At which height above earth's surface is the value of 'g' same as in a 100 km depp 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 will use the formulas for gravitational acceleration at a height above the Earth's surface and at a depth below the Earth's surface.

### Step-by-Step Solution:

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

2. **Formulas for Gravitational Acceleration**:
- The formula for gravitational acceleration at a height (H) above the Earth's surface is given by:
\[ g_H = g_0 \left(1 - \frac{2H}{R}\right) \]
- The formula for gravitational acceleration at a depth (d) inside the Earth is given by:
\[ g_d = g_0 \left(1 - \frac{d}{R}\right) \]
where \( g_0 \) is the acceleration due to gravity at the surface of the Earth, and \( R \) is the radius of the Earth.

3. **Setting the Equations Equal**: We want to find the height (H) such that:
\[ g_H = g_d \]
Thus, we can set the two equations equal to each other:
\[ g_0 \left(1 - \frac{2H}{R}\right) = g_0 \left(1 - \frac{d}{R}\right) \]

4. **Canceling \( g_0 \)**: Since \( g_0 \) is common on both sides, we can cancel it out:
\[ 1 - \frac{2H}{R} = 1 - \frac{d}{R} \]

5. **Simplifying the Equation**: This simplifies to:
\[ -\frac{2H}{R} = -\frac{d}{R} \]
Multiplying through by -1 gives:
\[ \frac{2H}{R} = \frac{d}{R} \]

6. **Cross Multiplying**: We can cross-multiply to eliminate \( R \):
\[ 2H = d \]

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

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

### Final Answer:
The height above the Earth's surface where the value of 'g' is the same as in a 100 km deep mine is **50 km**.

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