The earth is a solid sphere of radius 6400 km, the value of acceleration due to gravity at a height 800 km above the surface of the earth is

Options

A. `5.35m//s^(2)`
B. `6.35m//s^(2)`
C. `7.35m//s^(2)`
D. `8.35m//s^(2)`

Explanation

To find the acceleration due to gravity at a height of 800 km above the Earth's surface, we can use the formula for gravitational acceleration at a height \( h \) above the Earth's surface:

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

where:
- \( g' \) is the acceleration due to gravity at height \( h \),
- \( g \) is the acceleration due to gravity at the Earth's surface (approximately \( 9.8 \, \text{m/s}^2 \)),
- \( R \) is the radius of the Earth,
- \( h \) is the height above the Earth's surface.

### Step 1: Convert the radius and height to meters
Given:
- Radius of the Earth, \( R = 6400 \, \text{km} = 6400 \times 10^3 \, \text{m} \)
- Height above the surface, \( h = 800 \, \text{km} = 800 \times 10^3 \, \text{m} \)

### Step 2: Substitute the values into the formula
Now we substitute the values into the formula:

\[ g' = 9.8 \left( \frac{6400 \times 10^3}{6400 \times 10^3 + 800 \times 10^3} \right)^2 \]

### Step 3: Simplify the expression
Now simplify the expression inside the parentheses:

\[ g' = 9.8 \left( \frac{6400 \times 10^3}{7200 \times 10^3} \right)^2 \]

\[ g' = 9.8 \left( \frac{6400}{7200} \right)^2 \]

### Step 4: Calculate the fraction
Now calculate \( \frac{6400}{7200} \):

\[ \frac{6400}{7200} = \frac{64}{72} = \frac{8}{9} \]

### Step 5: Substitute back into the equation
Now substitute back into the equation:

\[ g' = 9.8 \left( \frac{8}{9} \right)^2 \]

### Step 6: Calculate \( \left( \frac{8}{9} \right)^2 \)
Calculate \( \left( \frac{8}{9} \right)^2 \):

\[ \left( \frac{8}{9} \right)^2 = \frac{64}{81} \]

### Step 7: Final calculation
Now substitute this back into the equation for \( g' \):

\[ g' = 9.8 \times \frac{64}{81} \]

Calculating this gives:

\[ g' = \frac{9.8 \times 64}{81} \approx \frac{627.2}{81} \approx 7.75 \, \text{m/s}^2 \]

### Conclusion
Thus, the acceleration due to gravity at a height of 800 km above the Earth's surface is approximately \( 7.75 \, \text{m/s}^2 \).