Fig. 153.
Fig. 154.
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.
Fig. 155.
Fig. 156.
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.