Question

A satellite of mass m is in a circular orbit of radius R2 around a spherical planet of radius R1 made of a material with density ρ. ( R2 is measured from the center of the planet, not its surface.) Use G for the universal gravitational constant.

A) Find the kinetic energy of this satellite, K

Express the satellite's kinetic energy in terms of G, m, π, R1, R2, and ρ.

B) Find U, the gravitational potential energy of the satellite. Take the gravitational potential energy to be zero for an object infinitely far away from the planet.

Express the satellite's gravitational potential energy in terms of G, m, π, R1, R2, and ρ.

C) What is the ratio of the kinetic energy of this satellite to its potential energy?

Express K/U in terms of parameters given in the introduction.

Answer 1

a)

get mass of planet:

ρ = M / V

V = 4/3 * R_1^3

M = ρ * V

M = ρ * 4/3 * R_1^3

equate force equations:
F = (GMm) / r^2     // r = R_2

F = ma

a = v^2/R_2

F = m * (v^2/R_2)

m * (v^2/R_2) = (GMm) / R_2^2

plug in and solve v^2:

m * (v^2/R_2) = (G * (ρ * 4/3 * R_1^3) *m) / R_2^2

v^2 = (G * ρ * (4/3) * π * R_1^3) / R_2

put into kinetic energy equation:

K = 1/2 * m * v^2

K = 1/2 * m * (G * ρ * (4/3) * π * R_1^3) / R_2

b) 

givens:

U = -(GmM) / R_2

plug in mass of planet:

U = -(G * m * ρ * 4/3 * R_1^3) / R_2

c)

K/U = (1/2 * m * (G * ρ * (4/3) * π * R_1^3) / R_2) * -(R_2 / (G * m * ρ * 4/3 * R_1^3))

K/U = -1/2