c. Write the three laws given by Kepler. How did they help Newton to arrive at the inverse square law of gravity?

Answer: Kepler's laws describe planetary orbits; using his third law (\(T^2 \propto r^3\)) with the centripetal force, Newton showed that the gravitational force \(F \propto \tfrac{1}{r^2}\).

Kepler's three laws

1. First law: The orbit of a planet is an ellipse with the Sun at one of the foci.
2. Second law: The line joining the planet and the Sun sweeps equal areas in equal intervals of time.
3. Third law: The square of the period of revolution of a planet around the Sun is directly proportional to the cube of the mean distance of the planet from the Sun, i.e. \(T^2 \propto r^3\) or \(\tfrac{T^2}{r^3} = K\) (constant).

How they helped Newton

Newton assumed, for simplicity, that a planet moves in a circular orbit of radius \(r\) around the Sun with uniform speed \(v\). The force needed to keep it moving in a circle is the centripetal force:

\(F = \dfrac{mv^2}{r}\) ... (1)

In one revolution the planet covers the circumference \(2\pi r\) in time \(T\), so \(v = \dfrac{2\pi r}{T}\). Putting this in (1):

\(F = \dfrac{m}{r}\cdot\dfrac{4\pi^2 r^2}{T^2} = \dfrac{4\pi^2 m r}{T^2}\) ... (2)

By Kepler's third law, \(T^2 = K r^3\). Substituting in (2):

\(F = \dfrac{4\pi^2 m r}{K r^3} = \dfrac{4\pi^2 m}{K}\cdot\dfrac{1}{r^2}\)

Since \(4\pi^2\), \(m\) and \(K\) are constants for a given planet, \(F \propto \dfrac{1}{r^2}\).

Thus Kepler's third law showed Newton that the force of attraction between the Sun and a planet must be inversely proportional to the square of the distance between them. This is the inverse square law of gravitation.