A planet of mass 4 times earth spins about itself and completes one rotation is 96 hours . The radius of a secondary stationary satellite about this planet in comparisons to the radius of the geostationary orbit around the earth is

Options

A. 4 times
B. `(1/4)` times
C. 2 times
D. `(1/2)` times

Answer

To solve the problem, we need to find the radius of a secondary stationary satellite around a planet with a mass that is 4 times that of Earth, given that it completes one rotation in 96 hours. We will compare this radius with the radius of a geostationary orbit around Earth.

### Step-by-Step Solution:

1. **Understanding the Geostationary Orbit**:
The formula for the radius of a geostationary orbit \( R \) is given by:
\[ R = \left( \frac{GM T^2}{4\pi^2} \right)^{1/3} \]
where \( G \) is the universal gravitational constant, \( M \) is the mass of the planet, and \( T \) is the time period of the orbit.

2. **Parameters for Earth**:
For Earth, we denote:
- Mass of Earth: \( M_E \)
- Time period for geostationary orbit: \( T_E = 24 \text{ hours} = 86400 \text{ seconds} \)

Thus, the radius of the geostationary orbit around Earth is:
\[ R_E = \left( \frac{G M_E T_E^2}{4\pi^2} \right)^{1/3} \]

3. **Parameters for the Planet**:
For the planet:
- Mass of the planet: \( M_P = 4M_E \)
- Time period for rotation: \( T_P = 96 \text{ hours} = 345600 \text{ seconds} \)

The radius of the secondary stationary satellite around this planet is:
\[ R_P = \left( \frac{G M_P T_P^2}{4\pi^2} \right)^{1/3} \]

4. **Substituting Values**:
Substitute \( M_P \) and \( T_P \) into the equation for \( R_P \):
\[ R_P = \left( \frac{G (4M_E) (345600^2)}{4\pi^2} \right)^{1/3} \]

5. **Comparing with Earth's Radius**:
Now, we can find the ratio \( \frac{R_P}{R_E} \):
\[ \frac{R_P}{R_E} = \frac{\left( \frac{G (4M_E) (345600^2)}{4\pi^2} \right)^{1/3}}{\left( \frac{G M_E (86400^2)}{4\pi^2} \right)^{1/3}} \]

This simplifies to:
\[ \frac{R_P}{R_E} = \left( \frac{4 (345600^2)}{86400^2} \right)^{1/3} \]

6. **Calculating the Ratio**:
Calculate \( \frac{345600^2}{86400^2} \):
\[ \frac{345600}{86400} = 4 \quad \text{(since } 345600 = 4 \times 86400\text{)} \]
Therefore:
\[ \frac{R_P}{R_E} = \left( 4 \times 4 \right)^{1/3} = (16)^{1/3} = 2^{4/3} \]

7. **Final Result**:
Thus, the radius of the secondary stationary satellite about this planet in comparison to the radius of the geostationary orbit around Earth is:
\[ R_P = 4 R_E \]

### Conclusion:
The radius of the secondary stationary satellite around the planet is **4 times** the radius of the geostationary orbit around Earth.