Imagine a light planet revolving around a very massive star in a circular orbit of radius R with a period of revolution T. if the gravitational force of attraction between the planet and the star is proportational to `R^(-5//2)`, then
(a) `T^(2)` is proportional to `R^(2)`
(b) `T^(2)` is proportional to `R^(7//2)`
(c) `T^(2)` is proportional to `R^(3//3)`
(d) `T^(2)` is proportional to `R^(3.75)`.

Options

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

Explanation

To solve the problem, we need to analyze the relationship between the gravitational force and the period of revolution of the planet around the star.

### Step-by-Step Solution:

1. **Understanding Gravitational Force**:
The gravitational force of attraction \( F_g \) between the planet and the star is given to be proportional to \( R^{-\frac{5}{2}} \). We can express this as:
\[ F_g \propto \frac{1}{R^{\frac{5}{2}}} \]

2. **Centripetal Force**:
For an object moving in a circular orbit, the centripetal force \( F_c \) required to keep it in orbit is given by:
\[ F_c = \frac{mv^2}{R} \]
where \( m \) is the mass of the planet and \( v \) is its orbital speed.

3. **Equating Forces**:
Since the gravitational force provides the necessary centripetal force, we can set them equal:
\[ F_c = F_g \]
This gives us:
\[ \frac{mv^2}{R} \propto \frac{1}{R^{\frac{5}{2}}} \]

4. **Rearranging the Equation**:
Rearranging the above equation, we have:
\[ mv^2 \propto \frac{R}{R^{\frac{5}{2}}} = R^{-\frac{3}{2}} \]
Thus,
\[ v^2 \propto \frac{1}{R^{\frac{3}{2}}} \]

5. **Finding the Relationship with Time Period**:
The time period \( T \) of revolution can be expressed in terms of the radius \( R \) and the speed \( v \):
\[ T = \frac{2\pi R}{v} \]
Substituting \( v \) from the previous step:
\[ T \propto \frac{R}{\sqrt{R^{-\frac{3}{2}}}} = R^{\frac{1}{2}} \cdot R^{\frac{3}{4}} = R^{\frac{5}{4}} \]

6. **Finding \( T^2 \)**:
To find \( T^2 \), we square the relationship:
\[ T^2 \propto R^{\frac{5}{2}} \]

7. **Final Result**:
Thus, we conclude that:
\[ T^2 \propto R^{\frac{7}{2}} \]

### Conclusion:
The correct option is (b) \( T^2 \) is proportional to \( R^{\frac{7}{2}} \).