A planet of uniform density and radius rotates slowly enough that its shape can be…
PHYSICS · Multi-part · A Level · Physics · Fields and their consequences · Fields and their consequences
Problem
A planet of uniform density ρ and radius R rotates slowly enough that its shape can be treated as a perfect sphere. A tiny spacecraft is placed in a circular orbit of radius r (measured from the planet’s center), where r>R. The spacecraft’s mass is negligible compared with the planet’s mass.
(1) Derive an expression for the planet’s mass M in terms of the density ρ and radius R.
(2) Using your result from part (a), obtain a symbolic expression for the orbital speed v of the spacecraft in terms of ρ, R, r, and universal constants as needed.
(3) Show that when the orbital radius is just above the planet’s surface (r→R+), the expression for v reduces to a simple form that depends only on G, ρ, and R. Provide that simplified form.
(4) Discuss how the orbital speed varies with r for r>R based on your general expression from part (b). Identify whether the dependence is proportional, inversely proportional, or follows another power relationship, and interpret the physical significance.