Satellite A of mass m orbits a planet of mass M at a distance of R from the center of the planet. The speed of the satellite is v. Answer in terms of m, M, R and v.

Question

Satellite B has mass m and orbits a planet of mass M. What should satellite B's orbital radius be so that its period is 8 times that of satellite A?

(The answer is 4 R I just don't know how to get there)

Answer 1

Use Kepler's 3rd law.

T = period

R = orbital radius

R³/T² = constant

R_A³ / T_A² = R_B³ / T_B²

T_B = 8T_A ⇒

R_B³ = (T_B / T_A)² R_A³ = 8² R_A³

R_B = (64)^1/3 R_A = 4R_A = 4R