and it means that if B gets larger, A gets smaller, and vice versa. Note that if A were directly proportional to B, then if B gets larger, A also gets larger. An example of' 'directly proportional to'' is the change in length of a stretched spring that increases as the stretching force increases. An example of "inversely pro-

Figure 3.15. Newton's second law of motion, (a) Effect of a force on acceleration for a small- and large-mass object, (b) Effect of application of constant force on a speed-time graph: 1—small mass or large force; 2—large mass or small force; 3—zero net force.

portional to" is the volume of a balloon full of air, which decreases as the pressure of the surrounding atmosphere increases. We will need to use each of these two different kinds of "proportional to" several times as we continue in this text.

Thus Newton's second law tells us that the acceleration of an object is directly proportional to the impressed force and inversely proportional to the mass of the object. This is reasonable, of course, for we know that if we push or pull harder on an object, it will speed up faster; and that if the object is heavier (more massive), the same amount of push or pull will not be as effective. Larger engines give greater accelerations, but more massive cars reduce the acceleration. But Newton's second law goes beyond just what we know must be true in a qualitative way and tells us how acceleration is related to impressed force and mass in a quantitative way. The law indicates that the acceleration is not directly proportional to the square root of the force, for example, or the cube, or whatever; but simply is proportional to the magnitude of the force (i.e., to the first power). Similarly, the second law indicates that the acceleration is inversely proportional to the mass, in a specific way.

This relationship of acceleration to force for large and small masses is shown graphically in Fig. 3.15. Newton's second law gives one the ability to calculate how much the acceleration of an object will be for a given force, if the mass is known. Scientists and engineers use this formula possibly more than any other in their various calculations. It applies to cars, spaceships, rockets, rubber balls, and subatomic particles. It is probably the most frequently used single equation in physics.

Optional Section 3.4 Examples of Newton's Second Law of Motion

Suppose we have a car of mass 1400 kg (about 3000 lbs) and we wish to be able to accelerate it from 0 to 100 km/h (62 mi/h) in 10 seconds. We can use Newton's second law to determine how powerful the engine must be.

We start with

The desired acceleration is

10 sec sec2

and the necessary force is

Here N stands for newton, which is the unit of force in the metric system. One newton is defined as the amount of force required to accelerate one kilogram of mass by an acceleration of one meter per second per second (m/sec2). One newton is about 0.22 pounds. For our example here, an engine of about 72 hp would be required, which is a reasonable-sized modern car engine.


The mutual actions of two bodies on each other are always equal and directed toward contrary parts.

Implicit in this law is the idea that forces represent the interaction between bodies. Whereas the first and second laws focus attention on the motion of an individual body, the third law says that forces exist because there are other bodies present. Whenever a force is exerted on one object or body by a second object, the first object exerts an equal and opposite force on the second object. The force relationship is symmetric.

For example, when a horse pulls a wagon forward along a road, the wagon simultaneously pulls back on the horse, and the horse feels that reaction force. The horse's hooves push tangentially against the surface of the road, if it is not too slippery, and simultaneously the surface of the road pushes tangentially against the bottom of the horse's hooves. It is important to remember which forces are acting on the horse. They are the reaction force of the road surface on the horse's hooves, which "pushes" forward, and the reaction force of the wagon as transmitted through the harness to the horse's shoulders, which "pulls" backward. In order for the horse to accelerate, the force of the road surface on the horse's hooves must be greater than the force of the harness on the horse's shoulders. But the force of the road surface is a reaction force and must be the same magnitude as the force exerted by the horse's hooves on the road. If the road is slippery there will not be enough friction between the road surface and the bottom of the horse's hooves, and the horse will slip. The horse will be unable to use its

Figure 3.16. Force pairs for a horse pulling a wagon, (a) Horse pulls on wagon (through the harness), (b) Wagon pulls back on horse, (c) Horse pushes against road surface, (d) Road surface pushes back against horse's hooves.

powerful muscles to push sufficiently hard against the road surface to generate a reaction force that will push the horse forward. These various forces, which are all involved in the horse pulling a wagon, are shown (by arrows) in Fig. 3.16. It is essential to recognize, strange as it may seem, that it is the reaction force of the road surface against the horse's hooves that propels the horse forward.

A further example of the role of reaction forces and the third law of motion is supplied by a rocket engine. In a rocket, high-speed gases are expelled from the exhaust of a combustion chamber. The rocket in effect exerts a force on the gases to drive them backward. Simultaneously the gases exert a reaction force on the rocket to drive it forward. It is this reaction force that accelerates the mass of the rocket. Indeed, the corporate name of one of the major American manufacturers of rocket engines in the 1950s was Reaction Motors, Inc.

2.1 Law of Conservation of Mass. In addition to his three laws of motion, Newton considered it important to indicate two conservation laws that he concluded must be true. In physics, a quantity is said to be conserved if the total amount is fixed (always the same). Newton's simplest conservation law is that for mass (quantity of matter). He believed that the total amount of mass in a carefully controlled, "closed" system is a fixed amount. A closed system simply means that nothing is allowed to enter or leave the system being studied. Certainly mass is not conserved in a system if more material is being added. The real import of this law is that Newton believed that mass could be neither created nor destroyed. He recognized that matter can change in form, such as from a solid to a gas in burning. But Newton believed that if one carefully collected all the smoke and ashes from the burning of an object, one would find the same total amount of matter as there was in the beginning. We now recognize that this conservation law is true only if we also capture all of the heat and light which escape.

2.2 Law of Conservation of Momentum. Using his second and third laws of motion, Newton was able to show that whenever two bodies interact with each other, their total momentum (i.e., the sum of the momentum of the first body and the momentum of the second body) is always the same, even though their indi vidual momenta will change as a result of the forces they exert on each other. Thus if two objects (e.g., an automobile and a truck) collide with each other, their paths, speeds, and directions immediately after the collision will be different than they were before the collision. Nevertheless, if the momentum of the first object before the collision is added to the momentum of the second object before the collision, and compared to the sum of the momentum of the first object and the momentum of the second object after the collision, the results will be the same. This is called the law of conservation of momentum, and it can be generalized to apply to a large number of objects that interact only with each other.

Although Newton derived the law of conservation of momentum as a consequence of his laws of motion, it is possible to use this law as a postulate, and derive Newton's laws from it. In fact, the modern theories of relativity and of quantum mechanics show Newton's laws to be only approximately correct, particularly at high speeds and for subatomic particles, but the law of conservation of momentum is believed to be always exact.

Optional Section 3.5 Conservation of Momentum

As an example of how total momentum is conserved, let us consider the collision between a large and small automobile. Suppose the large car has a mass of 2000 kg (weight ~ 4400 lb) and the small car a mass of 1000 kg (weight ~ 2200 lb). Suppose that they are each traveling at 65 km/h (~ 40 mi/h) and collide head-on in an intersection and stick together. From the fact that the total momentum is fixed, we can calculate the velocity (i.e., speed and direction) of the two cars immediately after the impact.

We recall that momentum is quantity of motion, or mass times velocity. Mathematically, we write this as

If we call the original direction of the large car the positive x-direction, then the smaller car was originally moving in the negative x-direction. Hence the total momentum just before the collision was

Immediately after the collision, the momentum must be pf = mlvl + m2v2

The law of conservation of momentum requires p, = pi or


in the positive x-direction.

3. The Universal Law of Gravitation

As we have discussed, Galileo showed that if the effects of buoyancy, friction, and air resistance are eliminated, all objects fall to the surface of the earth with exactly the same acceleration, regardless of their mass, size, or shape. He showed also that even projectiles (i.e., objects that are thrown forward) also fall while they are moving forward. Their forward, or horizontal, motion is governed by the law of inertia; but their simultaneous falling motion occurs with exactly the same acceleration as any other falling object. It was left to Newton to analyze the nature of the force causing the acceleration of falling objects.

Newton considered that falling objects accelerate toward the Earth because the Earth exerts an attractive force on them. It is said that the basic idea for the law of gravity occurred to Newton when an apple fell on his head. This story is almost certainly untrue; but Newton did remark that it was while thinking about how gravity reaches up to pull down the highest apple in a tree, that he first began to wonder just how far up gravity extended. He knew the Earth's gravity still existed high on the mountains and concluded it likely had an effect in space. Finally, he wondered if gravity reached all the way up to the Moon, and if one could, in fact, demonstrate that gravity was responsible for keeping the Moon in its orbit. He recognized that the Moon is in effect also ' 'falling'' toward the Earth, because its velocity is always changing, with the acceleration vector perpendicular to its circular path and therefore pointing to the Earth at the center of the Moon's orbit. This force, because it changes the direction of the Moon, is a centripetal force, as discussed earlier. Similarly, the planets are "falling" toward the Sun because in their curvilinear elliptical paths their acceleration vectors point toward the Sun. Because accelerations can exist only when there are impressed forces, the Sun must exert a force on the planets. Therefore Newton concluded that the same kind of force, the force of gravity, might be acting throughout the universe.

By the use of Newton's laws of motion, as well as experimental measurements of the orbits of the Moon and the planets, it is possible to infer (reason by induction) the exact form of the gravitational force law. According to Newton's second law, if the force were constant, more massive objects should fall to Earth with a smaller acceleration than less massive objects. Therefore the Earth must exert a greater force on more massive objects—they weigh more (weight is a force). In fact, the gravitational force on an object must be exactly proportional to the mass of the object in order that all objects have the same acceleration. Therefore the Earth exerts a force on an object proportional to the mass of the object. But according to Newton's third law, the object must exert an equal and opposite force on the Earth and this force must be proportional to the mass of the Earth, because from the object's "point of view" the Earth is just another object. Both proportionalities must apply, and therefore the force of gravitation must depend on both the mass of the Earth and the mass of the object.

Newton knew that the acceleration of the Moon, while ' 'falling'' toward the Earth (he considered the Moon to be like a projectile) was much less than the acceleration of objects falling near the surface of the Earth. (Knowing the dimensions of the Moon's orbit, and the length of time it takes the Moon to go around its orbit once—2l\ days—he was able to calculate the acceleration of the Moon.) He eventually concluded that the Earth's force of gravitation must depend on the distance between the center of the Earth and the center of the object (the Moon in this case). The distance from the center of the Earth to the center of the Moon is 60 times greater than the distance from the center of the Earth to an object on the surface of the Earth. The calculated "falling" acceleration of the Moon is 3600 times smaller than the falling acceleration of an object near the surface of the Earth. But 3600 equals 60 times 60. Therefore it is a good guess that the force of gravity decreases with the square of the distance from the center of the attracting object. Putting all these considerations together, Newton proposed his law of gravitation.

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