Calculate the distance from the surface of the earth at which above the surface, acceleration due to gravity is the same as below the earth

Explanation

To solve the problem of finding the distance from the surface of the Earth at which the acceleration due to gravity is the same as that below the Earth's surface, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We need to find a height \( h \) above the Earth's surface where the acceleration due to gravity \( g_h \) equals the acceleration due to gravity \( g_d \) at a depth \( d \) below the Earth's surface.

2. **Formulas for Acceleration due to Gravity**:
- The acceleration due to gravity at a height \( h \) above the Earth’s surface is given by:
\[ g_h = g \left(1 - \frac{h}{R}\right) \]
- The acceleration due to gravity at a depth \( d \) below the Earth’s surface is given by:
\[ g_d = g \left(1 - \frac{d}{R}\right) \]
- Here, \( R \) is the radius of the Earth, and \( g \) is the acceleration due to gravity at the surface.

3. **Setting the Equations Equal**:
- Since we want \( g_h = g_d \), we set the two equations equal to each other:
\[ g \left(1 - \frac{h}{R}\right) = g \left(1 - \frac{d}{R}\right) \]

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

5. **Relating Depth and Height**:
- Since \( d \) is the depth below the surface, we have \( d = R - h \). Substituting this into the equation gives:
\[ 1 - \frac{h}{R} = 1 - \frac{R - h}{R} \]

6. **Simplifying the Equation**:
- This simplifies to:
\[ 1 - \frac{h}{R} = \frac{h}{R} \]
- Rearranging gives:
\[ 1 = \frac{2h}{R} \]
- Thus, we find:
\[ h = \frac{R}{2} \]

7. **Final Result**:
- Therefore, the distance from the surface of the Earth at which the acceleration due to gravity is the same as that below the Earth is:
\[ h = \frac{R}{2} \]
- If we take \( R \) to be approximately \( 6400 \) km, then:
\[ h \approx 3200 \text{ km} \]