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Question:

A test particle is moving in a circular orbit in the gravitational field produced by a mass density ρ(r)=Kr². Identify the correct relation between the radius R of the particle's orbit and its period T.

T²/R³ is a constant

T/R² is a constant

TR is a constant

T/R is a constant

Solution:

Correct option is D. T/R is a constant
m = ∫₀ᴿ ρ 4πr² dr
m = 4πkR⁴/4
m = πkR⁴
V² = GM/R = G(πkR⁴)/R = GπkR³
T = 2πR/V = 2πR/√(GπkR³)
T = 2π√(R/(GπkR³))
T = 2π√(1/(GπkR²))
T² = 4π²(1/(GπkR²))
T²/R = 4π²/(GπkR³)
T/R is a constant.