Using Newton's law of gravitation, prove Kepler's `III^(rd)` law of planetary motion for circular orbits.

Explanation

Planet of mass 'm' moves around the Sun of mass 'M' in a circular orbit of radius 'r' with orbital velocity v.
`therefore(mv^2)/(r)=(GMm)/(r^2)`
`v^(2)=(GM)/(r)" "...(1)`
But,
Orbital speed `(v)=("circumference")/("period of revolution")=(2pir)/(T)`
Squaring,
`v^(2)=(4pi^(2)r^(2))/(T^2)" "...(2)`
Comparing of equations (1) and (2), we get
`(GM)/(r)=(4pi^(2)r^(2))/(T^2)`
In cross multiplication, `(GMm)/(r^2)`
`T^(2)propr^(3)`