Acceleration due to gravity as same at height h from surface and at depth d from surface, then find value of d

Options

A. Option1 `d = h`
B. Option2 `d = 2h`
C. Option3 `d = 3h//2`
D. Option4 `d = h//2`

Answer

To solve the problem, we need to find the value of depth \( d \) at which the acceleration due to gravity is the same as that at a height \( h \) above the Earth's surface.

### Step-by-Step Solution:

1. **Understand the formulas for acceleration due to gravity**:
- The formula for acceleration due to gravity at a height \( h \) from the Earth's surface is given by:
\[ g_h = g_0 \left(1 - \frac{2h}{R}\right) \]
where \( g_0 \) is the acceleration due to gravity at the surface and \( R \) is the radius of the Earth.

- The formula for acceleration due to gravity at a depth \( d \) from the Earth's surface is given by:
\[ g_d = g_0 \left(1 - \frac{d}{R}\right) \]

2. **Set the two expressions equal**:
Since the problem states that the acceleration due to gravity at height \( h \) is equal to that at depth \( d \), we can set the two equations equal to each other:
\[ g_0 \left(1 - \frac{2h}{R}\right) = g_0 \left(1 - \frac{d}{R}\right) \]

3. **Cancel \( g_0 \)**:
Assuming \( g_0 \) is not zero, we can divide both sides by \( g_0 \):
\[ 1 - \frac{2h}{R} = 1 - \frac{d}{R} \]

4. **Simplify the equation**:
Now, we can simplify the equation:
\[ -\frac{2h}{R} = -\frac{d}{R} \]
This leads to:
\[ \frac{2h}{R} = \frac{d}{R} \]

5. **Multiply both sides by \( R \)**:
By multiplying both sides by \( R \), we eliminate \( R \):
\[ 2h = d \]

6. **Conclusion**:
Therefore, the value of \( d \) is:
\[ d = 2h \]

### Final Answer:
The value of \( d \) is \( 2h \).

---