135. Conservation of angular momentum. Newton's first law of motion may be stated thus: A body maintains a constant linear momentum unless acted upon by some external force. Similarly, it is true that a rotating body tends to maintain a constant angular momentum unless acted upon by some external torque. Since angular momentum is a vector quantity, to say that the angular momentum tends to remain constant means not only constant in magnitude but also constant in direction. For example, in revolving around the sun the earth tends to keep the angular momentum about its own axis constant. Hence the earth's axis always points in the same direction,* approximately toward the North Star. A man throwing a quoit or a horseshoe, and wishing to prevent its turning over while in flight, gives it a spin. It will then travel with the axis of rotation always pointing in the same direction. In the same way, a coin when given a spin about an axis perpendicular to its face will travel without turning over. The barrel of a rifle or a cannon is "rifled" to give the projectile a rotation. A rotating projectile will travel without "tumbling" and will strike an object with its nose. The same thing is beautifully shown by a kicked or thrown spinning football, often called a spiral. There are a large number of simple cases which are readily explained by this principle, which states that angular momentum tends to remain constant. Take the case of a stone on the end of a string, the other end of the string being held by the right hand. Let the string and the stone be given a considerable angular velocity. If the string is drawn with the left hand through the right hand, so that the part rotating with the stone is shortened, the angular velocity of the stone and string will increase. This increase is due to the fact that the angular momentum tends to remain constant. Shortening the string decreases the moment of inertia of the stone. Since the angular momentum is the product of the moment of inertia and the angular velocity, the angular momentum can remain constant only in case the angular velocity increases at the same rate that the moment of inertia decreases. *This is only approximately true, for there are small external moments of force acting on the earth. These produce a slow precession of the axis (sect. 129). Nearly every child learns to "work up" in a swing, but few people know the principle involved. The child usually makes many unnecessary motions and usually does not know which are the essential ones. The necessary motions are (1) to raise the center of gravity of the body as the swing moves through the middle position, where it has the greatest velocity; (2) to lower the center of gravity of the body with respect to the swing at the extreme end of the motion. The action is explained as follows: When one raises his center of gravity at the middle part of the motion, where the speed is the greatest, he shortens the radius of the angular motion and hence decreases his moment of inertia. But in this part of the motion the resultant of the external forces is zero; hence the angular momentum is constant. If the angular momentum is not to change as the swing moves through the central part, the angular velocity must increase when the moment of inertia is decreased. At the end of the swing, where the velocity is zero, the angular -momentum is also zero, and the lengthening of the radius by lowering the center of gravity does not affect the action.* (If one lowers his center of gravity at the middle point and raises it at the end of the swing, the motion is quickly arrested. Why?) When water which is escaping from the central outlet in the bottom of a washbowl is given a slight rotational motion, the angular velocity of the water as it nears the center increases. This is due to the fact that the angular momentum of the water tends to remain constant, and this can be done for shortening radii only by an increasing angular velocity. In this case the action of the centrifugal force of the water is also clearly shown. It has the effect of an outward pressure banking the water up, so that often there is a central cavity down through the moving mass of water. The motion of tornadoes is somewhat similar to that of the water in the washbowl. In the center of the tornado there is a strong uprush of air, and the whirling mass of air is moving inward, just like the water in the bowl. The tendency for the angular momentum of the air to remain constant explains the enormous velocities which are frequently developed. * When one raises himself as the swing moves with its maximum speed through the middle part, he does work against centrifugal force. This is the work that supplies the energy. Tumbling in gymnasium work furnishes many fine examples of this tendency of angular momentum to remain constant. For example, when a man jumps off a springboard, he can give himself a certain amount of angular momentum. If he now doubles up, bringing his head, arms, knees, and feet near his center of gravity, he will decrease his moment of inertia and thus increase his angular velocity. He may increase his angular velocity enough so that he can turn more than one somersault while in the air. Just before he reaches the floor he straightens out, increasing his moment of inertia and thus decreasing his angular velocity. 136. Total kinetic energy. A moving body may have kinetic energy both on account of linear translational motion and on account of rotational motion. If a sphere rolls down an incline, the total kinetic energy developed is given by the formula K.E. = + mv2 + + Ιω2, where v is the linear velocity of the center of the sphere, w the angular velocity about a line through the center, m the total mass, and I the moment of inertia about a line through the center. A sphere and a cylinder of equal masses do not have the same moment of inertia. They will therefore roll down an incline with different linear speeds. (Why?) Not all the energy given a rifle or a cannon ball goes into linear kinetic energy; some of it goes into rotational energy. The same is usually true of a thrown or batted baseball. PROBLEMS 1. A circular disk (or cylinder) has a radius of 6 in. and weighs 10 lb. Compute its moment of inertia about an axis perpendicular to the disk. 2. How many foot-pounds of energy will the disk of Problem 1 have when making 4 revolutions per second? 3. How much work would be required to give a spherical ball, mass 4 kg. and radius 5 cm., an angular speed of 4 revolutions per second? 4. A flywheel which is a uniform circular disk of 12 in. radius weighs 320 lb. How much energy is stored in it when it makes 600 revolutions per minute? 5. A rod, the mass of which may be neglected, has a mass which weighs 4 lb. fastened on one end, and one which weighs 5 lb. on the other. The distance between the masses is 4 ft. Compute the moment of inertia of rod and weights when rotated about a line perpendicular to the middle point. 6. A light rod 30 cm. long has a mass of 1 kg. on each end. How much torque would be required to give this rod in 20 sec. an angular speed of 2 revolutions per second about an axis perpendicular to the middle point of the rod? 7. Two iron balls, weighing 25 lb. each, are connected by a very light rod so that their centers are 3 ft. apart. (a) Compute the moment of inertia about an axis perpendicular to the rod at its center. (b) How much torque would be required to give this an angular speed of 360 revolutions per minute in 20 sec.? 8. The entire weight, 200 lb., of a wheel 3 ft. in diameter is practically concentrated at the rim. (a) Compute its moment of inertia. (b) What torque will be necessary to give it a speed of 1 revolution per second in 5 sec.? CHAPTER XII GRAVITATION Gravitation, 137. Gravitational attraction is universal, 138. Newton's law of gravitation, 139. Mass of the earth, 140. Value of the gravitational constant, 141. Variations in the value of g, 142. Tides, 143. Cause of tides, 144. 137. Gravitation. Nearly everyone knows that the reason a stone falls to the ground is that the earth attracts the stone. There are, however, things about this which not all know. Is this attractive power possessed only by the earth? Do all masses attract each other? Will a plumb line hung near a mountain be pulled out of its vertical position by the attraction of the mountain? Is Newton's third law of motion true for this attraction? That is, if the earth attracts a stone, will the stone attract the earth? Everyone at some time or other wonders about the mysterious forces which hold the moon and the planets in the sky. Can these be explained? Largely through the aid of the imagination and the keen mind of Sir Isaac Newton these questions have been answered. 138. Gravitational attraction is universal. Newton was the first to see that the attraction of the earth for bodies on its surface was only a special case of a general law. According to his idea this attraction was not something possessed by the earth alone but was common to all bodies; every body attracted every other body, it made no difference how large nor how small. Bouguer, in Peru, about 1740, was the first to show the attractions of masses smaller than the earth. He found that a plumb line hung near a mountain was not parallel to another hung at some distance from the mountain. This deviation was due to the attraction between the plumb bob and the mountain. A later modification was to use two plumb lines hung on opposite sides of |