Question

A test particle is moving in a circular orbit in the gravitational field produced by a mass density $\rho(r) = \frac{K}{r^2}$. Identify the correct relation between the radius $R$ of the particle's orbit and it's period $T$ :

• A. $T/R^2$ is a constant
• B. $TR$ is a constant
• C. $T^2/R^3$ is a constant
• D. $T/R$ is a constant

Answer 1

Answer: (D)

$T/R$ is a constant

Answer 2

$m = \int^{R}_{0} \rho4\pi r^{2}dr$
$m = 4\pi KR$
$v \propto \sqrt{4\pi K}$
$\frac{T}{R} = \frac{2\pi}{\sqrt{4\pi K}}$