Question

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

• A. $\frac{T}{R^2}$ is a constant
• B. $TR$ is a constant
• C. $\frac{T}{R}$ is a constant
• D. $\frac{T^2}{R^3}$ is a constant

Answer 1

The correct option is C $\frac{T}{R}$ is a constant

Gravitational force experienced by the test particle is given by,

$F=\frac{GMm}{r^2}=\int \frac{G\rho \left(dV\right)m}{r^2}$

$=mG{\int }_0^R\frac{k}{r^2}\frac{4\pi r^{2}dr}{r^2}$
$=-4\pi kGm{\left[\frac{1}{r}\right]}_0^R$
$=-\frac{4\pi kGm}{R}$
Using newton's second law, we have

$\frac{m{v}_0^2}{R}=\frac{4\pi kGm}{R}$

$\Rightarrow v_0=C\left(\text{const.}\right)$

Time period, $T=\frac{2\pi R}{v_0}=\frac{2\pi R}{C}$

$\Rightarrow \frac{T}{R}=\text{const.}$

<!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> Hence, $\left(A\right)$ is the correct answer.