A small planet is revolving around a massive star in a circular orbit of radius R with a period of revolution T. If the gravitational force between the planet and the star were proportional to `R^(-5//2)`, then T would be proportional to

Options

A. `r^(3//2)`
B. `r^(5//3)`
C. `r^(7//4)`
D. `r^3`

Answer

To solve the problem, we need to analyze the relationship between the period \( T \) of a planet revolving around a star and the radius \( R \) of its orbit when the gravitational force is proportional to \( R^{-5/2} \).

### Step-by-Step Solution:

1. **Understanding Gravitational Force**:
The gravitational force \( F \) between the planet and the star can be expressed as:
\[ F \propto \frac{1}{R^{5/2}} \]
This implies that:
\[ F = G \cdot \frac{M \cdot m}{R^{5/2}} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the star, and \( m \) is the mass of the planet.

2. **Centripetal Force Requirement**:
For a planet in circular motion, the centripetal force required to keep it in orbit is given by:
\[ F_{centripetal} = m \cdot \frac{v^2}{R} \]
where \( v \) is the orbital speed of the planet.

3. **Relating Orbital Speed to Period**:
The orbital speed \( v \) can be expressed in terms of the period \( T \):
\[ v = \frac{2\pi R}{T} \]
Substituting this into the centripetal force equation gives:
\[ F_{centripetal} = m \cdot \frac{(2\pi R/T)^2}{R} = m \cdot \frac{4\pi^2 R}{T^2} \]

4. **Setting Forces Equal**:
Since the gravitational force provides the necessary centripetal force, we can set the two expressions for force equal to each other:
\[ G \cdot \frac{M \cdot m}{R^{5/2}} = m \cdot \frac{4\pi^2 R}{T^2} \]

5. **Canceling Masses**:
We can cancel \( m \) from both sides (assuming \( m \neq 0 \)):
\[ G \cdot \frac{M}{R^{5/2}} = \frac{4\pi^2 R}{T^2} \]

6. **Rearranging for \( T^2 \)**:
Rearranging the equation to solve for \( T^2 \):
\[ T^2 = \frac{4\pi^2 R^{7/2}}{G \cdot M} \]

7. **Finding Proportionality**:
From the equation \( T^2 \propto R^{7/2} \), we can deduce that:
\[ T \propto R^{7/4} \]

### Conclusion:
Thus, the period \( T \) is proportional to \( R^{7/4} \).