Newton’s second law says, “If a body has an acceleration a, then you need a force
F= ma (3.1)
to produce that acceleration.”
In this chapter we will focus on one dimension and write
F= ma (3.2)
where F and a are along the x-axis.
A few words about units. Acceleration is measured in ms−2. Mass is measured in kilograms or kg. So force has units kilogram meters per sec-ond squared. But we get tired of saying that long expression, so we call that a Newton, denoted by N. If you had invented mechanics, we’d be calling it by your name, but it is too late for you now.
Here is a typical problem that you may have solved in your first pass at Newton’s laws. A force of 36N is acting on a mass 4 kg. What’s the accel-eration? You divide 36 by 4 and you find it is a= 9 ms−2. You say, “Okay, I know Newton’s laws.”
It’s actually more complicated than that. Take yourself back to the seventeenth century, when Newton was inventing these laws. You have an intuitive definition of force: when somebody pushes or pulls an object we
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say a force is acting on it. Suddenly, you are told there is a law F= ma.
Are you better off in any way? Can you do anything with this law? What does it help you predict? Can you even tell if it’s true? Here’s a body that’s moving. Is Newton right? How are we going to check? Well, you want to measure the left-hand side and you want to measure the right-hand side.
If they’re equal, you will say the law is working. What can you measure in this equation?
Let’s start with acceleration. What’s your plan for measuring accel-eration? What instruments will you need? If you say a watch and ruler, that would be correct provided by ruler you don’t mean Queen Elizabeth.
Here is your ruler and here is a Rolex. Tell me exactly how you plan to measure acceleration. Everyone seems to know the answer. First, let it go a little distance, and take the distance over time. That gives you the velocity now. Let it go a little more, and repeat the velocity measurement.
Take the difference of the two velocities and divide by the difference of the two times, and you have got the acceleration. Since the body has moved a finite distance in a finite time, this gives the average acceleration. You want to make these three positional measurements more and more rapidly. In the end, as all the time intervals shrink to 0, you will measure what you can say is the acceleration now, the second derivatived2xdt2 = a(t) defined in calculus.
Back to testing if what Newton told you is right: You see an object in motion, you measure a, and you get a certain numerical value, say 10 ms−2. But that’s not yet testing the equation, because you still have to find F and m. What’s the mass of this object? One common idea is to take a standard mass and balance the unknown mass on a seesaw by adjusting its position.
But suppose you were in outer space. There’s no gravity. Then the seesaw will balance even if you put a potato on one side and an elephant on the other side. What you are doing now is appealing to the notion of mass as something that’s related to the pull of the earth on the object. You have got to go back and wipe out everything you know. If F= ma is all you have, there is no mention of the earth in these equations. You only know how to measure a, but not the other two. So you have a problem. You cannot say that since F= ma, it follows that m =Fa; that is circular reasoning since you have not told me how to measure F either.
Let me give you a hint. How do we decide how long a meter is? You seem to know that it is arbitrary. A meter is not deduced from anything.
Napoleon or somebody said, “The size of my ego is one meter.” That’s a
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new unit of length. Seriously, at the National Bureau of Standards there used to be a rod made of some special alloy in a glass case, and that defined the meter. There are fancier definitions now, but let us stick to this simple one. (See http://physics.nist.gov/cuu/Units/meter.html for more details on definitions of units.) Then I ask you, “What is two meters and what is three meters?” We have ways of handling that. You take the meter and attach it to a duplicate, and that’s two meters. You can cut it in half, using dividers and compasses; you can split the meter into any fractions you like.
Likewise for mass, we will take a chunk of some material and we will call it a kilogram. That is a matter of convention, just like one second is some convention.
I’m going to give you a glass case that contains a block of some metal defined as one kilogram. Then I give you another object, an elephant.
What’s the mass of the elephant? Here is a hint: I also give you a spring. We cannot do the seesaw experiment because it requires gravity. A spring, on the other hand, will exert a force even in outer space. Here’s what we do.
We hook one end of the spring to a wall and we pull the other end from its equilibrium position by some amount and we attach the one-kilogram mass to it. We don’t know what force it exerts, but it will not matter. We let it go and measure the initial acceleration, a1. Then we bring the elephant (another point particle) of unknown mass mE, pull the spring by the same amount so it can exert the same force, and find the acceleration aE of the elephant. Assuming only that the same extension produces the same force in the two cases, we have
1· a1= mEaE (3.3)
mE= 1 · a1
aM
. (3.4)
Once you have the mass of the elephant you can use it to measure any other mass moby using
mo= mE
aE
ao
(3.5) where aEand aoare produced by the same force.
There are subtleties even here. For example, how do we know that when we pull the spring the second time with the elephant, it will exert the same force as the first time when the 1 kg was attached to it? After all,
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springs wear out. That’s why you change the shock absorbers in your car.
So first, we have to make sure the spring exerts a fixed force every time (for a given extension). How are we going to check that? We don’t have the definition of force yet. But we can do the following. We pull the one-kilogram mass and let it go, and we note the acceleration. Then, we pull it again, by the same amount, and let it go; we do it ten times. If every time we get the same acceleration, we are convinced this is a reliable spring that is producing the same force under the same conditions. On the eleventh time we pull the spring and attach the elephant. With some degree of con-fidence, we can say we are applying the same force on the elephant as on the one-kilogram mass.
Why is this discussion so important? Because you need to know that everything you or I write down in the notebook or on the blackboard as a symbol is actually a measurable quantity, or, as they say in France, Les Mesurables. You should know at all times how you measure anything that enters your theory or calculation. If not, you are just doing math or playing with symbols. You are not doing physics.
This discussion also tells you that the mass of an object has nothing to do with gravitation but with how much it hates to accelerate in response to a force. Newton tells you forces cause acceleration. But the acceleration is not the same on different objects for a given force. Certain objects resist it more than others. They are said to have a bigger mass. We can be pre-cise about how much bigger by saying, “If the acceleration of a body in response to a given force is 101 that of a 1-kilogram mass, then the mass of the body is 10 kilograms.”