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07 December 2001

What a Diet Can't Fix

by George E. Hrabovsky, President of MAST

News from MAST

Hello again. I know that I claimed that we would have some results from a crystallography experiment done last night. I am sorry to announce that the experiment did not take place. We have been unable, thus far, to acquire the necessary parts for a hot stage microscope. We are working on this and will report our results when we get them.

So far we have considered motion without thinking about what causes it. This week we will begin a study of what causes motion to occur, or change.

Mass and the Acceleration Due to Gravity

Galileo Galilei was one of the greatest physicists ever, the list of his accomplishments is truly remarkable. In his day there was an interesting idea that make perfect sense. Recall that at that time it was believed that the Earth was the center of the universe. The idea was that the center of the Earth drew all things to it, and that all things had their place. This attraction is called gravitation and was seen as a pervasive throughout the universe. Another idea was that the heavier an object is the faster it falls. Again, this makes perfect sense.

Once you think about the idea carefully a contradiction occurs. Galileo recognized this and exposed the fallacy for all to see. If a heavier object falls faster than a lighter object, and a lighter object is tied to a heavy one the heavy object should then slow down due to the action of the lighter object. On the other hand, the lighter object tied to the heavier object that creates a compound object that is heavier still than the heavy object, so it must fall faster. Whenever an idea generates two conclusions that are contradictory, that idea is incorrect.

The only correct conclusion is that all objects fall equally fast. Based on the last few columns we can put this into more modern language, the acceleration due to gravity is a constant.

But what causes the acceleration due to gravity? What causes an object to fall as soon as we let go of it? The answer to that is a remarkable story that brought us into the modern age of science.

It turns out that the planet Earth itself generates a pull on every other object in the entire universe! This pull is stronger on some objects than others. It is this pull that creates weight. It is this pull that causes an object to fall. Such a pull is called a force and there are many kinds of forces. The force that causes an object to fall is the force due to gravity.

From this we can make the following inference, a force applied to an object causes an acceleration in that object. If we think about this for a while we come to the realization that the force of gravity is what causes the acceleration due to gravity. How does it do that? This is a deep mystery that was only solved by Albert Einstein in 1915. The precise details of this are still a mystery today.

The Earth somehow exerts a force on other objects. In essence the Earth grabs hold of these other objects in some way. But what does the Earth grab hold of with the force of gravity?

The answer is that the force of gravity grabs onto the material body of the object. The more material the body has the greater the force that pulls on it. What is this measure of the material in the body? We call this mass. Mass can be thought of (at least for now) as the amount of stuff in the object. The more stuff, the greater the mass.

Like distance and time we need to develop a unit of mass, with which we can compare all other masses. We will use the kilogram as our mass. We now have three measurement scales that form the basis for a system of measurement called the SI system (System International); meters for length, kilograms for mass, and seconds for time (this is also called the mks system).

If we measure the relationships between force, mass, and acceleration in repeated experiments we find the following fact: The quantity of force acting on an object is equal to the product of the object's mass and its acceleration. We can write this using as the symbol representing the quantity of force and as the mass of the object,

This is called Newton's Equation of Motion and we will go into great lengths to describe what it means and what it does. For now this will be enough.

Theory Challenge Answer for Last Week's Column

We begin where we ended last week. We derived the expression,

If we rewrite as then we get,

If we separate variables again we get,

We can now integrate both sides, including the limits of integration for from to , and for from 0 to ,

Integrating the left-hand side of the equation we have,

where the right-hand side is,

We now have to find a functions whose derivative is . If we look in a table of derivatives we will eventually notice a rule, called the power rule that states,

If we apply this in reverse we see that, for ,

For our specific case we need to make the coefficient of 2 vanish, so we introduce a constant of ,

Putting both halves of the equation back together, we get.

or,

in the downward direction. If this were in the positive direction then the sign of would be reversed.