Question

If a satellite orbits very near the surface of a planet with period T, derive an algebraic expression for the density (mass/volume) of the planet. Express your answer in terms of the variable T and the gravitational constant G. Part B Estimate the density of the Earth, given that a satellite near the surface orbits with a period of about 85 min.

Answer 1

Let angular velocity be w

balancing the forgravitational force and centrifugal force

GMm/r^2 = mw^2*r

M is mass of earth

m is mass of satellite

r is distance of satellite from center of earth which is approx equal to radius of earth as satellite is very close

G(M/r^3) = w^2

Volume of earth = 4/3*pi*r^3

multiply and divide by (4/3)pi

G*(4*pi/3)*(M/(4pir^3/3)) = w^2 = (2*pi/T)^2

this gives 4.19G*d = 39.44/T^2

d = 9.41/GT^2 ----------- ANS

b.)

d = 9.41/(6.67 * 10-11*(85*60)^2) = 5424.05 Kg/m^3