Lecture 4

Why Things Move

In addition to believing that heavier bodies fall faster, Aristoteles and his fellow scientists/philosophers from antiquity also believed that to keep an object in motion one has to apply to it a force continuously. This sounds quite reasonable, every fool could have seen that to move a cart a horse has to keep pulling it, or that a sailboat will move as long as there is some wind pushing it. Anyone stating that a cart can move forever even if nobody is pulling it would have been considered at best a dimwit.

The problem with such thinking was in not realizing that, in addition to the pulling or pushing agent, there are other forces, typically friction and air or water resistance, involved in the phenomena mentioned.

If we try to eliminate such spurious agents, then we can see that, once a body is in motion with a certain velocity, it tends to maintain its velocity: in ideal, albeit abstract, conditions, it will move forever at the same velocity (it is a fact that one would not easily encounter frictionless condition in ancient Greece; I sometime wonder whether their thinking would have been different if they had been able to skate on a frozen pond....)

Generalizing the idea we have just presented, we can state Newton's first law of motion or Law of Inertia :

an object will maintain a state of rest or of motion at constant velocity unless some external agent intervenes to modify it.

Any such agent that can modify the state of motion of an object is what we call a Force.

When you stop to think about it, there are plenty of examples of the law of inertia in everyday life :

• if you (very unwisely) sit in the front seat of a car without fastening the seat belts, and the driver jams the brakes, you will be projected against the windshield. Why? Is anyone throwing you against the shield? No, what is happening is that, while the car is slowing down, you instead tend to continue moving with whatever velocity you (and the car) had
• when a car going around the bend encounters an icy patch or an oil slick, it will take off along the tangent to the curve (i.e. in the direction of its instantaneous velocity), since the force that was keeping it on the curved trajectory, i.e. the tire friction on the ground, stops acting.
• etc.

We then see that a force is something that causes the velocity of an object to change. By definition, whenever an object changes its velocity, it undergoes an acceleration. We can then infer that there has to be a strict correlation between forces and accelerations.

Rememeber : a change in velocity can occur either in magnitude or in direction (or both). A body moving with constant speed (i.e. constant magnitude of velocity) along a curved trajectory is undergoing acceleration.

Can we say more about the relation between forces and accelerations? The first obvious statement is that, the greater the force, the greater the acceleration. Careful measurements would show that, more precisely, the acceleration is actually proportional to the force. Moreover, it is equally intuitive that the acceleration (i.e. the change in velocity) will be in the same direction as the force. We can then write

$\vec{F}\propto\vec{a}$

What else can we say ? We can try applying the same force to different objects, and see what happens . Obviously a more massive object will be accelerated less than a less massive one: try to give the same kick to a soccer ball or to a bowling ball.....Or try to stop (i.e. to change the velocity of) a bowling ball or a soccer ball moving at the same speed...

These considerations allow us to give a quantitative definition to the concept of mass : the mass of an object is a measure of its inertia, i.e of the resistance a body opposes to have its state of motion changed. Mass is therefore the "missing link" in the force vs. acceleration equation, that we can now write

$\vec{F}= m\vec{a}$

This is Newton's second law of motion, and, in spite of its apparent simplicity and almost obvious appearance, it has incredibly far-reaching consequences.

Masses are measured (in the SI system) in kilograms. In British units, masses are measured in slugs : now that I said it, you can quickly forget it; in many years of work in Physics I have never encountered such unit, and probably many physicists never heard of it, unless they teach Introductory Physics from the same textbook as us. But what is important to remember, as we will soon discuss, is that mass and weight are not the same thing.

Newton's law allows me to define the unit for forces, the newton (N): a force of 1 newton will impart to the mass of 1 kg the acceleration of 1 m/s². Dimensionally:

[F] = [MLT^-2]

Combination of Forces

Until now, I have been a little casual talking about the effect of a force on a body, and in a sense I was implicitely assuming that there was only one force. In the more general case, there can be many different forces applied to a given body and, in order to obtain the final effect (i.e. what will the acceleration be) I have to consider the net force, i.e. the (vector) sum of all the forces acting onto the body. Most obvious example : if I apply two forces which are equal and opposite, their sum will be zero and no net force will be acting on the body. Does this mean that the object is at rest?

No, if an object has no net force acting upon it, it will be either at rest or moving with constant velocity.

We can then conclude :

no net force $\longrightarrow$ constant velocity (possibly =0)

constant velocity (including 0) $\longrightarrow$ no net force