# Physics 125, Fall 1995 -- Lecture 7: Mass, Momentum and Force

## 7.1 Free Body Diagrams

It is convenient to introduce the concept of free body diagrams to make sure that you have considered all of the forces acting on an object. Such a diagram is nothing more than a picture of the object in question isolated from other objects except for arrows representing the vector forces acting on the object. Figure 1: A Free Body Diagram for an object suspended from a scale in an elevator is shown at the right.

The only forces are that of the scale and gravity. Note that m is not itself a force, rather it is the mass times the acceleration due to the net force.

As another example, consider an object of mass m on an incline at an angle above the horizontal. The free body diagram is shown to the right (see Figure 2) where N is the force exerted by the plane normal or perpendicular to it and f is the frictional force exerted by the plane parallel to it. In this case, two dimensions are involved and the vector algebra needed to determine requires a coordinate system and the vectors' components. The convenient coordinate system is that with axes parallel and perpendicular to the plane since the object will accelerate (move) parallel to the plane. The only force that does not point along one of the axes is = mg (down). The components are: mg sin parallel and mg cos perpendicular.

From this we have  In many cases, the frictional force for an object in motion is proportional to the magnitude of the normal force, = , so . Figure 2: A Free Body Diagram for an object on an inclined plane.

## 7.2 Newton's Three Laws of Motion

Newton expressed the concepts of the last two sections as his three laws of motion which are reiterated here:

NI: An object will maintain uniform motion unless a net force acts on it.

NII: The rate of change of momentum of an object of mass m due to a net force is given by .

NIII: The body upon which a force is exerted by a second body exerts a force of equal magnitude but opposite direction on the second: (the force on 1 by 2 is equal in magnitude, opposite in direction, to the force on 2 by 1.)

## 7.3 Conservation of Momentum

The law of conservation of momentum is a concept so powerful that every field of physics relies on the law never being broken! The total momentum of a system does not change unless a net force acts on the system from outside the system. This is the statement of the law of Conservation of Momentum. To a physicist conservation of a quantity means not just that none of that quantity is lost but also that none is gained, thus the quantity does not change. Conservation laws of momentum, energy, angular momentum, electric charge and several other physical quantities are crucial to physics and provide very powerful tools with which to examine the behavior of physical systems.

Conservation of momentum is an experience we have all had. For example, remember playing medicine ball with a friend? The object of the game is to throw a ball at your partner so hard that he is driven backward by the ... momentum of the ball. It is apparent qualitatively to an observer that the one catching the ball recoils with momentum in the same direction as the ball's momentum just before it was caught. What about the one throwing the ball? Try this. Stand with your feet very close together and throw a ball such as a basketball, as hard as you can with two hands, but don't move your feet. Lose your balance and step backward? This is because the forward momentum or impulse you impart to the ball by applying a force for a short duration of time is balanced by the backward momentum you gain due to the force of the ball pushing back at you for the same duration of time. The total momentum of you and ball before you throw it (0) is equal to the total momentum of the ball (moving forward) and you (moving backward) after the ball is thrown.

A more subtle question is: how does momentum conservation manifest itself during the free falling flight of the medicine ball. There are really two ways out of this, one cheap, the other neat, but both correct. The cheap way is to note that gravity is an external force and therefore the requirement of momentum conservation that no net external force act is not satisfied. The neat way is to consider the earth and medicine ball as part of the same system with no external force acting. Only the internal force of gravity acts to pull the medicine ball toward the earth and the earth toward the medicine ball. In this case, as expected, momentum is conserved. When your partner catches the ball, the downward component of momentum is transfered to the earth which had been moving toward the medicine ball. After the ball is caught, the earth is no longer moving toward the ball!

Now consider two other cases: 1) A 0.01 kg ping-pong ball with initial velocity 10 m/s collides with a 4 kg bowling ball initially at rest. The final velocity of the ping-pong ball is 9.9 m/s at 45 to the left of the initial direction. 2) An 800 kg Porsche with initial velocity 30 m/s north collides with and sticks to a second Porsche initially at rest so that the recoiling object is a 1600 kg biPorsche. In both cases, you should be able to find the unknown recoil velocity.

Finally, try to picture a pool table just after the cue ball is sent rushing toward the full rack for the break. The only momentum in the system of 16 billiard balls is carried by the cue ball. Where did this momentum come from? After the cue ball hits the rack and the 15 numbered balls scatter, bounce, and perhaps fall into the pockets, how has the momentum of the system changed? If momentum is conserved, where did it go after all has settled down? Thinking about this example and answering the queries will give you a good feel for the concept of momentum conservation.

## 7.4 Momentum Conservation and Forces

We have already introduced the concept of momentum ( , the quantity of motion). For an object whose mass does not change (for instance it does not break up into pieces) the rate of change of motion, is m = m if is constant over the interval . Thus the net force tells us the rate of change of an object's momentum.

Consider a collision between two objects, for example two billiard balls. During the short interval of contact, the momentum of each object changes because of the force exerted by the other. If there is no external force, we know that the total momentum of the two is a conserved quantity, that is the vector sum of momenta is a constant vector. Thus we know (you should convince yourself of this) The interval during which object 1 exerts a force on object 2 is of course equal to the interval during which object 2 exerts a force on object 1. Thus we can write and . Finally we see that that is, the force exerted on 1 by 2 is equal but opposite to that exerted on 2 by 1. This relationship between the two forces exerted on each other by any two objects is always true, regardless of the change of momentum and of the duration, . Thus as you sit in your chair and the earth exerts a downward force on you through gravity (somewhere between about 500 N and 900 N), you are exerting a force of equal magnitude but opposite direction (upward) on the earth.