show that theoretically the period of rotation of a satelite which circles a planet of negligible distance firm its surface depends only on the density of the planet

Answer 1

G m M/r^2 = m v^2/r
so
G M/r = v^2

M = (4/3)rho r^3

G(4/3) rho r^2 = v^2

T = 2 pi r/v
v^2 = (2 pi)^2 r^2/T^2

so
G(4/3) rho = (2 pi/T)^2