of the rider and bicycle (Fig. 68). The vector OW represents the downward force, the weight. The vectors OC and OD are two components of the weight. There is another force acting, the one with which the ground pushes on

the tire (not shown in the figure). This acts in the direction AO and is equal and opposite to the force OC, thus annulling the effect of OC. The resultant force acting on the rider is OD. It is the central, or centripetal, force that is used to hold the bicycle and rider in the curved path.

The outer rail on a curve of a railroad track is made higher, in order to make the trains lean. A race track is

A

FIG. 68

"banked" at the curves for the same reason. On the ordinary street-crossing the turn cannot be banked; hence an automobile must depend on the friction between the road and the tires to furnish a central force.

79. Summary. This chapter contains some of the most important principles of physics, and the student should persist until he understands them clearly.

When different forces act on the same mass at different times, the accelerations are proportional to the forces.

When equal accelerations are produced in different masses, the forces are proportional to the masses.

When equal forces act on different masses, the accelerations are inversely proportional to the masses.

If the student will not attempt to commit these laws to memory but will imagine actual experiments showing the facts in question, he will have no difficulty in remembering the laws.

The above statements are included in formula (20),

F = kma,

which is a statement of one of the great principles of physics. In both the C.G.S. and B. E. systems the units are so chosen that k = 1. Hence (equation (21))

F = ma.

The dyne is the unbalanced force that will give 1 gram an acceleration of 1 centimeter per second per second. The weight of a gram is 980 dynes. The weight of a mass m is mg, or (approximately) 980 m dynes. The B. E. unit of mass, the slug, is of such a size that

weight in pounds

Mass of a body =

g

W 32

slugs (approximately).

It is convenient when using B. E. units to use formula (25),

Momentum is equal to mass times velocity. It is a vector quantity having the same direction as the velocity.

Impulse is equal to force times the time the force acts. An impulse when unbalanced will produce a change in momentum numerically equal to the impulse.

Newton's first law states that when the resultant of all forces acting on a body is zero, if the body is at rest it will stay at rest, and if in motion it will continue in motion with a uniform speed in a straight line. Stated in other words: The momentum of a body will remain constant unless the body is acted upon by an unbalanced force.

Newton's second law may be stated: The rate of change of the momentum of a body is proportional to the impressed force and is in the same direction as that force.

The equation F = ma covers both the first and second laws. Newton's third law is as follows: Every force has an equal and opposite reaction. Forces always exist in pairs, but the two never act on the same body. If a body A exerts a force on a body B, then B exerts an equal and opposite force on A.

When A exerts a force and changes the momentum of B, there will be an equal and opposite change in the momentum of A unless other forces act on A to prevent it. When no external forces act on a group of bodies, their total momentum remains constant (conservation of momentum).

When a body is forced to move in a curved path, an acceleration must be given to it. If the body moves with a speed v in a path of radius r, the acceleration is v2/r.

The centripetal, or central, force is mv2/r; or, if the B. E. system is ωυ2 used, it is where w is the weight in pounds and g is approximately equal to 32 feet per second per second.

gr

,

The centrifugal force acts away from the center, being always equal and opposite to the central force. These two never act on the same body. In the case of a stone whirling on the end of a string, the pull of the string on the stone is the central force, and the pull of the stone on the string is the centrifugal one.

PROBLEMS

1. (a) What is the acceleration when a force of 40 dynes acts upon a mass of 5 gm.? (b) How far will the mass move in 4 sec. if it starts from rest?

2. A body of mass 500 gm. is started sliding along a fairly smooth floor with an initial speed of 100 cm./sec. It is brought to rest by friction in 5 sec. Find (a) the value of the (negative) acceleration and (b) that of the force that brings the body to rest.

W

3. (a) What force (state the name of the unit as well as the numerical value) is required to give 200 gm. a speed of 200 cm./sec. in 2 sec.? (b) After it has acquired a speed of 240 cm./sec., what force would be required to stop it in 4 sec.?

4. Compute the force, in dynes, of the attraction of the earth for a mass of 20 gm.

5. Find the mass, in slugs, of a man who weighs 160 lb.

6. A certain body weighs 30 lb. on the earth. Suppose that this body were far away in space, and that a force of 15 lb. acted on it. What would happen?

7. What force in pounds will give to a body weighing 120 lb. an upward acceleration of 40 ft./sec.2 ?

8. How many seconds would it take an engine pulling with a force of 2 tons more than friction to get a 640-ton train from rest up to a speed of 60 mi. an hour (88 ft./sec.) on a level track?

9. A boy and sled with a total weight of 96 lb. are pushed on very smooth ice by a horizontal force of 6 lb. (a) What speed is gained in 3 sec. ? (b) What distance will be traversed in 5 sec.?

10. A horizontal force of 10 lb. acts on a baseball weighing 5 oz. for one fifth of a second. What velocity will be given the ball?

✓ 11. Two masses of 40 and 50 gm., respectively, are attached to the opposite ends of a perfectly flexible string which passes over a frictionless pulley, the mass of which may be neglected. (a) What force will be active in setting the two masses in motion? (b) What will be the acceleration? 12. A 150-pound man is hanging by a rope from a balloon which is falling with an acceleration of 4 ft./sec.2 What is the tension on the rope ?

13. A man who weighs 150 lb. gives an impulse to a 75-pound canoe as he steps out of it. If the canoe is given a velocity of 2 ft./sec., compute the following: (a) the momentum given to the canoe, (b) the momentum the man acquires, and (c) the velocity the man acquires.

✓ 14. Two masses of 200 and 600 gm., respectively, move toward each other on smooth ice at the same speed, 20 cm./sec. (a) Compute the total momentum before collision. (b) If they stick together on impact, compute their common velocity. (c) Compute the numerical value of the impulse.

15. A 1000-pound animal is running toward a hunter at a speed of 10 ft./sec. A bullet weighing 1 oz. is fired, with a speed of 2000 ft./sec., into this animal. (a) Would the bullet have momentum enough to stop the animal? (b) If not, how much speed would the animal have left?

16. A horizontal force of 10 lb. acting for one fifth of a second is used to throw a stone weighing half a pound. (a) What is the numerical value of the impulse? (b) the momentum given the stone? (c) the velocity given the stone?

17. A mass of 1000 gm. is moving in a circular path of radius 20 cm. at a speed of 30 revolutions per minute. What force is required to hold it in its path?

✓ 18. What force is necessary to keep 600 gm. moving in a circle of radius 40 cm. at 120 revolutions per minute?

19. A 3000-pound automobile goes around a curve of 100-foot radius at 30 mi. per hour. Compute the side-thrust on the roadway.

20. Compute the centrifugal force exerted by a 150-pound man while he is being carried by an automobile around a curve of 50-foot radius at a speed of 20 mi. per hour.

CHAPTER VII

WORK AND ENERGY

Work, 80. Units of work, 81. Examples, 82. General method for computing work, 83. Energy, 84. Measurement of energy, 85. Two forms of energy, 86. Transformation of energy, 87. Method of computing kinetic energy, 88. Method of computing potential energy, 89. Power, 90. Conservation of energy, 91. Availability of energy, 92. Sources of energy, 93. Comparison of the C.G.S. and B. E. systems of units, 94. Summary, 95.

The concepts of work and energy are among the most important in the whole realm of physical science. The laws of energy developed in the subject of mechanics and originally applied only to bodies of finite size are now known to be far more general, applying not only to the smallest subdivisions of matter, such as the molecule, the atom, and the electron, but also to such things as electrical and magnetic fields and to the radiations of light-waves and electrical waves; in fact, so far as we know, to everything. Many of our common, daily observations can be clearly understood only when interpreted in terms of work and energy; therefore a clear understanding of these concepts is necessary for many persons and desirable for all.

80. Work. The following are simple cases of work. A man carries a hod of bricks up a ladder. Another shovels dirt out of a ditch. Another lifts stone into a wagon. In each of these cases force is exerted. In measuring the amount of work done the magnitude of this force must be taken into account. Another factor enters; namely, the distance through which the stone or dirt moves while acted upon by this force. If the man exerts force through a greater distance (for example, if he lifts the stone higher), more work is done. The man who digs a deeper ditch does more work in lifting a certain amount of dirt than one digging in a shallow ditch. Two factors always enter when one wishes to