Question

Using Kepler's 3rd law and Newton's law of universal gravitation, find the period of revolution P of the planet as it moves around the sun. Assume that the mass of the planet is much smaller than the mass of the sun. Use G for the gravitational constant.

Answer 1

Answer:

Explanation:

According to the Kepler's third law, the square of the time period of a planet is directly proportional to the length of cube of semi major axis.

Let M is the mass of sun, m is the mass of planet and r is the radius of orbit and T is the time period of the planet around the sun. Here ω is the angular velocity of the planet around the sun.

G is the universal gravitational constant.

The centripetal force is balanced by the gravitational force between the planet and the sun.

$mr\omega^{2}=\frac{GMm}{r^{2}}$

$r\times \frac{4 \pi^{2}}{T^{2}}=\frac{GM}{r^{2}}$

$T^{2}=\frac{4 \pi^{2}r^{3}}{GM}$

$T=\sqrt \frac{4 \pi^{2}r^{3}}{GM}$