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A test particle moves in a circular orbit within the gravitational field generated by a mass distribution of density rho(r) = K/r². Find the correct relation between the orbital radius R and the orbital period T.

1. T/R is a constant
2. T/R² is a constant
3. T R is a constant
4. T²/R³ is a constant

Correct answer: T/R is a constant

Solution

M(R) = integral₀^R (K/r²)(4 pi r²) dr = 4 pi K R, proportional to R. Circular orbit: v²/R = G M(R)/R² => v² = G M(R)/R = 4 pi G K = constant, so v is independent of R. Then T = 2 pi R / v => T proportional to R => T/R = constant.

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