NEWTON’S LAW OF UNIVERSAL GRAVITATION

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The motion of the moon around the earth is accelerated motion, as is the motion of the earth around the sun. So there must be a force on the moon and one on the earth. If there were no force on the earth, it would just move past the sun in a straight line at constant speed, assuming it could somehow get started in the first place.

Force on the earth Path of the earth

The
slide show on motion pointed out that an object in
circular motion needs a force pointed toward the center of the circle. Newton was able to show that the force on
any planet in a curved orbit – not necessarily a circle but an ellipse or any
other curve it could follow – had to be governed by a force pointed directly
toward the sun. Furthermore, the moon
needed a force pointed directly toward the earth in order to stay in its
orbit. Newton proved this using
Kepler’s Second Law, though we will not go into the details.

Now what kind of force could there be on the moon pointed directly toward the earth? Well, it could be the same force that pulls an apple (or a rock, or a human, or anything else) directly toward the center of the earth – in other words -- down. It could be gravity, conjectured Newton. This was not a new idea, but Newton managed to work it out mathematically and predict the consequences of the idea much better than anyone else had done.

For one thing, there would also have to be a gravitational force on the earth due to the sun. So other objects besides the earth could attract things. In fact, it developed, anything will attract other things to it with a gravitational force as long as the attracting object has any mass at all. So this law became known as Newton’s Law of Universal Gravitation because it could apply to anything.

Newton also worked out a formula for the gravitational force of one object on another. It turned out that there was a specific formula for the gravitational force that had to be true, if gravity was going to explain Kepler’s Laws.

This formula involves the mass of each object. Mass is a concept that tells you several things about the object that possesses it. Newton at first called it “quantity of matter”, which could be loosely thought of as “how much stuff is in something”. In the next file, on Newton’s Second Law, mass will turn out to tell about the inertia of an object – its resistance to changing its motion. In any case, it is measured in kilograms, and it can be measured.

Newton found the gravitational force due to one object on another object to be this:

(A constant called G) x (mass of first object) x (mass of second object)

Force = --------------------------------------------------------------------------------------------

(divided by the square of the distance between them)

In this formula, suppose we use the mass of the earth for
the first mass, my mass for the second one, and the radius of the earth for the
distance. Then the force of gravity
would turn out to be my weight. Of
course, there needs to be units of measurement such as feet, miles, meters, or
something for distance. The radius of
the earth is 6.38 million meters (about 4000 miles), and the mass of the earth
is huge, 5.97x10^{24} kilograms.
Suppose that my mass is 100 kilograms (it is not really that
large!). The constant G is something
that is the same for all objects, and it has been measured to be 6.67x10^{-11}
in standard metric system units. The
gravitational formula gives the gravitational force due to the earth on me as
about 978 in standard metric system units of force called “newtons”, named
after Sir Isaac, of course. This is
exactly the same thing as my weight. For
you, me, or anyone else, the weight is the force with which the earth’s gravity
pulls on us. In our everyday units of
weight, the 978 newtons would be 220 pounds.

Notice that the mass of 100 kilograms is not the same thing as the weight of 220 pounds. The mass stays the same no matter where you go in the universe. If I went to the moon I would still have the same 100 kilograms of mass. However, to figure the force of the moon’s gravity on me I would have to use the mass of the moon and the radius of the moon in the formula. The answer for weight turns out to be about one sixth of the value on the earth. So your weight will depend on what planet you are on. Your mass just stays the same.

For the force of gravity of the earth on a person with only a 50 kilogram mass would be half of the 220 pounds mentioned above. Weight goes up and down with mass, as long as you stay on the same planet, but weight is not the same thing as mass.

In Newton’s gravitational formula, you have to divide by the square of the distance between the two objects. Newton was able to show, after a lot of difficulty, that this means the distance between the centers of spherical objects just as planets. When comparing a planet with an object that is much smaller (such as me), it doesn’t matter too much whether you use the distance to my head, my feet, or to some part of me in between. It comes out the radius of the earth anyway – about 4000 miles.

Suppose, though, that I move twice as far away from the center of the earth – 8000 miles. That is 4000 miles above the surface. Then my weight will turn out to be one fourth of its value on the surface. That is what happens when you divide by the square of the distance as in the above formula.

Suppose, for example, you divide 220 by 1^{2}. You get 220. But then suppose you divide 220 by 2^{2}. That means to divide by 4, and you get
55. If you triple the distance, you
might divide by 3^{2}, or 9.
Then you would get one ninth of 220, or 24.4 pounds.

In the same way, if I were to triple my distance from the
center of the earth, I would be 12000 miles from it, or 4000 miles from the
surface. My weight would be one ninth
of its value on the surface. If I went
as far away as the moon, I would be about 60 times by distance from the center
of the earth. So my weight would be 220
divided by 60^{2}. That is only
0.0611 pounds. This represents the
force of the earth on me at the distance to the moon (240,000 miles). It is not the pull of the moon’s gravity on
me.

A law that behaves this way is called an inverse square law.

This law of gravitation is supposed to apply to any object with mass. Suppose there were two humans each with 50 kilograms of mass separated by one meter. They would, in fact, attract one another with a gravitational force, but it would be a very small one. Since humans have a funny shape, it is hard to figure out where the center is. So the distance is a little ambiguous. But we can get an approximate value by using 1 meter. If you put 50 kilograms into the above formula for each mass and one meter for the distance, you get:

Force of either person on the other = 0.000000167 newtons = 0.000000037 pounds.

They will not exactly stick together because of this force!

If you use 0 for the mass of either object, you get zero for the gravitational force. So an object has to have mass to have gravity. However, a lack of mass is not known to be a problem for any known object. Certain elementary particles are thought to be massless, or at least nearly so, but we won’t deal with that here.