Find the value of g at a height of 400 km above the surface of the earth . Given radius of the earth , R = 6400 km and value of g at the surface of the earth = `9.8 ms^(-2)`.

Answer

To find the value of \( g \) at a height of 400 km above the surface of the Earth, we can use the formula for gravitational acceleration at a height \( h \) above the Earth's surface:

\[ g' = \frac{g \cdot R^2}{(R + h)^2} \]

where:
- \( g' \) is the gravitational acceleration at height \( h \),
- \( g \) is the gravitational acceleration at the surface of the Earth (given as \( 9.8 \, \text{m/s}^2 \)),
- \( R \) is the radius of the Earth (given as \( 6400 \, \text{km} = 6400 \times 10^3 \, \text{m} \)),
- \( h \) is the height above the Earth's surface (given as \( 400 \, \text{km} = 400 \times 10^3 \, \text{m} \)).

### Step-by-step Solution:

1. **Convert the radius of the Earth and height into meters:**
\[ R = 6400 \, \text{km} = 6400 \times 10^3 \, \text{m} = 6.4 \times 10^6 \, \text{m} \]
\[ h = 400 \, \text{km} = 400 \times 10^3 \, \text{m} = 4.0 \times 10^5 \, \text{m} \]

2. **Calculate \( R + h \):**
\[ R + h = 6.4 \times 10^6 \, \text{m} + 4.0 \times 10^5 \, \text{m} = 6.8 \times 10^6 \, \text{m} \]

3. **Substitute the values into the formula for \( g' \):**
\[ g' = \frac{9.8 \, \text{m/s}^2 \cdot (6.4 \times 10^6 \, \text{m})^2}{(6.8 \times 10^6 \, \text{m})^2} \]

4. **Calculate \( R^2 \) and \( (R + h)^2 \):**
\[ R^2 = (6.4 \times 10^6)^2 = 40.96 \times 10^{12} \, \text{m}^2 \]
\[ (R + h)^2 = (6.8 \times 10^6)^2 = 46.24 \times 10^{12} \, \text{m}^2 \]

5. **Now substitute these values back into the equation:**
\[ g' = \frac{9.8 \cdot 40.96 \times 10^{12}}{46.24 \times 10^{12}} \]

6. **Simplify the expression:**
\[ g' = 9.8 \cdot \frac{40.96}{46.24} \]

7. **Calculate the fraction:**
\[ \frac{40.96}{46.24} \approx 0.884 \]

8. **Finally, calculate \( g' \):**
\[ g' \approx 9.8 \cdot 0.884 \approx 8.675 \, \text{m/s}^2 \]

### Final Result:
The value of \( g \) at a height of 400 km above the surface of the Earth is approximately \( 8.675 \, \text{m/s}^2 \).