Change in acceleration due to gravity is same upto a height h from the surface of the earth and below the surface at a depth x, then relation between x and h is (h and `x lt lt lt R`)

Options

A. `x=h`
B. `x=2h`
C. `x=h/2`
D. `x=h^(2)`

Explanation

To solve the problem, we need to establish the relationship between the height \( h \) above the Earth's surface and the depth \( x \) below the Earth's surface, given that the change in acceleration due to gravity is the same in both cases.

### Step-by-Step Solution:

1. **Understanding the Acceleration due to Gravity at Height \( h \)**:
- The acceleration due to gravity at a height \( h \) from the surface of the Earth is given by the formula:
\[ g_h = g \left(1 - \frac{2h}{R}\right) \]
where \( g \) is the acceleration due to gravity at the surface of the Earth, and \( R \) is the radius of the Earth.

2. **Understanding the Acceleration due to Gravity at Depth \( x \)**:
- The acceleration due to gravity at a depth \( x \) below the Earth's surface is given by the formula:
\[ g_x = g \left(1 - \frac{x}{R}\right) \]

3. **Setting the Two Expressions Equal**:
- According to the problem, the acceleration due to gravity at height \( h \) is equal to the acceleration due to gravity at depth \( x \):
\[ g \left(1 - \frac{2h}{R}\right) = g \left(1 - \frac{x}{R}\right) \]

4. **Canceling \( g \)**:
- Since \( g \) is common on both sides, we can cancel it out:
\[ 1 - \frac{2h}{R} = 1 - \frac{x}{R} \]

5. **Simplifying the Equation**:
- By simplifying the equation, we get:
\[ -\frac{2h}{R} = -\frac{x}{R} \]
- Removing the negative signs and multiplying through by \( R \):
\[ 2h = x \]

6. **Final Relation**:
- Therefore, the relationship between \( x \) and \( h \) is:
\[ x = 2h \]

### Conclusion:
The relation between the depth \( x \) below the surface of the Earth and the height \( h \) above the surface of the Earth is:
\[ x = 2h \]