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}?
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.
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