205. Communication of Motion in Elastic Bodies.—Momentum is transferred from one body to another very differently in elastic from what it is in non-elastic bodies. As you saw in § 201, when one non-elastic body strikes upon another the momentum is divided between them, and both move on together. Now if a and b, Fig. 133, were elastic bodies, as ivory balls, and b should be let fall against a, it would give all its momentum to a. Therefore b would stop, and a would move on to the same height from which b came. The reason is, that the velocity lost by b and received by a is just double what it would be if the balls were non-elastic. For the same reason, if a and b, being elastic, meet each other from equal heights on the arc, they will both rebound, and return to the same heights from which they came. But if non-elastic they simply destroy each other's momentum and stop. The effect produced in the former case is just twice as great as in the latter, as you may see by reckoning on the arc. For the same reason, too, if you have a row of elastic balls, as in Fig. 134, and let a fall from the point i upon b, it will stop there; and communicating all its momentum to b, this momentum will pass from b to c, and so on through all the row of balls to e, the last one, which will fly off to the point h, at the same height with i, the point from which a fell. If b be held still, and a be let fall upon it, a will rebound to the height from which it fell, for then the compressed elastic spring (§ 39) of each ball, as b is immovable, communicates all the motion to a. It is for this reason that an elastic ball, on being thrown against any thing fixed, rebounds. If what it is thrown against be perfectly elastic it rebounds with a force equal to that with which it is thrown.
206. Reflection of Motion.—If an elastic body be thrown perpendicularly upon a surface it rebounds in the same path in which it is thrown. But if it hit the surface obliquely it is thrown off or reflected in a different direction. Thus a ball thrown from b upon c, Fig. 135 (p. 158), will return in the line drawn to b. But if it be thrown from d it will be reflected in the line c a. Now the angle d c b, called the angle of incidence, is always equal to b c a, the angle of reflection. The same, you will find in other parts of this book, is true of sound and light and heat.
207. Uniformity of Motion.—Since motion, when once begun, is disposed to continue unless arrested by obstacles, it is naturally uniform both in its velocity and its direction. I will speak now only of velocity. Suppose a body to be set in motion, and to meet with no opposition from friction, or the resistance of air, or attraction, it would move on forever, and with the same velocity with which it began. Now precisely these circumstances we have in the motion of the heavenly bodies in their orbits. They are, it is true, under the influence of attraction, but in such a way, as you will soon see, as not to interfere with the uniformity of their motion. Were it not for this uniformity we should have no regularity of times and seasons. It is only by the uniform motion of the earth round the sun, and round its own axis, that we can calculate for to-morrow, or next week, or next year. If these motions were irregular it would throw confusion into all our calculations for the future and all our recollections of the past. We can measure time by nothing else but regular motion, and were there no regular motion we should have merely the very inaccurate measure furnished by our sensations. To measure time with accuracy we take some great and extensive uniform motion as our standard. Thus, the revolution of the earth around the sun we take as one division of time, and call it a year. We observe that during this time it whirls around on its own axis 365 times, and the time occupied by each of these revolutions we call a day.
208. The Pendulum.—Various modes of measuring time have been adopted by mankind. At first time was inaccurately divided by merely observing the sun. But after a while man resorted to various contrivances to measure short periods of time with accuracy. All of these depend upon the uniformity of motion alone. The sun-dial measures time by the uniform movement of the shadow on its face, caused by the uniform movement of the earth in relation to the sun. The hour-glass measures time by the uniform fall of sand produced by the attraction of gravity. The best measurement of time is by the comparatively modern invention of clocks and watches, in which time is divided into very minute periods by the uniform motion of the pendulum or the balance-wheel. The pendulum furnishes an interesting example of motion kept up by the influence of gravity. It was not till the time of Galileo, less than three centuries ago, that its operation was understood and appropriated to the measurement of time. He observed that chandeliers hanging from lofty ceilings vibrated very long and uniformly after they were accidentally agitated, and the thought of the philosopher evolved from this phenomenon the most important results. Though it had been before men's eyes in some shape or other since the creation, it was reserved for Galileo to observe its significance, and the result is that the pendulum has become man's time-keeper over the whole earth.
209. Explanation of its Operation.—A pendulum consists commonly of a ball or weight at the end of a rod suspended so as to vibrate with little friction at the point of the suspension. Let a b, Fig. 136, represent such a pendulum. When it is at rest it makes a plumb-line hanging toward the centre of the earth. If it be raised to c and be left to fall, the force of gravity will not only carry it to b, but, by the accelerated velocity or accumulated momentum which it gives it in its descent, it will carry it to d. The same would be true of its return from d. And it would vibrate forever in this way if it could be entirely freed from the resistance of the air and friction. But, as it is, the pendulum left to itself gradually loses its motion from these obstacles. In the common clock the office of the weight is to counteract the influence of these obstacles, and keep the pendulum vibrating. In the watch the mainspring performs the same office to the balance-wheel.
The times of the vibrations of a pendulum are nearly equal whether the arc it describes be great or small. For when the vibration is a large one the velocity which the pendulum acquires in falling is greater than when the vibration is of small extent. The reason is that the higher it rises the more steep is the beginning of its descent. Thus a c, Fig. 137, is steeper than c b.
210. Gridiron Pendulum.—The longer a pendulum is the longer time does its vibration occupy. It requires a pendulum of the length of a little over thirty-nine inches to vibrate seconds. Cold weather, by contracting the pendulum, makes it vibrate quicker than in summer, and so makes the clock go faster. Various contrivances have been resorted to in order to counteract the variation of length in pendulums by heat and cold, but what is called the gridiron pendulum is the best. In this pendulum an ingenious use is made of the fact that heat expands brass nearly twice as much as it does steel. A simple form of this pendulum is given in Fig. 138. The middle rod is made of brass, and the side rods, b and c, of steel. Suppose that the brass rod expands or increases in length half an inch. The rod c would be drawn upward by it, and the rod b downward, each one quarter of an inch; but this effect is counteracted by the expansion of each steel rod, which is half that of the brass, that is, one quarter of an inch. The ball d, therefore, always retains the same distance from the point of suspension, e. In Fig. 139 you have a gridiron pendulum of a more compound character, a part of the bars being steel, and a part brass.
211. Motion Disposed to be Straight.—When a body is set in motion, if it be left to itself—that is, if nothing interfere with its motion—it will move in a perfectly straight line. It requires some interference from some force to bend the motion. You will readily see from the views which I have given you that there never is any motion that is, strictly speaking, straight, because every motion is in some measure compound; that is, each cause of motion is modified in its action by other causes of motion. But we can approximate very nearly to straight motion by making one cause preponderate very much over other causes. This I will illustrate. If we fire a bullet horizontally from a gun it is acted upon by three forces: the propulsive force of the powder, the resistance of the air, and the attraction of the earth. The action of the second of these is in direct opposition to the first, and therefore only retards the motion, and does not tend at all to turn it from its straight course. This is seen in the fact that the ball is turned neither to the right hand nor to the left. But the third force tends to make the ball bend its course toward the ground. It does this from the instant that the ball leaves the gun throughout its flight, but so slightly that practically we can consider the ball as going straight for short distances. When we take a long range we must make allowance for this bending down of the motion. Accordingly, for the sake of precision, a double sight is provided in modern guns, as seen at A and B, Fig. 140. This you see secures the pointing of the gun a little above the level of the object aimed at, that level being indicated by the dotted line.
The greater is the propulsive force the more nearly to a straight line is the path of the propelled body. This may be seen very clearly in Fig. 141, representing the issuing of water at different points from a vessel. As pressure in a liquid is as depth, § 121, the force with which the water is thrust out is greater at C than at B, and at D than at C. The issuing stream, therefore, is most nearly straight at the lowest point, D.
The motion of projectiles, thus alluded to, will be more particularly noticed farther on.
212. Compound Straight Motion.—We call that motion compound which is produced by two or more forces acting upon the body. This may be straight or curved. I will first speak of the straight. If a man attempt to row a boat straight across a river, the point which he will reach will not be directly opposite to that from which he started, but below. Two forces act upon the boat: the current tending to carry it straight down the stream, and his rowing tending to carry it straight across. The boat will go in neither of these directions, but in a line between them. Let A B, Fig. 142, represent the bank of the river, from which he starts at A, with the bow of the boat pointing to C, on the opposite bank. Suppose now that in the time that it takes him to row across the current would carry him down to B if he did not row at all. He will in this time, by the two forces together, reach the point D, opposite to B, his course being the line A D. So if the wind blow upon a vessel in such a way as to carry it eastward, and a current is pushing it southward, the vessel will run in a middle line, viz., southeast. For the same reason if a boy kick a foot-ball already in motion, it will not be carried in the direction in which he kicks it, but in a line between that direction and the direction in which its former motion was carrying it. In swimming, flying, rowing, etc., we have examples of compound motion, the middle line between the directions of the forces always being taken by the body moved.
If we take Fig. 142, illustrating the movement of the boat, and draw two lines, one from A to C and the other from B to D, we shall have the parallelogram A C D B, Fig. 143, in which the line A C represents the force of the rowing, A B the force of the current, and A D the path of the boat. You see, then, that if we wish to find in what direction and how far in a given time a body acted upon by two forces will move, we are to draw two lines in the direction of these forces, and of a length in proportion to the distances to which they would move it in that time; then by drawing two lines parallel to these we shall have a parallelogram, and the diagonal of this will represent the distance and the course of the moving body. If a body be acted upon by two equal forces and at right angles to each other, the figure described will be a square, as you see in Fig. 144. If they vary from being at right angles to each other the figure will vary in the same proportion from the square figure, as seen in Figs. 145 and 146. In the three figures A B and A D represent the two forces, and A C the resulting motion. You observe by these diagrams that the nearer the two forces come to being in the same direction the farther will they move the body. You see this in the different lengths of the diagonals in Fig. 144 and Fig. 146. The more nearly, therefore, the wind coincides with the current the more rapidly will a vessel be carried along before the wind. When, on the other hand, the angle at which two forces act upon a body is much greater than a right angle, they will propel it but a small distance. Thus if two forces act on a body in the directions D A and D C, Fig. 147, they will move it only the distance represented by the diagonal D B. This diagram represents the motion of a vessel sailing almost directly against a current by a wind the force of which is equal to that of the current, while Fig. 146 represents the motion of a vessel where wind and current being of equal force very nearly coincide. In the above diagrams I have supposed the forces to be equal; but the same truth can be shown in regard to unequal forces as seen in Fig. 143.
213. Curved Motion.—No single impulse can produce a curved motion. Neither can two or more impulses communicated at one time. In both of these cases the motion would be in a straight line. Curved motion may be produced by two forces, one of which gives it a single impulse, and the other acts upon it continuously. A familiar example you have in a ball whirled around at the end of a string. You can give it an impulse, and then, holding it in your hand, let it whirl. Here the impulse you give the ball is one force, and the tension of the string is the other, the latter acting continuously. Your hand holding the end of the string is the centre about which the motion revolves; the impulse which you have given the ball tends to make it fly away from the centre in a straight line, and hence is called the centrifugal force; the tension of the string keeps it from thus flying off, and so is called the centripetal force. When the earth, at the creation, was put in motion it would have moved in a perfectly straight line, were it not constantly drawn toward the sun by attraction, the continuous action of this latter force being the same as the tension of the string in the case of the whirling ball. The force of attraction, then, is the centripetal force of the earth, and the impulse which was given to it by the Creator in the beginning is its centrifugal force; and, balanced between these two forces, the earth and all the heavenly bodies move uniformly onward in their orbits. The centrifugal force you see in these illustrations is simply the tendency of motion to a straight line from the inertia of matter; and this is constantly counteracted by the centripetal force.
214. Illustrations of Centrifugal Force.—When a wet mop is whirled the water flies off in every direction by its centrifugal force. On the same principle a dog, coming out of the water, shakes off the water by a semi-rotary motion.—When a suspended bucket of water is turned swiftly around the water rises high on its sides, and leaves a hollow in the middle. It is the tendency to fly away from the centre of motion that causes this.—Large wheels, revolving with great velocity, have been broken by the centrifugal force of its particles, and hence the necessity of having such wheels made very strong. The immense grindstones used in gun-factories have sometimes been broken through in the middle, or have flown into pieces from the same cause.—A man riding horseback on turning a sharp corner inclines his body toward the corner, to avoid being thrown off by the centrifugal force. So, in the feats of the circus, a man standing on a horse running at full speed around the ring inclines his body strongly inward, as you see in Fig. 148 (p. 167). The horse also instinctively inclines in the same direction for the same reason. If the rider finds himself in danger of falling, by making the horse go a little faster, thus adding to the centrifugal force, the difficulty is relieved.—The centrifugal force is made use of in milling. The grain is admitted between two circular stones by a hole in the centre of the upper one, and as the stone revolves it constantly moves toward the circumference, and there escapes as flour.
215. Bends in Rivers.—We see the operation of the centrifugal force in the bends of rivers. When a bend has once commenced in a river it is apt to increase, for as the water sweeps along the outer bank of the bend it presses strongly against it, just as the water in the whirled bucket, § 214, presses against its sides, by its centrifugal tendency, or, in other words, its tendency to assume a straight motion. Of course the result is a wearing away of this outer bank, and in proportion to the looseness of the material of which it is composed and the velocity of the river's current. And when one bend is formed another is apt to form below, but in an opposite direction. The water, by sweeping along the bend a, Fig. 149, is directed by it toward the opposite bank at b, and makes a bend there also. It is in this way that a river, running through a loose soil, the Mississippi, for example, acquires a very serpentine course. As the water in the whirled bucket rises around the sides, so in the river the water will be higher against the bank a than on the opposite side. Eddies and whirlpools are produced on the same principles, when water is obliged to turn quickly around some projecting point. If a current were moving swiftly along the shore a toward the point b, Fig. 150, it would be directed outward by the resistance of this projection, and so a depression would be left at c, just behind it, and this depression would be surrounded by a revolving edge of water.
216. Application of the Centrifugal Force in the Arts.—Much use is made of the centrifugal force in the arts, but I will give but two examples. In the art of pottery the clay is made to revolve on a whirling table, the workman at the same time giving the clay such shape as he chooses with his hands and various instruments. In doing this he constantly has reference to the centrifugal force, giving the table a velocity proportioned to the amount of this force which is needed in each stage of the operation. The most beautiful application of this force that I have ever witnessed is in the manufacture of common window-glass. The glass-blower gathers up on the end of his iron tube a quantity of the melted glass, and blows it out into a large globe. When it is of sufficient size and thinness he places it on a rest, as you see in Fig. 151 (p. 169). A second man now comes with a rod having some melted glass on the end, and attaches this to the globe at a point opposite to that where the tube of the first man is joined to it. There now comes a boy, and, giving this tube a quick blow, severs its connection with the globe, leaving a hole in the globe where the glass breaks out. The second man, having the globe attached to his rod, carries it to a blazing furnace, and resting the rod on a bar at its mouth, puts the globe directly into the flame. The glass is soon softened, and he whirls the globe continually around. The hole in the globe enlarges by the centrifugal force, and at length by this force the globe is changed into a flat, circular disk. Panes of glass which are called bull's-eyes are cut from the centres of these disks.
217. Steam-Governor.—The operation of the centrifugal force is beautifully exemplified in this regulator of the steam-engine. It consists of two heavy balls, Fig. 152, suspended by bars from a vertical axis, the bars being connected to the axis by hinges. The bars have also a hinged connection at their lower ends with two smaller bars, and these latter have a similar connection with a collar that slides up and down on the axis. Now the faster the axis turns the farther the balls fly out from it, from the centrifugal force, and the higher the collar slides up on the axis. From the collar extends, as you see, a lever. This is connected with a valve in the steam-pipe, and so regulates the amount of steam that enters the working part of the engine. The object of this ingenious contrivance is to make the engine regulate its own velocity. When it is not working too fast the valve in the steam-pipe is wide open. But the moment that it works too rapidly the balls extend out far from the axis, so that the collar rises, and by the lever partly closes the valve. Less steam, therefore, can come to the engine, and the engine working in consequence less rapidly, the balls fall again, opening the valve. You see, then, that the regulation of this valve by the governor effectually prevents the action of the engine from becoming too rapid.
218. Shape of the Earth Influenced by the Centrifugal Force.—If the potter should make a ball of soft clay revolve rapidly around on a stick run through it, the ball would bulge out at the middle, where the centrifugal force is greatest, and would be flattened at the ends where the stick runs through it. This is precisely what has happened to the earth. At the equator, where the centrifugal force is greatest, it has bulged out about thirteen miles, while it is flattened at the poles. This shape was of course assumed before the earth became solid. In Fig. 153 we have the shape of the earth represented, N S being the polar diameter, and E E' the equatorial diameter. The tendency to take this shape from the centrifugal force may be illustrated by the instrument represented in Fig. 154. It consists of a set of circular hoops of brass, with an axis, b a. The hoops are fastened to the axis at a, but are left free at b. By a little machinery at the top they can be made to revolve rapidly, and bulging out at the sides by the centrifugal force, they slide down on the axis at b.
219. Projectiles.—I have already spoken of projectiles in § 211. You saw there that any body, as a cannon-ball, which is projected horizontally, falls to the earth in a curved line. Two forces act on the ball; viz., the projectile force given by the powder and the force of gravitation. The force of gravity being always the same, the shape of the curve which the projected body describes must depend on the force with which it is projected. This is very strikingly exemplified in the curves described by the different streams of water in Fig. 141. But whether the projectile force be great or small, the moving body thrown horizontally will in every case reach the ground in the same time. Thus if two cannons stand side by side on a height, one of which will send a ball a mile and the other half a mile, the two balls, if fired together, will reach the ground at the same instant, though at first thought it would seem that the ball which travels twice as far as the other would take a longer time to do it in. This is because the horizontal force of the ball does not oppose in the least the downward force of gravity. If it were thrown upward instead of horizontally, the projectile force would be opposed to gravity, and in proportion as the direction came near to being vertical. As horizontal force does not interfere with the action of the force of gravity, it follows that a ball dropped at the instant at which another is fired will reach the ground at the same instant that the fired ball does. This can be made clear by Fig. 155. Suppose it takes three seconds for a ball to fall from the top of a tower to its foot. In the first second it falls to a. The ball projected horizontally from the cannon, being operated upon by the same force of gravity, will fall just as far, and will be on a level with it at b. Both balls fall farther and farther each second, both being accelerated in the same degree because it is done by the same force. The projected ball will reach d when the falling ball is at c, and the plain at f when the falling ball is at e, the foot of the tower. The same holds true in all cases. A bullet dropped from a level with the barrel of a gun, paradoxical as it may seem, will fall to the ground no sooner than one which is shot from the gun.
220. All Falling Bodies really Projected.—When a body falls from any height, it does not, as you have already seen in § 187, fall in a straight line, as it appears to do. It falls in a curved line, for, like all projectiles, it is acted upon by a horizontal force as well as the force of gravity. But what is this horizontal force? It is the motion which the body has in common with the earth in its rotation on its axis. In this rotation the height from which the body falls goes to the eastward 1500 feet in a second. If, therefore, the body did not partake of the motion of the earth, and went to the ground in a straight line in a second, it would be when it reached the ground 1500 feet westward from the foot of the height from which it fell. But it does partake of the earth's motion, and goes eastward as fast as the height does, and so describes the curved line of a projectile. Suppose a ball falls from a height A, Fig. 156, and in a second of time that height passes to C. The forward or projectile force would tend to carry the ball to C, and the force of gravity would tend to carry it to B. But both forces acting together, it pursues a middle path, and this path is a curved line, because one of the forces is a continued force, § 213. For the same reason, if a ball be dropped from a railway car in motion, and it takes a second for it to fall, it will be at the end of that second just under that part of the car from which it fell. Although the car may have moved a considerable distance, the dropped ball, partaking of its motion, goes along with it in its fall. So a ball dropped from a mast-head when a ship is in motion goes along with the ship in its fall. The ball in each of these cases describes in its fall a curved line.
221. Motion in Orbits.—Why is it, let us ask, that a cannon-ball shot horizontally from some great height will not revolve around the earth like the moon. It has the same two forces acting upon it as the moon has—viz., a projectile force, and the attraction of the earth—and both ball and moon describe a curve in their motion. But the curve of the ball bends to the earth, while that of the moon ever sweeps around the earth. Why is this? First, there is the resistance of the air continually retarding the velocity of the ball. But, secondly, even if the ball could be projected from an elevation sufficiently high to be outside of the atmosphere, the force of the projection would not be great enough. We know, from the rate of progress of the heavenly bodies in their orbits, that it would require an immense velocity to keep the ball from being brought to the earth by its attraction. The Creator of these worlds, when he launched them into their orbits, gave them precisely that impulse which is needed to balance the centripetal force of attraction, and so they pursue a middle course between the two directions in which these two forces tend to carry them. And as their velocities have never been retarded by the resistance of air or any other substance, they have been ever the same from the beginning.