At a height of 10 km above the surface of earth, the value of acceleration due to gravity is the same as that of a particular depth below the surface of earth. Assuming uniform mass density of the earth, the depth is,

Options

A. 1 km
B. 5 km
C. 10 km
D. 20 km

Explanation

To solve the problem, we need to find the depth below the Earth's surface where the acceleration due to gravity is the same as that at a height of 10 km above the Earth's surface. We will use the formulas for gravitational acceleration at a height and at a depth.

### Step-by-Step Solution:

1. **Identify the given values:**
- Height above the Earth's surface (h) = 10 km = 10,000 m
- Acceleration due to gravity at the surface (g₀) = 9.8 m/s²
- Radius of the Earth (R) = approximately 6400 km = 6,400,000 m

2. **Formula for acceleration due to gravity at a height (h):**
The formula for gravitational acceleration at a height above the Earth's surface is given by:
\[ g_h = g₀ - \frac{2gh}{R} \]
However, for small heights compared to the Earth's radius, we can approximate it as:
\[ g_h = g₀ \left(1 - \frac{h}{R}\right) \]

3. **Formula for acceleration due to gravity at a depth (d):**
The formula for gravitational acceleration at a depth below the Earth's surface is given by:
\[ g_d = g₀ \left(1 - \frac{d}{R}\right) \]

4. **Set the two expressions for gravity equal:**
Since we want to find the depth (d) where the acceleration due to gravity is the same as that at a height of 10 km, we set the two equations equal:
\[ g₀ \left(1 - \frac{h}{R}\right) = g₀ \left(1 - \frac{d}{R}\right) \]

5. **Cancel out g₀ from both sides:**
Since \(g₀\) is common in both terms, we can cancel it out:
\[ 1 - \frac{h}{R} = 1 - \frac{d}{R} \]

6. **Rearranging the equation:**
This simplifies to:
\[ \frac{h}{R} = \frac{d}{R} \]

7. **Substituting the height (h):**
We know \(h = 10,000\) m, so:
\[ d = 2h \]

8. **Calculating the depth (d):**
Substituting the value of \(h\):
\[ d = 2 \times 10,000 \text{ m} = 20,000 \text{ m} = 20 \text{ km} \]

9. **Final Answer:**
The depth below the Earth's surface where the acceleration due to gravity is the same as that at a height of 10 km is **20 km**.