If `g` is same at a height `h` and at a depth `d`, then

Options

A. `R=2d`
B. `d=2h`
C. `h=d`
D. none

Explanation

To solve the problem, we need to analyze the gravitational acceleration \( g \) at a height \( h \) above the Earth's surface and at a depth \( d \) below the Earth's surface. We will derive the relationship between \( h \) and \( d \) based on the given condition that \( g \) is the same at both locations.

### Step-by-Step Solution:

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

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

3. **Setting the Two Expressions Equal**:
Since it is given that \( g_h = g_d \), we can set the two expressions equal to each other:
\[ g \left(1 - \frac{2h}{R}\right) = g \left(1 - \frac{d}{R}\right) \]

4. **Cancelling \( g \)**:
We can cancel \( g \) from both sides (assuming \( g \neq 0 \)):
\[ 1 - \frac{2h}{R} = 1 - \frac{d}{R} \]

5. **Simplifying the Equation**:
Subtracting 1 from both sides gives:
\[ -\frac{2h}{R} = -\frac{d}{R} \]
Multiplying through by -1 results in:
\[ \frac{2h}{R} = \frac{d}{R} \]

6. **Removing \( R \)**:
Since \( R \) is a common factor, we can multiply both sides by \( R \) (assuming \( R \neq 0 \)):
\[ 2h = d \]

7. **Final Relationship**:
Rearranging gives us the final relationship between depth \( d \) and height \( h \):
\[ d = 2h \]

### Conclusion:
The correct answer is that the depth \( d \) is equal to twice the height \( h \), which corresponds to option 2.