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

The Correct Option is D

To determine the correct relationship between the radius \( R \) of a particle's orbit and its period \( T \), let's analyze the given information:

Given mass density is \( \rho(r) = \frac{K}{r^2} \).

This mass density suggests a spherically symmetric mass distribution. The gravitational field inside a material with such mass distribution can be derived from the integrated mass enclosed within the radius \( r \).

The gravitational force acting on the particle provides the necessary centripetal force required for circular motion. Therefore, we can relate the gravitational force to the centripetal force:

\[ F_{\text{gravity}} = F_{\text{centripetal}} \]

We start by calculating the mass enclosed \( M(r) \) within a radius \( r \):

\[ M(r) = \int_0^r \rho(r') \cdot 4\pi r'^2 \, dr' = \int_0^r \frac{K}{r'^2} \cdot 4\pi r'^2 \, dr' \]

This gives us:

\[ M(r) = 4\pi K \int_0^r 1 \, dr' = 4\pi K r \]

The gravitational force is then given by:

\[ F_{\text{gravity}} = \frac{G M(r) m}{r^2} = \frac{G \cdot 4\pi K r \cdot m}{r^2} \]

\[ F_{\text{gravity}} = \frac{4\pi G K m}{r} \]

For circular motion, the centripetal force \( F_{\text{centripetal}} \) is:

\[ F_{\text{centripetal}} = \frac{m v^2}{r} \]

Equating both forces:

\[ \frac{4\pi G K m}{r} = \frac{m v^2}{r} \]

Solving for \( v \), we have:

\[ v^2 = 4\pi G K \]

For circular motion, the period \( T \) is given by:

\[ T = \frac{2\pi r}{v} \]

Substitute \( v = \sqrt{4\pi G K} \):

\[ T = \frac{2\pi r}{\sqrt{4\pi G K}} \]

This implies:

\[ T \propto r \]

Thus, \( \frac{T}{R} \) is a constant. Therefore, the correct relationship between the radius \( R \) of the particle's orbit and its period \( T \) is:

\(\frac{T}{R}\) is a constant.