At what height from the ground will the value of ‘ g ’ be the same as that in 10 km deep mine below the surface of earth

Options

A. `20` km
B. `10` km
C. `15` km
D. `5` km

Answer

To solve the problem of finding the height from the ground where the value of 'g' is the same as that in a 10 km deep mine below the surface of the Earth, we can follow these steps:

### Step 1: Understand the relationship of 'g' at different depths and heights
The acceleration due to gravity 'g' changes with depth and height relative to the Earth's surface. The formula for 'g' at a height 'h' above the Earth's surface is given by:

\[ g_h = g \left( \frac{R}{R+h} \right)^2 \]

And the formula for 'g' at a depth 'd' below the Earth's surface is:

\[ g_d = g \left( 1 - \frac{d}{R} \right) \]

where \( R \) is the radius of the Earth.

### Step 2: Set the equations equal
We need to find the height 'h' such that the value of 'g' at that height is equal to the value of 'g' at a depth of 10 km (which is 10,000 m). Thus, we set the two equations equal:

\[ g \left( \frac{R}{R+h} \right)^2 = g \left( 1 - \frac{10000}{R} \right) \]

### Step 3: Cancel 'g' from both sides
Since 'g' is common on both sides, we can cancel it out:

\[ \left( \frac{R}{R+h} \right)^2 = 1 - \frac{10000}{R} \]

### Step 4: Cross-multiply and simplify
Cross-multiplying gives:

\[ R^2 = (R+h)^2 \left( 1 - \frac{10000}{R} \right) \]

Expanding the right side:

\[ R^2 = (R^2 + 2Rh + h^2) \left( 1 - \frac{10000}{R} \right) \]

### Step 5: Distributing the terms
Distributing gives:

\[ R^2 = R^2 + 2Rh + h^2 - 10000R - \frac{20000h}{R} - \frac{10000h^2}{R} \]

### Step 6: Rearranging the equation
Rearranging the equation to isolate terms involving 'h':

\[ 0 = 2Rh + h^2 - 10000R - \frac{20000h}{R} - \frac{10000h^2}{R} \]

### Step 7: Solve for 'h'
To solve for 'h', we can substitute \( h = 2h \) from the earlier relationship derived from the equations. We know that \( h1 = 10 \) km, hence:

\[ h = \frac{h1}{2} = \frac{10000}{2} = 5000 \text{ m} = 5 \text{ km} \]

### Conclusion
Thus, the height from the ground where the value of 'g' is the same as that in a 10 km deep mine is:

\[ \boxed{5 \text{ km}} \]