If `g prop 1/R^(3)` (instead of `1/R^(2)`), then the relation between time period of a satellite near earth's surface and radius R will be

Options

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

Answer

To solve the problem, we need to derive the relationship between the time period \( T \) of a satellite and the radius \( R \) when the acceleration due to gravity \( g \) is proportional to \( \frac{1}{R^3} \).

### Step-by-Step Solution:

1. **Understanding the gravitational force**:
The gravitational force acting on a satellite of mass \( m \) at a distance \( R \) from the center of the Earth can be expressed as:
\[ F_g = m \cdot g \]
Given that \( g \propto \frac{1}{R^3} \), we can write:
\[ g = \frac{k}{R^3} \]
where \( k \) is a constant.

2. **Centrifugal force**:
For a satellite in circular motion, the centrifugal force acting on it is given by:
\[ F_c = \frac{m v^2}{R} \]
where \( v \) is the orbital speed of the satellite.

3. **Setting the forces equal**:
For the satellite to remain in orbit, the gravitational force must equal the centrifugal force:
\[ m \cdot g = \frac{m v^2}{R} \]
Substituting \( g \) from the earlier equation:
\[ m \cdot \frac{k}{R^3} = \frac{m v^2}{R} \]
We can cancel \( m \) from both sides (assuming \( m \neq 0 \)):
\[ \frac{k}{R^3} = \frac{v^2}{R} \]

4. **Solving for \( v^2 \)**:
Rearranging the equation gives:
\[ v^2 = \frac{k}{R^2} \]

5. **Finding the time period \( T \)**:
The time period \( T \) of the satellite is the time it takes to complete one full orbit. It can be calculated using the formula:
\[ T = \frac{2\pi R}{v} \]
Substituting \( v \) from the previous step:
\[ T = \frac{2\pi R}{\sqrt{\frac{k}{R^2}}} \]
Simplifying this expression:
\[ T = \frac{2\pi R}{\frac{\sqrt{k}}{R}} = 2\pi R \cdot \frac{R}{\sqrt{k}} = \frac{2\pi R^2}{\sqrt{k}} \]

6. **Establishing the relationship**:
From the final expression, we can see that:
\[ T \propto R^2 \]
This means that the time period \( T \) is directly proportional to the square of the radius \( R \).

### Conclusion:
Thus, if \( g \propto \frac{1}{R^3} \), then the relation between the time period \( T \) of a satellite near the Earth's surface and the radius \( R \) is:
\[ T \propto R^2 \]