Question

9. Derive Kepler’s 3rd Law?.

Answer 1

Consider a mass m (planet) rotating around another mass M (sun). We assume that m is rotating in a circular orbit.
According to Newton's law of gravity, force between these two masses is,
F = GMm/r².
Where G is the unversal gravitational constant and r is the distance between the two masses.
Any mass, moving in circular motion experiences another force called centripetal force.
F = mv²/r
Where v is the velocity of the orbiting mass m.

Equating these tow forces,
GMm/r² = mv²/r
Canceling m and r,
GM/r = v².                   ...(1)
Velocity can be written as v = distance/time
For a circular path, distance is the circumference of the circle = 2r
time can be taken as T, which is the time taken to travel one full circle.
So, v = 2r/T
Substituting v in (1),
GM/r = [2r/T]2
GM/r = 4²r²/T2
T² = [4²/GM] r³