. Set it on fire, and put over it an inverted tumbler. The heat of the combustion will expand the air in the tumbler and force FIG. l EXPANSION BY HEAT. it out under the edge ; what is left will quickly cool and con- tract, so that almost immediately the water will rise into the tumbler. 10. Expansion by Cold. Heat does not always expand bodies. Experiment 5. Fill a bottle with water, and cork tightly. Leave in a cold place till the water is frozen. The bottle will be cracked. The cold here caused expansion. At 39.2 Fahrenheit a MATTER. given weight of pure water takes up least room and ex- pands with a change of temperature either way. 11. Some Bodies can be Hammered into Plates and Drawn into Wires, When certain solid bodies are ham- mered out into plates or drawn into wires the molecules slide past one another and arrange themselves differently. This motion of the molecules is not possible in all solid bodies, and some possess it in a much higher degree than others. Gold may be hammered out into sheets less than 2"f)oYoo~o f an mc ^ i n thickness. Copper, silver, and tin can also be beaten out into very thin foil. One of the substances which may most readily be drawn out into wires is glass. Experiment 6. Heat in an alcohol flame, or hot gas flame, a small glass rod or tube. When red and soft, it may be drawn out into a very fine thread. Metal wire is made by drawing the soft metal through holes, each one smaller than the preceding. Platinum wire can be reduced so that it will be finer than the finest hair. 12. Matter Elastic. All bodies are more or less elastic. By this it is meant that when compressed within certain limits the molecules tend to come back to their original position with respect to one another. When a ball is allowed to fall on a hard floor, there is a compression of the molecules of the ball near the point of contact with the floor. The elasticity of the ball causes an immediate restoration to the original form of the ball, and this produces the rebound. When gases are com- pressed, they recover their former state immediately when the pressure is withdrawn. They are said to be perfectly elastic. Although liquids can be compressed but slightly, they are also perfectly elastic. 13. Tenacity. When the molecules of a solid adhere so closely that they strongly resist a force tending to pull them apart, it is said to be tenacious. The amount of tenacity depends on the structure of the substance. 12 NATURAL PHILOSOPHY. "Wrought iron, being fibrous, has much more tenacity than cast iron, which is granular. Steel is very tenacious. A bundle of wires will support much more weight than the same material in solid form. Hence the cables of suspen- sion-bridges, which have to hold up immense weights, are usually made up of bundles of fine steel wire. Experiment 7. Place a piece of stick on two supports some dis- tance apart, and break it by a weight applied in the middle. Ex- amine the fracture. The lower fibres will be found to be separated. 14. Bridges. When a weight rests on a bridge, it has to stand the same kind of strain as the stick. The tendency is to pull it apart at the bottom. Hence an iron bridge has its lower " chord" made of tenacious wrought iron rather than of cast iron. The upper chord is compressed, and as cast iron will stand more compression than wrought iron, it is frequently used there. 15. Hardness. Hardness is another property of solid bodies, depending on the closeness with which the mole- cules stick together and resist the entrance of another body which tends to penetrate them. Hard bodies are not always tenacious. Diamond is the hardest of substances, being able to scratch everything else. This, ability to scratch is the test of hardness. Experiment 8. Scratch a piece of glass with the edge of a quartz crystal or piece of flint. Attempt to do the same with a penknife- blade. Quartz is harder than glass, and glass is harder than steel. 16. Density. There is more matter in the same space in some bodies than in others. This is either because the molecules are closer together, or because each molecule contains more matter. We express this by saying that some bodies are more dense than others. 17. Volume. The volume of a body is the amount of space it occupies. 18. Mass. The mass of a body is the quantity of mat- ter which it contains. If a gas be heated so as to expand it, the mass remains the same, as no new molecules are formed, but the density decreases. The mass therefore de- pends on two things, the volume and the density. The MATTER. 13 number of molecules in a unit (as a cubic inch) of a body multiplied by the number of cubic inches gives the whole number of molecules. In other words, the product of the volume by the density gives the mass, or Mass = volume X density. 19. Unit of Length. The English units of length are the inch, the foot, and the yard ; the French are the metre and its decimal divisions. It is convenient to remember that a metre is about 40 inches, a decimetre about 4 inches, a centimetre about -^ of an inch, and a millimetre -^ of an inch. 1 20. Unit of Surface. For square measure we have in English the square inch, square foot, and square yard, and in French the square metre, square decimetre, and square centimetre. The cubic units are derived in the same way. 21. Unit of Mass. The unit of mass in the English system is the pound avoirdupois ; in the French it is the mass of a cubic centimetre of water at its greatest density, 39.2 F. This is called a gram, and is about 15 i grains. This is divided and multiplied decimally for smaller and larger weights. 22. Unit of Density. The unit of density for solids and liquids is the density of water at 39.2 F. 23. Affinity, Cohesion, Attraction. The force which holds together the atoms in a molecule is called affinity. The force which holds together the molecules in a body is called cohesion. The force which holds together the different bodies of the universe is called attraction. Hence affinity makes substances ; cohesion makes bodies; attraction makes systems. Attraction is also used to express the force which draws one body to another, as in the case of magnets, etc. 1 The metric system possesses great advantages, especially for scien- tific people. Appendix I. gives it in part, and should be studied. 2 14 NATURAL PHILOSOPHY. 24. Solids. In solid bodies the molecules preserve their positions with considerable firmness, resisting attempts to displace them. Hence these retain their form and size. The force of cohesion in them is strong. 25. Liquids. In liquid bodies there is perfect freedom of the molecules among themselves, so that the bodies adapt their form to the surrounding vessel. They retain their size, but change their form with the slightest force exerted upon them. The force of cohesion in them is weak. 26. Gases. In gaseous bodies there is no cohesion, the molecules have a repellent action upon one another, so that an unrestrained gas will expand indefinitely. 27. Motion of Molecules. The molecules of all bodies are be- lieved to be in rapid motion. In solids this is restrained by cohesion, so that a molecule has only a short vibratory motion. In liquids the molecules slide over one another without resistance, restrained only when they reach the sides of the enclosing vessel. This contact pro- duces the pressure against the sides. In gases the molecules are strongly repelled from one another, and dash about with great ve- locity. Hence there are constant collisions among them and with other bodies. Our bodies are subject to this incessant battering by the little molecules of the atmosphere, but, the force being the same on both sides of the tissues, we do not notice it. 28. Adhesion. Adhesion differs from cohesion in that it acts between molecules of different bodies. The force which causes mortar to stick to bricks, which causes a pencil to leave a mark on paper, which enables glues and pastes to be effective, is adhesion. It is also something like adhesion which causes water to rise in a small tube or on the side of a glass plate. 29. Weight. The weight of bodies results from attrac- tion. All bodies attract all other bodies. The more mole- cules a body contains, the greater is its attraction for others, and the attraction of others for it. The pull of all the par- ticles of the earth on the objects on its surface is the same as if one strong pull drew them to its centre. Hence a MATTER. 15 plumb-line points to the centre of the earth, 1 and different plumb-lines are not parallel, but converge downward. 30. Gravity, The attraction of the earth is called gravity. 31. Weight Proportional to Mass. The earth pulls every particle of a body. If we suppose a string attached to each molecule, and all the strings pulled by equal forces, we would have the case of attraction. Hence the more molecules the greater the attraction. But the mass is determined by the number of molecules. Hence we have the law, Under the same conditions the weights of bodies (or the total attractions) are proportioned to their masses. 32. Weight Inversely Proportional to Square of Distance. The position of the body affects the weight. The attrac- tion diminishes as the bodies recede from each other. If the distance doubles, the attraction is only one-fourth, and if the distance trebles, one-ninth, of the original amount. We express this by saying, The attraction varies inversely as the square of the distance. 33. Mass Constant. The position of the body does not affect the mass. It might be removed far from the earth and the mass would be the same. The number of mole- cules i.e., the mass would be constant if carried to the sun ; but as there is so much more mass in the sun than in the earth, the attraction, and consequently the weight of the body, would be greatly increased. 34. Unit of Weight. The unit of weight is the same as the unit of mass, the gram. 35. Mobility and Inertia. Bodies will not move unless some force is exerted on them from without, and they yield to the slightest force impressed which is not counter- balanced by some other force. This brings us to two other 1 This is very slightly modified by the fact that the earth is not a perfect sphere. 16 NATURAL PHILOSOPHY. properties of matter, mobility, which induces it to yield freely to impressed forces, and inertia, which prevents it from moving itself, or from changing any motion which may be given it. Matter has no power to move or to resist an unbalanced force. Examples of inertia are numerous. It requires more force to start a car than to keep it in motion. When sud- denly stopped by another force, the contents are thrown forward by their inertia. A ball projected upward stops, not because it has power to stop itself, but because another force, gravity, is constantly pulling against its motion. A marble thrown swiftly through a pane of glass will make a small round hole, because the inertia of the other parts of the glass prevents them from yielding to the sudden impression. Experiment 9. Place a card on the end of a finger, and a cent on the card. By a quick stroke with the forefinger of the other hand the card may be shot out, leaving the cent resting on the finger. 36. Ether. We have spoken of the three forms of matter, solid, liquid, and gaseous ; we have also said that the molecules of matter do not fill up the whole space, but that pores, which are large compared with the size of the molecules themselves, exist in all substances. This inter- molecular space is supposed to be filled with something called ether, which is as far separated from gases by its properties as gases are from liquids. It also fills the pores of the air, and the spaces between the planets and between the stars, outside the bounds of the atmospheres which surround them. It is highly elastic, without weight or color, or any other properties which can be perceived by the senses. It is supposed to be the agent which by its vibratory motion conveys the rays of light from the sun to the earth, and which carries them between the molecules through transparent substances. 37. Radiant Matter. Dr. William Crookes 1 has found 1 An English scientist, now living (1883). MATTER. 17 that by exhausting the air in a tube so as to leave not more than one-millionth the ordinary amount, the remaining sub- stance has such peculiar properties that he feels justified in giving it a new name. He calls it radiant matter, and considers it to be the fourth form. Solid, liquid, gaseous, and radiant would then be the four aggregate states, each having properties which widely separate it from the others. By passing electric sparks through radiant matter some of its properties have been determined. 1 Of the properties of ether we know nothing by direct experiment, but it is considered likely that it is a form of radiant matter. 38. Summary. Matter is made up of a countless number of minute molecules. It is perfectly inert, but each par- ticle possesses the property of attracting every other particle. It has extension in three directions, and has three (probably four) forms of aggregation. 39. Natural Philosophy. Natural Philosophy treats of the laws of cohesion, the molecular properties of matter, and the effects of the action of forces upon matter. 40. Astronomy. Astronomy treats of matter in large masses, and of the laws of gravitation. 41. Chemistry. Chemistry treats of the atomic proper- ties of matter, and of the laws of affinity. Exercises. 1. Is matter destroyed when water is dried up? when gunpowder explodes ? when house gas burns ? Where does it go to? 2. To what property of matter do blotting-pads owe their utility ? rubber bands ? watch-springs ? pop-guns ? putty ? hammers ? piano- strings ? water-filters ? 3. Why does not the addition of a little sugar to a full cup of coffee cause it to overflow ? 4. When we fix the head of a hammer on the handle by striking the end of the handle on a block, what property do we use? 5. Why does a foot-ball, nearly empty, become full when we ex- haust the air from around it ? why does it soon collapse ? 6. One sixteen-thousandth of a cubic inch of indigo dissolved in sulphuric acid can color two gallons of water. What property of matter is here shown ? 7. How would you test the relative hardness of two minerals ? 1 These will be further explained, page 314. b 2* 18 NATURAL PHILOSOPHY. 8. When water is converted into steam, are the molecules enlarged or separated? is its mass increased or diminished? its density? its weight ? its volume ? 9. Name a substance which is often found in all three forms. 10. If you knew the volume and mass of a solid, how would you obtain its density ? if you knew its mass and density, how would you obtain its volume? 11. Give an instance of a hard body which has little cohesion. 12. Why does not a large stone fall to the earth more rapidly than a small one ? 13. If a body were removed to a distance of 8000 miles from the surface of the earth, how much less would it weigh than at the sur- face? Ans. ^ as much. 14. What would a 100-pound weight weigh if moved to the dis- tance of the moon (60 radii of the earth) ? Ans. ^ pound. 15. Suppose a sphere were one-half the diameter of the earth and of the same density, what would a body which weighed 100 pounds on the earth weigh at its surface? Ans. 50 pounds. Note. Its mass would be one-eighth that of the earth, and dis- tance of the body from its centre one-half. MOTION AND FORCE. 19 CHAPTER II. MOTION AND FORCE. 42. Rest and Motion. A body is at rest when it does not change its place. It is in motion when it does change its place. No body with which we are acquainted is at rest. The earth and all that is on it move with great velocity. The sun moves, and so do the stars. But when a book lies on the table it does not move with respect to the surrounding bodies or the earth. It is at relative rest, but in absolute motion. 43. Kinds of Motion. When a body in motion passes over equal spaces in equal times, its motion is uniform. When it passes over unequal spaces in equal times, its motion is varied. When the spaces in successive times become greater, its motion is accelerated, and when less, re- tarded. This acceleration or retardation may also be uni- form or varied. 44. Velocity. The velocity of a motion is the space traversed in the unit of time. It may be in miles per hour, feet per second, etc. Feet moved in Successive Seconds. Kinds of Motion. 30 30 30 30 Uniform. 10 15 20 25 Uniformly accelerated. 20 18 16 14 Uniformly retarded. 20 14 16 4 Varied, not uniformly. Questions. When a train starts from a station, what kind of motion is it ? when stopping ? when a ball is thrown upward ? when it falls ? What kind of motion in the hands of a watch ? in the cur- rent of a river ? in the winds ? 45. Force. Force is anything which tends to produce, change, 20 NATURAL PHILOSOPHY. or destroy motion. If it acts on a body at rest, it produces motion. If it acts on a body in motion, it may change the direction or velocity of the motion, or destroy it. Two or more forces may act on a body at rest so as to balance each other and cause no motion. But each one tends to produce motion. In bridges and buildings we have cases of bal- anced forces. Gravity is a force always acting upon them, and upon everything they sustain. This produces other forces acting along the various timbers and pieces. If the structure is well built, the strains from these forces are exactly balanced, every part is sufficiently strong to do its work, and there is no motion except such as is due to the elasticity of the materials. 46. Kinds of Force. A force may act for an instant and then cease, in which case it is said to be an impulsive force ; or it may act for some time, when it is a continuous force. The striking of a ball by a bat is an example of an impul- srve force, and the pulling of a train by a locomotive, of a continuous force. 47. Impulsive Force produces Uniform Motion. An im- pulsive force tends to produce uniform motion, and a continuous force accelerated motion. This would seem to be contradicted by experience. For the motion of a ball is soon destroyed, and the, continual pull of the engine may only keep the train moving uniformly. But the force of the bat or of the locomotive does not act alone. Were it not for gravity, the resistance of the air, and friction, which are modifying forces, the ball would move on forever with uniform velocity, and the velocity of the train would be accelerated so long as the engine pulled it ever so slightly. 48. Newton's Laws of Motion. All the circumstances of motion are embraced in three laws, first enunciated by Sir Isaac Newton. These cannot be proved mathemati- cally. They should be looked upon as fundamental prin- ciples, which depend on the properties of matter, and which may be shown to be true by experiment. MOTION AND FORCE. 21 1. A body at rest remains at rest, and a body in motion con- tinues to move forward in a straight line, until acted on by force external to it. 2. Motion or change of motion is proportional to the force impressed, and is in the straight line in which the force acts. 3. When bodies act on each other, action and reaction are equal and in opposite directions. The first law is the result of the inertia of matter, and the second, of its mobility. The first says matter can do nothing itself, and the second, that the slightest force will have its corresponding effect. The third law may be made clear by some illustrations. The earth attracts an apple and causes it to fall. The apple attracts the earth just as strongly, and the earth moves to meet it, but the greater mass of the earth makes it move so little that the motion is not noticed. When you hold up a body in your hand, the hand presses up just as hard as the body presses down. The reaction of the wfcjer on the oar, and on the fins of a fish, causes the boat or the fish to advance ; the reaction of the air on the wings causes the bird to sustain itself and to move forward. 49. Momentum. Momentum is the quantity of motion. The momentum of the earth was the same as the momentum of the apple. For while its velocity was less, its mass was as many times greater. Hence mass and velocity together make up momentum. A body weighing two pounds has twice the motion of one of one pound which has the same velocity ; a body with twice the velocity of another has twice the motion, the mass being the same. In general we have the equation, Momentum = mass X velocity. 50. Measure of Forces. We may measure forces in two ways. One way is by the pressure necessary to resist them, weighing the forces, as it were. The unit would then be in the English system the pound, and in the French system the gram. These would vary as gravity varied, 22 NATURAL PHILOSOPHY. being greater nearer the level of the sea. A better way to measure forces is by the velocity they would produce. We have here also two systems. In the English, the unit of force is the force which, acting for one second, will cause a pound of matter to have a velocity of a foot a second. In the French, it is the force which, acting for one second, will cause a gram of matter to have a velocity of a centi- metre a second. This unit is called the dyne (pronounced dine), and is coming into general use among scientific men. 51. Acceleration. The velocity which a force would produce in a unit of mass in a second is called its acceleration. 52. Illustrations. We will now illus- trate some of these terms. If a body weighing 20 grams has a velocity of 10 centimetres a second, its momentum is 200. (This is not foot-pounds or grams or centimetres; the unit of momentum has no name.) If this momentum is produced by a force acting for 1 second, it is a force of 200 dynes; if for 5 seconds, it is a force of 40 dynes. 53. Acceleration a Measure of Force. If a force acting on the body for 1 second will give it a velocity of 10 cen- timetres, the acceleration is 10. During every succeeding second which it acts, it adds 10 centimetres to its velocity. As its inertia keeps the body moving at its former velocity, this continual force constantly increases its velocity. The greater the force, the greater will be the velocity produced the first second. The acceleration is a measure of the force. 54. Dynamometer. A practical way of measuring some FIG. 2. DYNAMOMETER. MOTION AND FORCE. 23 forces is by a spring-balance placed in the line through which the force must act. A dynamometer (Fig. 2) is an instrument of the same kind, registering the amount of force expended. It is used to determine the resistance to motion of a train, wagon, plough, or other instrument. If a body be hung on a spring-balance, we weigh the force of 'gravity. If a spring-balance or dynamometer is placed between a horse and a plough, we weigh in the same manner the force of the pull of the horse. If the horse pulls with a force of 200 pounds, this means that the connection with the plough is strained just as a rope would be if sustaining a weight of 200 pounds. Questions. 1. What kind of force is gravity? what kind of force drives the bullet from the gun ? what kinds of motion would they produce if unmodified ? 2. Which of Newton's laws are illustrated by the breaking of an egg against a table ? by the tendency of a train to be thrown out- ward over a curve ? in the throwing of a ball ? in the fact that it is more difficult to start a train than to keep it in motion ? 3. A body weighing 20 pounds has a momentum of 400: what is its velocity in feet per second ? 4. Two bodies, one of 20 and one of 2 pounds, are drawn together by their mutual attractions : which will move the faster, and how much ? 5. A body of 20 grams and a velocity of 10 centimetres per second meets another body of 40 grams moving in the opposite direction with a velocity of 4 centimetres per second : in what direction will the bodies move after impact ? Ans. In the direction of the first. 6. How many dynes of force are required to produce a velocity of 500 centimetres per second in a body of 200 grams weight in 5 seconds? Ans. 20,000. 7. What would be the mass if 20 dynes of force would produce in 5 seconds a velocity of 5 centimetres per second ? Ans. 20. 8. In how many seconds would a force of 40 dynes produce a momentum of 400 units ? Ans. 10. 55. Representation of Forces. A force may be repre- sented by a straight b line. Thus, the line ab indicates that the force acts on a body at a in the direction ab. The length of the line may also represent the mag- nitude of the force. A line twice as long would represent twice as great a force. 24 NATURAL PHILOSOPHY. 56. Resultant. The resultant of two or more forces is the name given to a single force which would produce the same effects. If two forces, one of 2 pounds and one of 4, act on a FIG. 3. FORCES IN A LINE. body at a in the same direction, their resultant is evidently a force of 6 pounds acting in the same direction. If they act in opposite directions, their resultant is the difference of their forces (2 pounds), and acts in the direction of the greater. !By considering one direction as positive and the other as negative, we express both of these cases by a single law. 57. Resultant of Parallel Forces. The resultant of two or more parallel forces is their algebraic sum. If in one direction we have forces of 6, 2, 4, and in the other 3, 7, 1, the resultant is 6 -f 2 -f 4 3 7 1 = 4-1. The resultant is 1, and acts in the direction of the plus forces. 58. Parallelogram of Forces. If the forces do not act in the same line, they may still -have a single resultant. Let the forces p and q act on a at the same time in the direc- tions given in the figure. Ac- cording to Newton's second law, f the FIG. ^.-PARALLELOGRAM OF FORCES. full effect on the body. The force p would carry it somewhere in the line ab, but the force q is such as to take it over the space ac. Hence it would bring it into the line cd, parallel to ab. By the same reasoning the body would be shown to be brought into the line bd, parallel to ac. If it is brought into both of these MOTION AND FORCE. 25 lines it must be brought to their place of meeting at d. The figure abdc is a parallelogram, and is called in this case the parallelogram of forces. The body would move in a straight line, ad, which is the diagonal of the parallelo- gram, and its motion would be the same as if acted on by the single force r. Hence r is the resultant of p and q. 59. Triangle of Forces. If we consider the forces with- out reference to their point of application, ab, bd, and ad will represent them, and will form a triangle. A force act- ing at a, equal and opposite to ad, will balance ad. Hence the three forces ab, bd, and da (notice the order of the letters) will form a system which is balanced. We have a general truth that if three forces are represented in mag- nitude and direction by the sides of a triangle taken in order, the system is balanced. 60. Resultant of any Number of Forces. If more than two forces act on one point, we must find the resultant of two of them ; then of this resultant and a third force ; and so on. If ab, ac, ad, and ae are forces acting at a, then the re- sultant of ab and ac is ar ; of ar and ad, ar' ; and of ar' and ae, ar". It will be observed that abrr'r" is polygon, four of the sides of which are parallel to the forces, and the fifth represents the resultant. --- dT 61. Polygon of Forces. This prin- . -i i , , , , 7 FIG. 5. POLYGON OF FORCES. ciple is called the polygon of forces, and may be stated as follows : If a figure be constructed having the sides equal and parallel to the forces, the line necessary to close this polygon, drawn from the starting- point, will represent the magnitude and direction of the resultant. Also, if the forces acting on a body be represented in magnitude and direction by the sides of a polygon taken in order, the system is balanced. B 3 It i a h - 26 NATURAL PHILOSOPHY. 62. Forces not in a Plane. If the forces are not all in one plane, the method would produce the outlines of a solid body. If ab, ac, ad be three forces not in one plane, ar is the resultant of the first two, and ar' the final resultant. 1 b FIQ. 6. PARALLELOPIPED OF FORCES. Fio. 7. RESOLUTION OP FORCES. 63. Composition and Resolution. The force ar may be divided into two forces, ab and ac, or into ae and of, or, in general, into any two which with it would make a triangle. Combining forces so as to get a resultant is called the composition of forces, and separating single forces into sev- eral parts is called the resolution of forces. The parts are called components. Experiment 10. Fasten two pulleys against a vertical board so that they will turn freely. Arrange cords as in the figure, making the knot at a so as not to slip. Hang weights p, <7, and r, being careful not to get r greater than p and q com- bined. Measure off ab and ac pro- portional to the weights p and q, and draw lines on the board to complete the parallelogram abdc. Measure ad, and it will be found to be equal to r on the same scale that ab and ac were made ; also the point d will be found to be directly over a. This shows that the diagonal ad represents the resultant in magnitude and direction. FIG. 8. RESOLUTION OF FORCES. 1 If the forces do not all act on the same point in the body the problem becomes too complex for this treatise. MOTION AND FORCE. 27 between a FIG. 9. CROSSING A CURRENT. Experiment n. Place spring balances in the strings and the pulleys, and measure the forces in this way. 64. Rowing across a Current. If a man undertakes to row straight across a river 6 f in which there is a current, his course will be oblique. For let ab represent the force of his rowing, and ac the force of the current. Then the resultant ad will be the direction of his course, and he will land at / instead of at e. If he wants to go straight across, he will steer in the direction a'b', so that a'b' combined with a'c' will have a resultant in the direc- tion a'e'. 65. Sailing a Boat. In sailing a boat we have a good illustration of the resolution of forces. Let ab be the keel, cd the direction of the sail, and fe the force of the wind, fe may be resolved into two forces, fg, parallel to the sail, which would have no effect in driving the vessel forward, and ge, perpendicular to it. The force ge may again be resolved into gh, perpendicular to the keel, and he, in its direction. This latter force is all that is effective in propelling the boat. The force gh tends to upset it. In a complete analysis of forces the action of the rudder must also be taken into account. 66. Component Forces greater than the Original. It is possible to resolve a force into two components each of which shall be much greater than the original force. If in Fig. 8 the weight r should be very small, the line between the pulleys would be nearly straight, and by constructing the parallelogram it would be seen that the components along ab and ac would be much greater than r. The same principle is shown in the knee-joint (Fig. 11). This consists of a pair of levers, jointed together at b. One of them is Fia. 10. SAILING A BOAT. 28 NATURAL PHILOSOPHY. firmly fixed at the end a, the other is attached to a movable 6 slide. Any force, p, acting verti- cally on the joint will be resolved into two, one along each lever. The more obtuse the angle at FIG. H.-KNEE-JOINT. the joint, the greater will be the component forces as compared with the applied force. Experiment 12. Stretch a string tightly between two fastenings. Tie a weaker string to its middle point. By pulling this the stronger string breaks first For the component pull is stronger than the original. 67. Centrifugal Force. When a body is swung around by a string there are two forces acting on it. One is its inertia, which would tend to cause it to move in a line, ab, touching the curve. The other is #c, the pull of the string. The tendency would be to move in the diagonal ad. But as this pull is acting continuously, and the direc- tion continually changing, the line is a Fw ' 12< "CVE ION IN A curve - These are the forces which keep the earth and all the planets in their orbits. The outward pull on a string, which is the result of the inertia of the body tending to cause it to get farther from the centre, is centrifugal force. It is always opposite and equal to the force drawing towards the centre. 68. Centrifugal Force Apparatus. Its effect is shown in the centrifugal force apparatus of Fig. 13. Here the flexible bands are put in rapid rotation, and the centrifugal force makes them assume the form indicated by the dotted line. 69. Effects of Centrifugal Force. There are many other illustrations of centrifugal force. When the earth was a soft body, the centrifugal force caused by its rotation on its axis probably produced the bulging at the equator which we now notice. The centrifugal force is greater at the equator MOTION AND FORCE. 29 than elsewhere, because of the greater velocity of the earth there. Hence bodies are lighter there than at the poles. An equestrian leans inward in riding around a curve, to FIG. 13. CENTRIFUGAL FORCE APPARATUS. balance the centrifugal force. It is this force which causes mud to fly off moving carriage-wheels, or water from a grindstone, and which sometimes breaks a rapidly-revolv- ing fly-wheel. In sugar-refineries the syrup is separated from the crystals by being thrown outward, the sugar being retained by a wire gauze. Clothes are dried by a simi- lar arrangement. In a bicycle in motion the centrifugal force causes the particles to continue to move in the same plane. Hence the faster it is going the more difficult it is to overturn. 70. Moment of a Force. The moment of a force is its ability to produce rotation. If be be a lever attached to a body which has power to rotate about an axis at a, and a 3* FIG. 14. MOMENT OF A FORCE. 30 NATURAL PHILOSOPHY. force be applied at b, in the direction of the arrow-head, it will tend to produce rotation. This ability will depend on the magnitude of the force and the length of its lever- arm, and is equal to their product. Thus, the moment of p=pX ab. 1. A force of 10 pounds has a lever-arm of 2 feet: what is its moment? Ans. 20 foot-pounds. 2. A force of 16 grams has a lever-arm of 200 metres : what is its moment in kilogram-metres ? If a man attempt to overturn a heavy pillar, he will push against it some distance above the base ; for in this case his lever-arm will be greater, and consequently the mo- ment of the force which he exerts. It is familiar to every one how much is gained by a long lever in producing an effect ; that this effect is increased not only by increasing the force, but also by increasing the length of the arm through which it acts. Seeing that this was the case, Archimedes is reported to have said that with a lever long enough he could move the world. Exercises. 1. A current flows east at the rate of 4 miles an hour, and a vessel heads north at the rate of 10 miles an hour : draw a diagram showing the true direction and velocity of the vessel. 2. Four men pull at a rope with forces of 40, 50, 25, and 60 pounds in the same direction : what is the resultant pull ? If the two latter pull in an opposite direction from the others, what is the intensity and direction of the resultant ? 3. Two men carry a basket ; one pulls upward with a force of 20 pounds, and the other with a force of 40 pounds: what is their re- sultant and the weight of the basket? 4. A body is given simultaneously three blows, one eastward at the rate of 40 feet per second, one northward, 28 feet per second, one westward, 32 feet per second : which way does it move, and with what velocity ? 71. Work. Work consists in moving against resistance. A horse or an engine does work when it pulls a load, a bird when it propels itself through the air, a man when he lifts up a weight. Let us take the latter case. When a load is lifted, a cer- MOTION AND FORCE. 31 tain amount of work is done ; when it is lifted twice as high, twice as much work is done, or when the weight is twice as great, twice as much work is done when twice as great a weight is lifted through three times the height, six times the work is done ; or, "Work done = weight X height. In general, the work done by any force is the product of the force and the distance through which the point of application is moved. 72. Unit of Work. A unit of work is the work done in raising a unit of weight through a unit of height. In the English system the units are the foot and the pound, and the unit of work is called the foot-pound; in the French system the kilogram and metre are used, and the unit of work is the kilogram-metre. 73. Horse-Power. For large engines a larger unit is used, the horse-power. This is equivalent to 33,000 foot- pounds per minute. 1 An engine capable of lifting 33,000 pounds 1 foot in 1 minute, or 66,000 pounds 1 foot in 2 minutes, or 11,000 pounds 6 feet in 2 minutes, is an engine of 1 horse-power. Multiply weight in pounds by height in feet, divide by the number of minutes and by 33,000, and we have the horse-power. 74. Erg. As these units depend on gravity, which is variable, another, based on the French system, has been employed, called the* erg; the erg is the work done by a force of one dyne acting through one centimetre. Exercises. 1. How many foot-pounds of work are done in lifting 20 pounds through 10 feet ? how many kilogram-metres ? 2. An engine can lift 2 tons 20 feet in 40 seconds: what is its horse- power ? 3. An engine can lift 20 kilograms 20 metres in 20 seconds : what is its horse-power ? 4. How many ergs of work are done by a force of one dyne acting through a metre? 5. A force gives to a decagram of matter a velocity of 2 centimetres 1 The element of time enters into horse-power, but not into foot- pounds. 32 NATURAL PHILOSOPHY. a second. If this force acts through a metre, how many ergs of work are done ? 75. Energy. Energy is ability to do work. A moving body has this ability, hence it has energy. A body lifted up has this ability, hence it has energy. The units of energy are the same as the units of work. 76. Potential Energy. A weight held up by the hand has the power by virtue of its position to fall, and hence do work, if its support be withdrawn. A body of water held up by a dam has the power to do work on a water-wheel, if allowed to fall upon it. A wound-up spring has power to perform work in turning the machinery of a clock. This kind of energy is called energy of position, or potential energy. 77. Actual Energy. A weight descending, water falling on a wheel, a spring uncoiling, a bullet moving through the air, a muscle in use, have energy, energy of motion, or actual energy. 78. Formula for Potential Energy. The formula for potential energy is w X ^, where w represents the weight of a body, and h the height to which it is raised. For it is evident that increasing either of these quantities will proportionately increase the ability of the body to do work. 79. Formula for Actual Energy. The formula for actual en- ergy is ?m> 2 u where ra represents the mass, and v the velocity of the moving body. For its momentum is mv (Art. 49). Now, suppose it to be moving against a resistance which takes one unit from its momentum each second, it will then require mv seconds to bring it to rest, and its mean velocity will be %v, for it diminishes uniformly from v to nothing. The distance through which the body would move is mv X \v = %mv z . It therefore does \mv l units of work upon the resistance (for the resistance is supposed to be a unit of force), and its actual energy is %mv z . 80. Energy of a Projectile. When a ball is thrown up- ward, its energy of motion becomes gradually less and less, and its energy of position greater and greater. At its highest point the one is nothing, the other is the greatest possible. During the fall the conditions are reversed. We MOTION AND FORCE. say that the energy of motion is converted into energy of position in the ascent, and converted back in the descent. 81. Potential and Actual Energy equal. We have proved the two formulae, Potential Energy = wh. Actual Energy $mv*. In the section on falling bodies we will prove other formulae, which will show that the energy of motion which a body has at the begin- ning of its ascent is just equal to the energy of position at its highest point ; that is, that under these conditions wh -= $mv*. 82. Conservation of Energy. This brings us to the very important doctrine of the conservation of energy. This says that energy is always conserved or preserved ; that it is never destroyed, but may be converted into energy of an- other form ; that the sum of the energies of the universe, like the sum of the matter of the universe, is constant ; that energy is indestructible, as matter is. We cannot follow energy through all its transformations, any more than we can follow matter, but we have the best of grounds for believing in the truth of the theory. We will show in future chapters that heat, light, and electricity are motions of the particles of bodies or of the ether; hence we have other forms of energy in them. These are all convertible, without loss, into one another and into the two forms mentioned above. When a nail is struck by a hammer, it becomes hot, for the force of the blow is changed into heat, and sometimes, when sparks are struck, into light. When water falls on a wheel from a pond, its energy of position, first converted into energy of motion, moves the wheel ; but part of this energy produces heat by striking the wheel, part produces heat in the bearings, and part runs the machinery. If an electrical machine be connected with it, some of the energy will be converted into electricity with its attendant light. In a steam-engine the energy of position of the mole- cules of coal is converted into heat, and the heat finally into motion of the piston. c 34 NATURAL PHILOSOPHY. 83. Correlation and Conservation. The principle that one force can be converted into another is the correlation of forces, while the principle that in this correlation no energy is lost is the conservation of energy. These are long names, but they express truths which are of great im- portance in modern science, and should be thoroughly understood. Exercises. 1. How many foot-pounds of energy of position has a weight of 20 pounds 8 feet above the floor, with respect to the floor ? a table 3 feet high stands on the floor : how much has the weight with respect to this table ? Ans. 160, 100. 2. A bullet of 1 ounce is shot from a 20-pound gun with a velocity of 1600 feet per second : has the motion of the bullet or the recoil of the gun greater energy ? and how much ? Note. Because the momentum of the bullet equals the momentum of the gun, 1600 X xV = 20 X velocity of recoil. Velocity of recoil = 5 feet per second. W Energy of motion = \ mv 2 = J v 2 . For the bullet = $ X oifex (1600) 2 = 2484. -f . 20 For the gun = J X X 5 2 = 7.7 +. Note. From this we see the diiference between momentum and energy. It is the energy, not the momentum, which gives the power to the ball to penetrate bodies and to do harm. As we increase the velocity, we increase the momentum in the same proportion, but we increase the energy in the square ratio. As velocity is doubled, mo- mentum is doubled, but energy is quadrupled. As velocity is trebled, momentum is trebled, but energy is increased ninefold. Perhaps we can understand this better if we consider that as it goes twice as fast it will meet twice as many particles in the same time, and it will crowd them away twice as fast ; that is, it has four times the effect ; if it has three times the velocity, it will have nine times the effect ; and so on. 3. State what transformations of energy take place in sliding a body down a plane, in the electric light, in ringing a bell, in lighting a match, in a clock running down, in a pendulum swinging. 4. Which would be preferable, to carry a 40-pound trunk up 20 feet or a 60-pound trunk up 15 feet ? 5. How many pounds of water per minute will a 20 horse-power engine raise through 200 feet ? Ans. 3300. MOTION AND FORCE, 35 GKAYITY AND STABILITY. 84. Effects of Attraction, The earth attracts every par- ticle of matter towards itself. This gives us the phenom- ena of falling bodies, causes matter to have weight, makes the surface of still water level, and constantly operates in many ways we do not notice. f 85. How Attraction Acts. It does not require any time 'I for this force to act. Attraction traverses the great space Ibetween the sun and the earth, to the best of our knowledge, instantaneously. Nor does the interposition of another body affect it in any way. We can cut off sound or heat or light by the interposition of a wall, but attraction acts through it without diminution. Nor does the kind of mat- ter make any difference. Every molecule is attracted alike, the number of molecules determining the total attraction. 86. Law of Attraction. The law of gravity, which was discovered by Newton, is that every portion of matter in the universe attracts every other portion with a force directly pro- portional to the masses, and inversely proportional to the square of the distance between them. Questions. How much is the attraction between two masses changed by doubling the distance between them ? by increasing it 5 times? by doubling one mass? by doubling one mass, trebling the other, and trebling the distance between them ? 87. Decrease of Gravity Downward. The gravity is greatest at the surface of the earth. When we go down into the earth the gravity decreases, because some of the matter of the earth is attracting us upward. Were we to get half-way to the centre we should only have half the weight that we have at the surface. At the centre we should have no weight, being equally attracted in all directions. Were it possible for a body to fall freely towards the centre, it would increase its velocity continually and fly to the surface on the other side, thence back again. Were there no resistance, this would go on forever. Otherwise, 36 NATURAL PHILOSOPHY. the vibrations would become smaller and smaller, and the body would finally settle at the centre. 88. Centre of Gravity. The centre of gravity of a body is the point about which it will balance in every position. If a body has a uniform figure and the same structure throughout, the centre of gravity is in t.he centre of the figure, and is readily found. The centre of gravity of a homogeneous sphere is at its centre ; of a cylinder, at the centre of its axis ; of a uniform ring, not in the mass of the ring, but in the space in the centre ; of a rectangular block, where its- diagonals intersect. 89. How to find the Centre of Gravity. If a body is hung up by a string, the centre of gravity will be in the line of the string pro- longed down- ward. If a new point of suspen- sion be taken, and the line pro- Fia. 17. CENTRE OF GRAVITY. , longed down- ward, it will cut the first line in the centre of gravity. This enables us to find the centre of gravity in certain cases. In the case of a thin body, we may balance it over a ruler in two directions, or over the edge of a table, as in Fig. 17. If the lines of the ruler or the edge of the table be marked on it, their intersection will be the centre of gravity. In general, its position has to be found mathematically. Experiment 13. Lay a thin board on a ruler, and find its centre of gravity as described. Bore a hole here and insert an awl. Notice how the board is balanced in every position. Experiment 14. Bore a hole in a board, and insert an awl, on which hang a plumb-line. Mark the path of the line on the board. Do this again from some other point. The intersection of the lines is the centre of gravity. 90. Representation of Weight as a Force. A line down- MOTION AND FORCE. 37 ward from the centre of gravity of a body may represent its weight; that is, it will be the resultant of all the parallel pulls of the earth on its different particles. Hence in treating of the weight of a body as a force we must represent it by a line, in the direction of a plumb-line, downward from its centre of gravity. 91. Base of Support and Centre of Gravity. If a body rests on a support, and a line from the centre of gravity downward meets this support within the base of the body, it will remain in position ; if not, it will slide or overturn, for the downward pull meets with no resistance. Experiment 15. Find the centre of a board, as in Experiment 14. Tilt it sidewise, and notice that when the centre is exactly over the FIG. 18. CENTRE OF FIG. 19. CENTRE OF GRAVITY. GRAVITY AND BASE OF SUPPORT. point of support a, as indicated by a plumb-line, the body is just ready to turn either way. Experiment 16. Place a light piece of stick, a>, with one end resting on a table. At b notch it so that another stick, be, may fit in the notch and press against the handle of a bucket under the table. A string, cd, must also be attached to the handle. A great weight may now be placed in the bucket, for the centre of gravity of the weight comes under the support at a. If b is depressed, it raises the centre of gravity, and hence b is again quickly raised. A wagon at rest will overturn when the line drawn from the centre of gravity falls outside the wheels. The tower of Pisa * could be made to overturn by building it higher, for the centre of gravity would thus be thrown farther out. When a man stands erect, the line from his centre of 1 Where is this, and how constructed? 4 38 NATURAL PHILOSOPHY. gravity falls between his feet. In beginyiing to walk, he throws his body forward, so as to bring his centre of gravity FIG. 20. LINE FROM CENTRE OF GRAVITY MUST FALL INSIDE THE WHEELS. in front of his feet. He would now fall did he not catch himself by throwing one foot forward. The operation is then repeated with the other foot. He also throws his body from side to side, so as to keep the centre of gravity over the foot which is on the ground. In carrying a FIG. 21. UNSTABLE, NEUTRAL, AND STABLE EQUILIBRIUM. weight on his back, he leans forward, and in carrying it in one hand he leans sidewise, for the same reason. 92. Stability. The position of a body is stable when any MOTION AND FORCE. 39 overturning force beginning to act will tend to cause its centre of gravity to rise, as a brick lying flat ; for then it will of itself return to its original position when the force is withdrawn. It is unstable when the slightest overturning force causes its centre of gravity to fall, as a cane bal- anced on end, in which case it will not recover its position, but will go farther from it. It is neutral when the over- turning force causes motion in a horizontal line, as a ball on a floor; then it will come to rest in any position. 93. Measure of Stability. The more the centre of gravity has to rise in overturning, the more stable the body is. A brick on its flat side is more stable than a brick on end. To over- turn it the centre of gravity has to be raised through the verti- cal height ab, which is a much Fm 22- _ STABIUTY . greater distance in one case than in the other, and therefore a much greater force is required. The work done in overturning is the weight multiplied by ab. Exercises. 1. When will a body slide, and when roll, down an inclined plane? 2. In rising from a chair, why do we lean the body forward ? 3. Why is it easier to walk on a fence with a long stick in the hand? 4. When is a pendulum in stable equilibrium ? 5. A cone balanced on its apex is in what kind of equilibrium ? on its base ? on its side ? 6. Why cannot a person pick up an object from the floor in front of him when standing with his heels against a vertical wall ? 7. Should the centre of gravity of a ship be high or low ? of a wagon ? 8. Why is it easier to suspend an iron ring on a nail on the inside than to balance it on the outside ? 9. What would a 200-pound man weigh if moved to within 1000 miles of the centre of the earth ? Ann. 50 pounds. 40 NATURAL PHILOSOPHY. FALLING BODIES. 94. How much a Body will Fall in a Second. The attrac- tion of the earth is such that it will cause a body starting from rest to fall about 16.1 feet (about 4.9 metres) in one second. Its velocity at the beginning was nothing, and it in- creased uniformly during the second. Hence at the end it is moving at the rate of 32.2 feet (9.8 metres) ; that is, it has acquired a velocity which if continued uniformly would carry it over 32.2 feet (9.8 metres) the second second. But during this second second it has also the pull of the earth, adding 16.1 feet more to the space passed over, making 48.3 feet, or three times 16.1 feet, in that second. The third second it has an acquired velocity of 64.4 feet, and an additional pull of 16.1 feet, making 80.5 feet, or five times 16.1 feet, as the fall during the third second. In general, the fall through any second is found by taking the series of odd numbers, 1, 3, 5, 7, etc., and mul- tiplying 16.1 by the number in the series corresponding to the given second. Let it be required to find the fall during the sixth sec- ond. The sixth number of the series is 11. 11 X 16.1 = 177.1 feet. The space fallen through in the first three seconds will evidently be (1 -f 3 -f 5) X 16.1 ; in the first five, (1 + 3 _j_ 5 _f_ 7 _|_ 9) x 16.1 ; and so on. The sum of the numbers in the parenthesis is found by adding the end terms, and multiplying by half the number of terms, or the number of seconds. 1 1 If the ninth second is given, we find the odd number correspond- ing by multiplying 9 by 2 and subtracting 1. If we add the first two odd numbers, it gives the square of 2 ; if the first three, the square of 3; and so on. Thus, 1 + 3 = 4, 1 + 3 + 5 = 9, 1 + 3 + 5 + 7 = 16. We thus see a reason for the rule, to be announced farther on, that the spaces passed over vary as the squares of the times. MOTION AND FORCE. 41 To find the space through which a body would fall in nine seconds, we add the end terms 1 and 17, and multiply by -| and by 16.1, or (17 + 1) X f X 16.1 = 1304.1 feet. 95. Formulae for Falling Bodies. But it is better to work out some general formulas. Let s represent the space passed over ; " t " " time; " v " " velocity; " g " " acceleration produced by gravity in one second = 32.2 feet = 9.8 metres, which is taken as the measure of gravity. As g is the velocity acquired in one second, in t seconds we will have v = gt. (1) But as the velocity uniformly increases from nothing to gtj the mean velocity is J gt, and the space passed over in t seconds with this velocity is = }0fX* = Iff?. (2) From(l), t=l (3) Substitute in (2), = l