If the change in the value of g at a height h above the surface of the earth is the same as at a depth x below it, then (both x and h being much smaller than the radius of the earth)

Options

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

Answer

To solve the problem, we need to establish the relationship between the change in the value of gravitational acceleration (g) at a height \( h \) above the Earth's surface and at a depth \( x \) below it. We will use the formulas for gravitational acceleration at height and depth.

### Step-by-Step Solution:

1. **Understanding the Problem**: We need to find the relationship between height \( h \) and depth \( x \) such that the change in gravitational acceleration at height \( h \) is equal to the change at depth \( x \).

2. **Gravitational Acceleration at Height**: The formula for gravitational acceleration at a height \( h \) above the Earth's surface is given by:
\[ g_h = g \left(1 - \frac{2h}{R}\right) \]
where \( g \) is the gravitational acceleration at the surface and \( R \) is the radius of the Earth.

3. **Gravitational Acceleration at Depth**: The formula for gravitational acceleration at a depth \( x \) below the Earth's surface is given by:
\[ g_x = g \left(1 - \frac{x}{R}\right) \]

4. **Setting the Changes Equal**: According to the problem, the change in gravitational acceleration at height \( h \) is equal to the change at depth \( x \):
\[ g - g_h = g - g_x \]
This simplifies to:
\[ g_h = g_x \]

5. **Substituting the Formulas**: Now, we substitute the expressions for \( g_h \) and \( g_x \):
\[ g \left(1 - \frac{2h}{R}\right) = g \left(1 - \frac{x}{R}\right) \]

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

7. **Rearranging the Equation**: By rearranging the equation, we get:
\[ -\frac{2h}{R} = -\frac{x}{R} \]
This simplifies to:
\[ 2h = x \]

8. **Final Relationship**: Therefore, we find the relationship between depth \( x \) and height \( h \):
\[ x = 2h \]

### Conclusion:
The relationship between the height \( h \) above the Earth's surface and the depth \( x \) below it is given by:
\[ x = 2h \]