A satellite appears to be at rest when seen from. the equator of the earth. The height of the satellite from the surface of the earth is

Options

A. 36000km
B. 6400km
C. 40000km
D. 1000km

Answer

To solve the problem of finding the height of a satellite that appears to be at rest when viewed from the equator of the Earth, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Geostationary Satellites**:
A satellite that appears to be at rest relative to the Earth's surface must have a time period of revolution equal to the Earth's rotation period, which is 24 hours.

2. **Convert Time Period to Seconds**:
The time period \( T \) of the satellite is:
\[ T = 24 \text{ hours} = 24 \times 3600 \text{ seconds} = 86400 \text{ seconds} \]

3. **Use the Formula for the Time Period of a Satellite**:
The time period \( T \) of a satellite in circular orbit is given by the formula:
\[ T = 2\pi \sqrt{\frac{r^3}{g}} \]
where \( r \) is the distance from the center of the Earth to the satellite, and \( g \) is the acceleration due to gravity (approximately \( 9.8 \, \text{m/s}^2 \)).

4. **Rearranging the Formula**:
To find \( r \), we can rearrange the formula:
\[ T^2 = 4\pi^2 \frac{r^3}{g} \]
\[ r^3 = \frac{g T^2}{4\pi^2} \]

5. **Substituting Known Values**:
Substitute \( T = 86400 \, \text{s} \) and \( g = 9.8 \, \text{m/s}^2 \):
\[ r^3 = \frac{9.8 \times (86400)^2}{4\pi^2} \]

6. **Calculating \( r^3 \)**:
First, calculate \( (86400)^2 \):
\[ (86400)^2 = 7.46496 \times 10^9 \text{ s}^2 \]
Then calculate \( r^3 \):
\[ r^3 = \frac{9.8 \times 7.46496 \times 10^9}{4 \times \pi^2} \]
\[ r^3 \approx 7.5 \times 10^{21} \text{ m}^3 \]

7. **Finding \( r \)**:
Now take the cube root to find \( r \):
\[ r \approx (7.5 \times 10^{21})^{1/3} \approx 4.224 \times 10^7 \text{ m} \approx 42240 \text{ km} \]

8. **Calculate the Height Above Earth's Surface**:
The radius of the Earth \( R \) is approximately \( 6400 \text{ km} \). The height \( h \) of the satellite above the Earth's surface is:
\[ h = r - R = 42240 \text{ km} - 6400 \text{ km} = 35840 \text{ km} \]

### Final Answer:
The height of the satellite from the surface of the Earth is approximately **35840 km**.