Advanced — Period Change with DensityHardQuestion 30779
Question
A uniform sphere of density ρ has orbital period T for a surface-skimming satellite. If ρ doubles (same mass, smaller radius), T:
- ADoubles
- BHalvesCorrect
- CStays the same
- DIncreases by √2
Correct answer
Halves
Explanation
T = 2π√(R/g) = 2π√(R/(4πGρR/3)) = 2π√(3/(4πGρ)). T depends only on density: T ∝ 1/√ρ. If ρ doubles, T → T/√2. But surface-skimming orbit: T² ∝ R³/M ∝ (M/ρ)/M = 1/ρ, so T ∝ 1/√ρ → halving ρ doubles T.