A star of mass `m` revolves in radius `R` in galaxy in which density varies with distance `r` as `rho = k/r`, `k = constant`. Find the relation between time period and radius `R`

Options

A. `T xx R^3/2` is constant
B. `T xx R^(-1/2)` is constant
C. `T xx R^7/2` is constant
D. `T xx R` is constant

Answer

To solve the problem, we need to find the relationship between the time period \( T \) of a star revolving in a galaxy and the radius \( R \) of its orbit, given that the density \( \rho \) of the galaxy varies with distance \( r \) as \( \rho = \frac{k}{r} \).

### Step-by-Step Solution:

1. **Determine the Mass of the Galaxy**:
- The density of the galaxy is given as \( \rho = \frac{k}{r} \).
- We consider a spherical shell of radius \( r \) and thickness \( dr \). The volume \( dV \) of this shell is:
\[ dV = 4\pi r^2 dr \]
- The mass \( dm \) of this shell can be expressed as:
\[ dm = \rho \cdot dV = \frac{k}{r} \cdot 4\pi r^2 dr = 4\pi k r dr \]
- To find the total mass \( M \) of the galaxy up to radius \( R \), we integrate \( dm \) from \( 0 \) to \( R \):
\[ M = \int_0^R 4\pi k r \, dr = 4\pi k \left[ \frac{r^2}{2} \right]_0^R = 4\pi k \cdot \frac{R^2}{2} = 2\pi k R^2 \]

2. **Apply Newton's Law of Gravitation**:
- The gravitational force \( F \) acting on the star of mass \( m \) due to the mass \( M \) of the galaxy is given by:
\[ F = \frac{G M m}{R^2} \]
- Substituting the expression for \( M \):
\[ F = \frac{G (2\pi k R^2) m}{R^2} = 2\pi G k m \]

3. **Centripetal Force Requirement**:
- The centripetal force required to keep the star moving in a circular orbit is given by:
\[ F_c = \frac{m v^2}{R} \]
- Setting the gravitational force equal to the centripetal force:
\[ 2\pi G k m = \frac{m v^2}{R} \]
- Canceling \( m \) from both sides (assuming \( m \neq 0 \)):
\[ 2\pi G k = \frac{v^2}{R} \]
- Rearranging gives us:
\[ v^2 = 2\pi G k R \]

4. **Relate Velocity to Time Period**:
- The time period \( T \) of the star's orbit is given by the circumference of the circle divided by the velocity:
\[ T = \frac{2\pi R}{v} \]
- Substituting \( v \) from the previous step:
\[ T = \frac{2\pi R}{\sqrt{2\pi G k R}} = \frac{2\pi R}{\sqrt{2\pi G k} \sqrt{R}} = \frac{2\pi \sqrt{R}}{\sqrt{2\pi G k}} \]

5. **Final Relation**:
- Thus, we can express the relationship between the time period \( T \) and the radius \( R \):
\[ T \propto \sqrt{R} \]
- This means that the time period \( T \) is proportional to the square root of the radius \( R \).