If g is the acceleration due to gravity on the surface of the earth , its value at a height equal to double the radius of the earth is

Options

A. g
B. `g/2`
C. `g/3`
D. `g/9`

Answer

To solve the problem of finding the value of acceleration due to gravity (g') at a height equal to double the radius of the Earth (2R), we can use the formula for the variation of gravity with height.

### Step-by-Step Solution:

1. **Understand the Problem**:
We need to find the value of acceleration due to gravity at a height (h) equal to double the radius of the Earth (R). Thus, h = 2R.

2. **Use the Formula**:
The formula for the acceleration due to gravity at a height h is given by:
\[ g' = \frac{g}{(1 + \frac{h}{R})^2} \]
where g is the acceleration due to gravity at the surface of the Earth.

3. **Substitute the Value of h**:
Substitute h = 2R into the formula:
\[ g' = \frac{g}{(1 + \frac{2R}{R})^2} \]

4. **Simplify the Expression**:
Simplifying the term inside the parentheses:
\[ g' = \frac{g}{(1 + 2)^2} = \frac{g}{3^2} = \frac{g}{9} \]

5. **Conclusion**:
Therefore, the value of acceleration due to gravity at a height equal to double the radius of the Earth is:
\[ g' = \frac{g}{9} \]

### Final Answer:
The value of g at a height equal to double the radius of the Earth is \( \frac{g}{9} \).

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