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a second, it follows that in this case the momentums produced in the two bodies by the two forces are exactly proportional to the forces themselves. In all cases in which forces simply overcome inertias this rule is found to hold. Thus, a 3000-pound pull on an automobile on a level road, where friction may be neglected, imparts in a second just twice as much velocity as does a 1500-pound pull. In view of this relation Newton's second law of motion was stated thus: Rate of change of momentum is proportional to the force acting, and the change takes place in the direction in which the force acts.
103. The third law. When a man jumps from a boat to the shore, we all know that the boat experiences a backward thrust; when a bullet is shot from a gun, the gun recoils, or "kicks"; when a billiard ball strikes another, it loses speed, that is, is pushed back while the second ball is pushed forward. The following experiment will show how effects of this sort may be studied quantitatively.
Let a steel ball A (Fig. 86) be allowed to fall from a position C against another exactly similar ball B. In the impact A will lose practically all of its velocity, and B will move to a position D, which is at the same height as C. Hence the velocity acquired by B is almost exactly equal to that which A had before impact. These velocities would be exactly equal if the balls were perfectly elastic. It is found to be true experimentally that the momentum acquired by B plus that retained by A is exactly equal to the momentum which A had before the impact. The momentum acquired by B is therefore exactly equal to that lost by A. Since, by the second law, change in momentum is proportional to the force acting, this experiment shows that A pushed forward on B with precisely the same force with which B pushed back on A.
FIG. 86. Illustration of third law
Now the essence of Newton's third law is the assertion that in the case of the man jumping from the boat the mass
of the man times his velocity is equal to the mass of the boat times its velocity, and that in the case of the bullet and gun the mass of the bullet times its velocity is equal to the mass of the gun times its velocity. The truth of this assertion has been established by a great variety of experiments.
Newton stated his third law thus: To every action there is an equal and opposite reaction.
Since force is measured by the rate at which momentum changes, this is only another way of saying that whenever a body acquires momentum some other body acquires an equal and opposite momentum.
It is not always easy to see at first that setting one body into motion involves imparting an equal and opposite momentum to another body. For example, when a gun is held against the earth and a bullet shot upward, we are conscious only of the motion of the bullet; the other body is in this case the earth, and its momentum is the same as that of the bullet. On account of the greatness of the earth's mass, however, its velocity is infinitesimal.
104. The dyne. Since the gram of force varies somewhat with locality, it has been found convenient for scientific purposes to take the second law as the basis for the definition of a new unit of force. It is called an absolute, or C.G.S., unit because it is based upon the fundamental units of length, mass, and time, and is therefore independent of gravity. It is named the dyne and is defined as the force which, acting for one second upon any mass, imparts to it one unit of momentum; or the force which, acting for one second upon a one-gram mass, produces a change in its velocity of one centimeter per second.
105. A gram of force equivalent to 980 dynes. A gram of force was defined as the pull of the earth upon 1 gram of mass. Since this pull is capable of imparting to this mass in 1 second a velocity of 980 centimeters per second, that is, 980 units of momentum, and since a dyne is the force required to impart in 1 second 1 unit of momentum, it is clear that the gram of force is equivalent to 980 dynes of force. The dyne is therefore a very small unit, about equal to the force with which the earth attracts a cubic millimeter of water.
106. Algebraic statement of the second law. If a force Facts for t seconds on a mass of m grams, and in so doing increases its velocity v centimeters per second, then, since the total momentum imparted in a time t is mv, the momentum imparted per second is ; and since force in dynes is equal to momentum imparted per second, we have
But since is the velocity gained per second, it is equal to the acceler
ation a. Equation (8) may therefore be written
This is merely stating in the form of an equation that force is measured by rate of change of momentum. Thus, if an engine, after pulling for 30 sec. on a train having a mass of 2,000,000 kg., has given it a velocity of 60 cm. per second, the force of the pull is 2,000,000,000 × 80 4,000,000,000 dynes. To reduce this force to grams we divide by 980, and to reduce it to kilos we divide further by 1000. Hence the pull exerted by the engine on the train =4081 kg., or 4.081
QUESTIONS AND PROBLEMS
1. What principle is applied when one tightens the head of a hammer by pounding on the handle?
2. Why does not the car C of Fig. 87 fall? What carries it from B to D? 3. Why does a flywheel cause machinery to run more steadily? 4. Balance a calling card on the finger and place a coin upon it. Snap out the card, leaving the coin balanced on the finger. What principle is illustrated?
5. Is it any easier to walk toward the rear than toward the front of a rapidly moving train? Why?
6. Suspend a weight by a string. Attach a piece of the same string to the bottom of the weight. If the lower string is pulled with a sudden jerk, it breaks; but if the pull is steady, the upper string will break. Explain.
7. Where does a body weigh the more, at the poles or at the equator? Give two reasons.
FIG. 87. A very ancient loop the loops
8. If the trains A, B, and C (Fig. 88) are all running 60 mi. per hour, what is the velocity of A with reference to B? to C? 9. If a weight is dropped from the roof to the floor of a moving car, will it strike the point on the floor which was directly beneath its starting point?
10. Why is a running track banked at the turns?
11. If the earth were to cease rotating, would bodies on the equator weigh more or less than now? Why?
12. How is the third law involved in rotary lawn sprinklers? 13. The modern way of drying clothes is to place them in a large cylinder with holes in the sides, and then to set it in rapid rotation. Explain.
14. Explain how reaction pushes the ocean liner and the airplane forward.
15. If one ball is thrown horizontally from the top of a tower and another dropped at the same instant, which will strike the earth first? (Remember that the acceleration produced by a force is in the direction in which the force acts and proportional to it, whether the body is at rest or in motion. See second law.) If possible, try the experiment with an arrangement like that of Fig. 89.
16. If a rifle bullet were fired horizontally from a tower 19.6 m. high with a speed of 300 m., how far from the base of the tower would it strike the earth if there were no air resistance?
FIG. 89. Illustrating Newton's second law
FIG. 90. Hydraulic ram
17. The hydraulic ram (Fig. 90) is a practical illustration of the principle of inertia. With its aid water from a pond P can be raised
into a tank that stands at a higher level than the pond. With the aid of Fig. 91 explain how it works, remembering that the valve V will not close until the stream of water flowing around it acquires sufficient speed.
18. If two men were together in the middle of a perfectly smooth (frictionless) pond of ice, how could they get off? Could one man get off if he were there alone?
19. If a 10-g. bullet is shot from a 5-kg. gun with a speed of 400 m. per second, what is the backward speed of the gun?
20. If a team of horses pulls 500 lb. in drawing a wagon, with what force does the wagon pull backward upon the team? Why do the wheels turn before the hoofs of the horses slide?
21. Why does a falling mass, on striking, exert a force in excess of its weight?
22. A pull of a dyne acts for 3 sec. on a mass of 1 g. What velocity velocity! does it impart? 494 23. How long must a force of 100 dynes act on a mass of 20 g. to impart to it a velocity of 40 cm. per second? mv T How far has the gram
24. A force of 1 dyne acts on 1 g. for 1 sec. been moved at the end of the second?
A laboratory exercise on the composition of forces should be performed during the study of this chapter. See, for example, Experiment 11 of the authors' Manual.