At which height above earth's surface is the value of 'g' same as in a 100 km dip mine ?

Answer

To solve the problem of finding the height above the Earth's surface where the value of 'g' is the same as in a 100 km deep mine, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We need to find the height \( H \) above the Earth's surface where the acceleration due to gravity \( g \) is equal to the acceleration due to gravity at a depth of 100 km in a mine.

2. **Formula for Gravity at Depth**: The acceleration due to gravity at a depth \( d \) inside the Earth is given by:
\[ g_d = g \left(1 - \frac{d}{R}\right) \]
where \( g \) is the acceleration due to gravity at the surface, \( d \) is the depth, and \( R \) is the radius of the Earth.

3. **Formula for Gravity at Height**: The acceleration due to gravity at a height \( h \) above the Earth's surface is given by:
\[ g_h = g \left(1 - \frac{2h}{R}\right) \]

4. **Setting the Two Equations Equal**: According to the problem, we want to find \( H \) such that:
\[ g_h = g_d \]
Substituting the formulas we have:
\[ g \left(1 - \frac{2H}{R}\right) = g \left(1 - \frac{d}{R}\right) \]

5. **Canceling \( g \)**: Since \( g \) is the same on both sides, we can cancel it out:
\[ 1 - \frac{2H}{R} = 1 - \frac{d}{R} \]

6. **Simplifying the Equation**: Now, we can simplify the equation:
\[ -\frac{2H}{R} = -\frac{d}{R} \]
This leads to:
\[ 2H = d \]

7. **Substituting the Value of Depth**: We know that \( d = 100 \) km:
\[ 2H = 100 \text{ km} \]

8. **Solving for \( H \)**: Dividing both sides by 2 gives:
\[ H = \frac{100 \text{ km}}{2} = 50 \text{ km} \]

9. **Final Answer**: Therefore, the height \( H \) above the Earth's surface where the value of \( g \) is the same as in a 100 km deep mine is:
\[ H = 50 \text{ km} \]