Answer to Question #168733 in Physics for Pushpo

Question

Derive the Newton’s law of gravitation from the Kepler’s law

Answer 1

Suppose a planet of mass m is revolving around the sun of mass M in a nearly circular orbit of radius r, with a constant angular velocity ω. Let, T be the time period of revolution of the planet around the sun.

$$\omega=\frac{2\pi}{T}$$

The centripetal force acting on the planet for its circular motion is:

$$F=mr\omega^2=\frac{4\pi^2mr}{T^2}$$

According to Kepler’s Third Law:

$$T^2= Kr^3$$

Therefore,

$$F=mr\omega^2=\frac{4\pi^2m}{Kr^2}$$

This centripetal force is provided by the gravitational attraction exerted by the sun on the planet. According to Newton, the gravitational attraction between the sun and the planet is mutual. If force F is directly proportional to the mass of the planet (m). It should be directly proportional to the mass of the sun (M).

Hence,

$$\frac{4\pi^2}{K}=GM$$

Thus,

$$F=\frac{GMm}{r^2}$$

Which is Newton’s Law of Gravitation.