ALGEBRA

PROBLEM 7.

The statement has been made that Newton's derivation of his inverse-square law of gravity from Kepler's third law is among the most important calculations ever performed in the history of science. Kepler's third law, based on observation rather than theory, states that the squares of the periods of any two planets are to each other as the cubes of their average distances from the Sun. Derive Newton's law from Kepler's law.

Solution: If we represent the periods of any two planets by t and T and their distances from the Sun by r and R, respectively, then

T^2/t^2 = R^3/r^3

or

T^2 = (t^2 x R^3)/r^3

Assuming that we know the values of t and r, and substituting a constant C for the quantity t^2/r^3 the equation can be reduced to

T^2= CR^3.

Thus if we know either T or R for the second planet, we can solve for the unknown quantity. In this problem, however, we wish to use this equation to discover a new relationship, Newton's law of gravitation. For a body moving in a circular path, the acceleration toward the center is

a = v^2/r

Substituting in F = ma,

F= mv^2/r

The velocity of the body in the circular orbit is

v = 2(pi)r/T

So

F = mv^2/r = m/R [2piR/T]^2

Because T^2 = CR^3, we find by substitution in the previous equation that

F = [4(pi)^2m/C] x 1/R^2

= K/R^2

That is, the force holding a planet in orbit falls off as the square of the distance R to the Sun. Newton expressed the value of K and obtained his law of universal gravitation:

F = GMm/r^2.

This law applies not only to the attraction between a planet and the Sun but also to the attraction between any two bodies. G is the constant of universal gravitation, M and m are the masses of the two bodies, and r is the distance between their centers of mass.

In solving the next problem, two special techniques are needed. One is a frequently used approximation based on the fact that (l + x) (l - x) = 1 - x^2. If x is small (for example, suppose x = 0.01), then x^2 is very much smaller (for x = 0.01, x^2 = 0.0001), and in this case it is well within the limits of experimental error to use (l + x) (l - x) = 1, or 1/(1 + x) = 1 - x. The other is the substitution of a single variable for the ratio of two other variable names.

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