Question

Two spherically symmetric planets with no atmosphere have the same average density, but planet B has twice the radius of planet A. A small satellite of mass m_A has period T_A when it orbits planet A in a circular orbit that is just above the surface of the planet. A small satellite of mass m_B has period T_B when it orbits planet B in a circular orbit that is just above the surface of the planet. How does T_B compare to T_A?

Answer 1

let M_A and M_B are the mass of the planets.

mass = density*volume

= rho*(4/3)*pi*R^3

so,

M_B = 8*MA

we know,

Time period of a satellite,

T = 2*pi*r^(3/2)/sqrt(G*M)

so,

TA = 2*pi*RA^(3/2)/(G*M_A)

TB = 2*pi*RB^(3/2)/(G*M_B)

TB/TA = sqrt(M_A/M_B)*(RB/RA)^(3/2)

= sqrt(1/8)*(2)^(3/2)

= 1 <<<<-------------------Answer

so, TA and TB are same.