Question

Show that the period of a particle that moves in a circular orbit close to the surface of a sphere
depends only upon G and the average density ρ of the sphere. Find what this period would be
for any sphere having an average density equal to that of water.

Answer 1

F(r)=F_(s)+F_(g)

_where F_(s) force due to sphere

F_(g) force due to gravity

F_(s) = -$\dfrac{GM m}{r^2}$

m is the mass of sphere

r distance between sphere and orbit

F_(d) =$\dfrac{GM{\tiny d} m}{r^2}$

where M_d is the mass of particle

F(r) = -$\dfrac{GM m}{r^2}$ -$\dfrac{GM{\tiny d} m}{r^2}$

taking M_d=${4 \over 3}\pi r^3\rho$

$\therefore$ F(r)-=-$\dfrac{GM{\tiny d} m}{r^2}$ -${4 \over 3}\pi r^3\rho$ G_r