CHAPTER VI

EFFECTS OF FORCES ON MOTION

Newton's first law of motion, 59. Force proportional to acceleration, 60. Mass, 61. Equal accelerations produced in different masses, 62. Equal forces acting on different masses, 63. The general relation between force, mass, and acceleration, 64. Units of force: the dyne; the pound, 65. Relationship between mass and weight, 66. Examples in the B. E. and C. G. S. systems, 67. Remarks on mass and weight, 68. Momentum; impulse, 69. Newton's first law of motion, 70. Newton's second law of motion, 71. Newton's third law of motion: action and reaction, 72. Conservation of momentum, 73. Motion in a curved path; centripetal force, 74. Magnitude of the centripetal force, 75. Centrifugal force, 76. Derivation of the formula for centripetal and centrifugal forces, 77. Applications of the foregoing principles, 78. Summary, 79.

59. Newton's first law of motion. What would happen to an automobile at rest if no forces acted on it? All of us know that it would stay at rest. But if the car were in motion and if no forces acted on it, what would happen? Before we answer this we should recall the fact that there are usually many different forces acting on an automobile in motion; among them forces of the nature of friction, such as air resistance, friction at the axles, and friction between the wheels and the ground. If we could eliminate all these forces, what would happen? This cannot be answered directly from experience, for in no case are we able to eliminate entirely all such forces; but it is a matter of common observation that as frictional forces are made less and less, there is a tendency in a moving body to run longer and longer. All experience points to this conclusion: If all forces were eliminated from a moving body, it would continue to move with a constant velocity.

We always find that it requires the constant application of a force to keep a wagon or a car in uniform motion on a level road, but this is because there are retarding forces of friction which oppose the motion. The important thing to know is this: that if the applied forces are equal and opposite in direction to the retarding forces, the car will continue to move with uniform velocity. In that case the resultant force acting on the car is zero. A larger driving force is necessary to keep an automobile running at a higher speed, because the retarding forces are greater for higher speeds. But the driving force is never greater than the retarding forces so long as the velocity is constant. In all cases where the motion is uniform and in a straight line the applied force is exactly equal to the retarding forces, and the resultant of all acting forces is zero.

We can now understand Newton's first law of motion, which may be stated as follows: When the resultant of all forces acting on any body is zero, if at rest the body will remain at rest, if in motion it will continue in motion with a uniform speed in a straight line. This statement can be summed up in one word, inertia. On account of its inertia a body requires a force to put it in motion, but once in motion a force is necessary to change its velocity.

This law of Newton's is partly an assumption and partly the result of experiment. It is an assumption in the same sense that the axioms of geometry are assumptions. The strongest argument in favor of it is that conclusions based on it always agree with observation; no exception has ever been found. This law applies to all kinds of bodies: to the atom and molecule as well as to larger bodies, to those far out in space as well as to terrestrial ones. The concept of acceleration is very important in connection with the action of forces. A body which is "at rest" or "in motion with uniform speed in a straight line" is one which has no acceleration. Hence Newton's first law may be stated as follows: If the resultant force is zero, there will be no acceleration.

Whenever there is an acceleration, there must be an unbalanced force, and the direction of this force is the direction of the acceleration. It will not be difficult for the student to find many illustra

L

tions of this fact. Since the passengers in a car are accelerated when the motion of the car is changed, there must be forces acting on them. We ride with comfort in trains or cars as long as the

motion is uniform; but if there are irregularities (that is, accelerations), we are jerked first one way and then the other. The motion of the earth is so constant that the high speed brings no discomfort.

60. Force proportional to acceleration. This question at once arises: Is there any simple relation between the force acting on a car and the acceleration produced? For example, if the resultant force acting on the car be made twice as large, will the acceleration be twice as large? The answer is Yes. Experience shows that the acceleration is directly proportional to the resultant force. We may state this result of experience in the form of an equation and say that, for any one body,

F 1
1,
F2 42

(17)

where a1 is the acceleration produced by the force F1, and a2 the acceleration produced at another time by a force F2. Both F1 and F2 are resultant forces acting on the body.

1

As an example the weight of any body is a force which will give it a downward acceleration of approximately 32 feet per second per second. To make this same body have an acceleration of 64 feet per second per second, or twice as much, the force must be twice the weight of the body. To give it an acceleration of 16 feet per second per second the resultant force must be half the weight. If the frictional force acting on a body sliding down vertical ways is equal to one fourth of the weight of the body, the resultant downward force will be three fourths of the weight, and the downward acceleration will be three fourths of 32, or 24 feet per second per second.

Equation (17) is the result of experience and is therefore called an experimental law. The reason that quantitative laws are so commonly stated in the form of equations is that this method gives statements which are simpler and more definite than those expressed in words. But, on the other hand, the student will find that it will always add to his understanding of such an equation to translate the meaning into nonmathematical terms. Then the equation will mean something more than a mere collection of symbols. 61. Mass. The word weight is the name given to the attractive force exerted by the earth. This force acts on objects around us because they happen to be near the earth. If this book were taken to the top of a mountain, it would weigh less. If it could be taken far out into space, the weight would become insignificant. At the center of the earth it would be attracted equally in all directions, and the resultant force would be zero. The weight is not something inherent in a body itself, but is an "accident of its environment."

The mass of a body is the same wherever it may be. A piece of lead placed at the center of the earth would have no weight; but the quantity of lead (the mass) would remain the same. Mass may be said to be another name for matter; it is used whenever the quantity of matter is to be measured.

In commerce it is usually not necessary to make a distinction between mass and the weight of the mass, but in scientific work many cases arise where a distinction is necessary. A method of comparing or measuring masses is indicated in the next section.

62. Equal accelerations produced in different masses. If a man kicks a barrel lying on its side, and then another barrel, he may observe that the acceleration of the first barrel is much greater than that of the second. If so, he concludes that the quantity of matter included in the two barrels (that is, the mass) is different. To have produced the same acceleration in each, a greater force would have been necessary for the barrel of greater mass. Another way to state this is to say that large masses have greater inertia than smaller ones. Why is it rougher riding over a rough road in a light automobile than in a heavy one? A light local train does not run as smoothly as the heavy through trains. A heavy Pullman on the rear of a light train will steady the motion of the entire train.

The quantitative statement is as follows: To produce the same acceleration in two different masses the forces must be proportional to the masses; or, stated algebraically, to produce the same acceleration,

1

F
F2

1

m
1

,

m2

(18)

where F1 is the force applied to the mass m1, and F2 the force applied to m2. As before, F1 and F2 are resultant forces.

1

2

Everyone knows that a greater force is needed to throw (that is, to accelerate) a large stone than a small one. But this equation states more than this, for it states how much larger the force must be to give the same acceleration.

It should be noticed that forces are proportional to masses and not to weight. Far away from the earth bodies have practically no weight, yet it is believed that the same force is required to give a certain acceleration to a body out in space as would be necessary if that body were here on earth. This is assumed by astronomers in making computations of the masses and the accelerations of the moon and planets. The results confirm the validity of the assumption.

If the man had lifted the two barrels at a uniform speed, he would have compared their weights; but when he applied horizontal forces to them and observed the accelerations, he was comparing masses. Imagine this experiment to be repeated far out in space where the barrels would have no weight. There no force would be required to "lift" them, or move them away from the earth at a uniform speed; but the forces required to accelerate them would be the same as on the earth.

Equation (18) gives a method of comparing or measuring

masses.

63. Equal forces acting on different masses. Imagine two small toy carts placed on a smooth table and connected by a stretched spiral spring. When released the two carts will be accelerated toward each other. The force acting on one is equal to that acting on the other, for each force is equal to the tension of the spring. If the masses are equal, the accelerations will be equal; but if a large mass is placed in one of the carts, its acceleration will