A light planet is revolving around a massive star in a circular orbit of radius "R" with a period of revolution "T" .If the force of attraction between planet and star is proportional to `R^(-3/2)` then choose the correct option :

Options

A. `T^(2)prop R^(3/2)`
B. `T^(2)prop R^(5/2)`
C. `T^(2)prop R^(3)`
D. `T^(2)prop R^(7/2)`

Answer

To solve the problem, we start by analyzing the given information about the light planet revolving around a massive star.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have a light planet revolving around a massive star in a circular orbit of radius \( R \) with a period of revolution \( T \). The gravitational force of attraction between the planet and the star is proportional to \( R^{-3/2} \).

2. **Expressing the Gravitational Force**:
According to the problem, the gravitational force \( F \) can be expressed as:
\[ F \propto R^{-3/2} \]
This means we can write:
\[ F = k \cdot R^{-3/2} \]
where \( k \) is a constant of proportionality.

3. **Using Circular Motion**:
For a planet in circular motion, the gravitational force provides the necessary centripetal force. The centripetal force \( F_c \) is given by:
\[ F_c = m \cdot \frac{v^2}{R} \]
where \( m \) is the mass of the planet and \( v \) is its tangential velocity.

4. **Relating Velocity to Period**:
The tangential velocity \( v \) can be related to the period \( T \) as:
\[ v = \frac{2\pi R}{T} \]
Substituting this into the centripetal force equation gives:
\[ F_c = m \cdot \frac{(2\pi R/T)^2}{R} = m \cdot \frac{4\pi^2 R}{T^2} \]

5. **Setting Gravitational Force Equal to Centripetal Force**:
Since the gravitational force provides the centripetal force, we can set them equal:
\[ k \cdot R^{-3/2} = m \cdot \frac{4\pi^2 R}{T^2} \]

6. **Rearranging the Equation**:
Rearranging this equation gives:
\[ T^2 \propto R^{5/2} \]
This implies:
\[ T^2 = C \cdot R^{5/2} \]
where \( C \) is a constant.

7. **Conclusion**:
From the derived relationship, we can conclude that the period \( T \) is proportional to \( R^{5/4} \) when considering the square root:
\[ T \propto R^{5/4} \]

### Final Answer:
The correct option is the one that matches \( T^2 \propto R^{5/2} \).