Law of Gravitation

Law of Gravitation

A stone has weight. This is the gravitational force between earth and the stone. If the stone is very far from the earth, say as far as the moon, it still feels this force. Exactly what is the value then?

About 400 years ago, Kepler measured the orbits of the planets very accurately. Issac Newton found that if he assumed a certain law of gravitation, then he could calculate each orbit correctly.

Suppose the mass of the sun is m₁, and the mass of a planet is m₂, and the distance is r. The law Newton used, was that the force F was directly proportional to m₁, m₂, and inversely proportional to r.

So Newton's law of gravitation is F = Gm₁m₂/r², where G is a constant that is the same for any mass or distance. It works for any two masses, e.g. sun and earth, earth and stone, or stone and ball.

Exercise. I am 50 kg, and the radius of the earth is 64oo km. Find the mass of the earth. Given that G = ____. Answer. Let M = mass of earth, m = my mass, F = my weight, and R = radius of earth. The law of gravitation is F = GMm/R². Rearranging, M = FR²/(Gm) = 60 x 6400000² / ( __ x 50) = ___ kg.

Is the law always true? What if we think of some rather perverse examples, like a stone in a metal cylinder? Would the gravitational force between the two still obey the law? If the stone is right in the middle of the cylinder, the force between them would be zero. The law would not hold.

In fact, the law is only true for the following three cases:

1. When the sizes of both bodies are much smaller than the distance in between, e.g. for two planets far apart.

2. When the shapes of both bodies are perfectly spherical. Then the distance r is from centre of one sphere, to centre of the other.

3. When the shape of one body is perfectly spherical, and the size of the other body much smaller that its distance r from the centre of the sphere. E.g. between earth and me.

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