At a height equal to earth's radius, above the earth's surface, the acceleration due to gravity is

Options

A. g
B. `(g )/(2)`
C. `(g )/(4)`
D. `g/8`

Explanation

To find the acceleration due to gravity at a height equal to the Earth's radius above the Earth's surface, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We need to find the acceleration due to gravity (g') at a height (h) equal to the Earth's radius (R) above the Earth's surface.

2. **Identify the Total Distance from the Center of the Earth**:
- The distance from the center of the Earth to the height where the object is located is:
\[ \text{Total Distance} = R + h = R + R = 2R \]

3. **Using the Formula for Acceleration due to Gravity**:
- The formula for acceleration due to gravity at a distance (d) from the center of the Earth is given by:
\[ g' = \frac{GM}{d^2} \]
- Here, \( G \) is the gravitational constant, and \( M \) is the mass of the Earth.

4. **Substituting the Distance**:
- Since we found that the distance from the center of the Earth to the object is \( 2R \), we substitute this into the formula:
\[ g' = \frac{GM}{(2R)^2} = \frac{GM}{4R^2} \]

5. **Relating to the Acceleration due to Gravity on the Surface**:
- The acceleration due to gravity at the surface of the Earth (g) is given by:
\[ g = \frac{GM}{R^2} \]
- We can express \( g' \) in terms of \( g \):
\[ g' = \frac{GM}{4R^2} = \frac{1}{4} \cdot \frac{GM}{R^2} = \frac{g}{4} \]

6. **Final Answer**:
- Therefore, the acceleration due to gravity at a height equal to the Earth's radius above the Earth's surface is:
\[ g' = \frac{g}{4} \]

### Conclusion:
The acceleration due to gravity at a height equal to the Earth's radius above the Earth's surface is \( \frac{g}{4} \).

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