Question

Deduce Kepler's 3rd law from Newton's law of gravitation.​

Answer 1

Answer:

The square of the period of planet is proportional to the cube of the semi major axis of the orbit. For the circular orbit, this would translate to square of period being proportional to the cube of orbit radius.

Answer 2

Answer:

Let :

m = mass of planet

M = mass of sun

r = radius of orbit of the planet

ω = constant angular velocity of planet.

The centripetal force on the planet required for its orbital motion is provided by gravitational force of attraction between planet and sun as :

m ω² r = G M m / r²

We know :

ω = 2 π / T

= > m ( 4 π ² / T² ) r = G M m / r²

= > T² = ( 4 π² / G M ) r³

According to Kepler's law :

T² ∝ r³

= > T² ∝ r³  where ( 4 π² / G M ) is constant .

Therefore we get required answer.