A satellite which is geostationary in a particular orbit is taken to another orbit. Its distance from the centre of earth in new orbit is 2 times that of the earlier orbit. The time period in the second orbit is

Options

A. (a)24 h
B. (b)48 h
C. (c)`48 sqrt(2)h`
D. (d)`48/sqrt2h`

Answer

To find the time period of a satellite in a new orbit that is twice the distance from the center of the Earth compared to its original geostationary orbit, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Time Period Formula**:
The time period \( T \) of a satellite in orbit is given by the formula:
\[ T = 2\pi \sqrt{\frac{r^3}{GM}} \]
where:
- \( T \) = time period of the satellite
- \( r \) = distance from the center of the Earth (radius of the orbit)
- \( G \) = gravitational constant
- \( M \) = mass of the Earth

2. **Geostationary Orbit**:
For a geostationary satellite, the time period \( T_1 \) is 24 hours (or 86400 seconds). The radius \( r_1 \) of the geostationary orbit can be calculated using the formula:
\[ T_1 = 2\pi \sqrt{\frac{r_1^3}{GM}} \]

3. **New Orbit Radius**:
The problem states that the new orbit's distance from the center of the Earth is twice that of the original orbit. Therefore, we have:
\[ r_2 = 2r_1 \]

4. **Finding the Time Period in the New Orbit**:
We can express the time period \( T_2 \) for the new orbit using the same formula:
\[ T_2 = 2\pi \sqrt{\frac{r_2^3}{GM}} = 2\pi \sqrt{\frac{(2r_1)^3}{GM}} = 2\pi \sqrt{\frac{8r_1^3}{GM}} = 2\pi \sqrt{8} \sqrt{\frac{r_1^3}{GM}} \]
This simplifies to:
\[ T_2 = 2\pi \cdot 2\sqrt{2} \sqrt{\frac{r_1^3}{GM}} = 2\sqrt{2} T_1 \]

5. **Substituting the Time Period of the Original Orbit**:
Since \( T_1 = 24 \) hours, we substitute this value:
\[ T_2 = 2\sqrt{2} \cdot 24 \text{ hours} \]
To calculate \( T_2 \):
\[ T_2 = 48\sqrt{2} \text{ hours} \]

6. **Final Calculation**:
The approximate value of \( \sqrt{2} \) is about 1.414, so:
\[ T_2 \approx 48 \cdot 1.414 \approx 67.99 \text{ hours} \]

### Conclusion:
The time period in the second orbit is approximately \( 67.99 \) hours.