The communication satellites are parked at a height of (from surface of earth)

Options

A. 36 km
B. 360 km
C. 3600 km
D. 36000 km

Answer

To determine the height at which communication satellites are parked above the surface of the Earth, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Geostationary Satellites**:
Communication satellites are often geostationary, meaning they remain fixed relative to a point on the Earth's surface. This is achieved when the satellite's orbital period matches the Earth's rotational period, which is 24 hours.

2. **Use the Formula for Orbital Period**:
The relationship between the orbital period (T) of a satellite and its radius (r) from the center of the Earth is given by:
\[ T^2 = \frac{4\pi^2}{GM} r^3 \]
where \( G \) is the universal gravitational constant and \( M \) is the mass of the Earth.

3. **Convert Time Period to Seconds**:
Since the time period \( T \) is given in hours, convert it to seconds:
\[ T = 24 \text{ hours} = 24 \times 60 \times 60 = 86400 \text{ seconds} \]

4. **Rearrange the Formula**:
We need to express the radius \( r \) in terms of \( T \):
\[ r^3 = \frac{GMT^2}{4\pi^2} \]

5. **Substitute Known Values**:
- \( G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \)
- \( M = 6 \times 10^{24} \, \text{kg} \)
- \( T = 86400 \, \text{s} \)

Plugging these values into the equation:
\[ r^3 = \frac{(6.67 \times 10^{-11})(6 \times 10^{24})(86400^2)}{4\pi^2} \]

6. **Calculate \( r \)**:
First, calculate \( T^2 \):
\[ T^2 = 86400^2 = 7.46496 \times 10^9 \, \text{s}^2 \]
Now substitute \( T^2 \) into the equation:
\[ r^3 = \frac{(6.67 \times 10^{-11})(6 \times 10^{24})(7.46496 \times 10^9)}{4 \times (3.14)^2} \]

After calculating the right-hand side, we find \( r \).

7. **Find the Height \( h \)**:
The radius \( r \) is the distance from the center of the Earth to the satellite. To find the height \( h \) above the Earth's surface, subtract the Earth's radius \( R \) (approximately \( 6.4 \times 10^6 \, \text{m} \)):
\[ h = r - R \]

8. **Final Calculation**:
After performing the calculations, you will find that:
\[ h \approx 36000 \, \text{km} \]

### Conclusion:
The height at which communication satellites are parked above the surface of the Earth is approximately **36000 km**.