Deduction of Newton’s Law of Gravitation from Kepler’s Law
Newton’s Law of Gravitation is states that in this universe attracts every other body with a force which is directly proportional to the product of their masses and is inversely proportional to the product of the squares of the distance between them.
Newton’s Law of Gravitation can be easily obtained from Kepler’s Laws of Planetary Motion.
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.
\(\omega =\frac{2\pi }{T}\) … (1)
The centripetal force acting on the planet for its circular motion is:
\(F=mr{{\omega }^{2}}=mr{{\left( \frac{2\pi }{T} \right)}^{2}}=\frac{4{{\pi }^{2}}mr}{{{R}^{2}}}\),
According to Kepler’s Third Law:
T² α r³ (Or) T² = Kr³
Where, K = Constant of Proportionality.
Therefore,
\(F=\frac{4{{\pi }^{2}}mr}{K{{r}^{3}}}=\frac{4{{\pi }^{2}}}{K}\times \frac{m}{{{r}^{2}}}\) … (2)
\(F\,\,\propto \,\,\frac{m}{{{r}^{2}}}\) (∵ \(\frac{4{{\pi }^{2}}}{K}\) is a constant) … (3)
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, the factor \(\frac{4{{\pi }^{2}}}{K}\,\,\propto \,\,M\) (Or) \(\frac{4{{\pi }^{2}}}{K}=GM\) … (4)
Substitute equation (4) in the equation (2), we get:
\(F=\frac{GMm}{{{r}^{2}}}\), Which is Newton’s Law of Gravitation.
On the basis of Kepler’s Laws, Newton concluded the following:
1. A force acting on the planet due to sun is the centripetal force which is directed towards the sun.
2. The force acting on the planet must be inversely proportional to the square of the distance from the sun.
3. The force acting on the planet is directly proportional to the product of the masses of the planet and the sun.