Question

Planet of mass m moves around the sun with mass m in a circular orbit of radius r with angular velocity omega another planet of mass m moves around the sun in circular orbit of radius 4r with angular speed omega dash find the ratio of omega square upon omega dash square

Answer 1

GMm / r^2 = m ω^2 r

ω^2 = GM / r^3

As ,

ω'^2 = GM / ( 4r )^3

ω'^2 = GM / 64r^3

ω^2 / ω'^2 = GM / r^3 /  GM / 64r^3

ω^2 / ω'^2 = 64 r^3 / r^3

ω / ω' = ( 4 / r ) ^3/2

ω / ω' = ( 2 ) ^3

ω / ω' = 8