How much below the surface of the earth does the acceleration due to gravity becomes 70% of its value at the surface of earth ? (Take `R_(e)=6400` km)

Explanation

To solve the problem of how much below the surface of the Earth the acceleration due to gravity becomes 70% of its value at the surface, we can follow these steps:

### Step-by-Step Solution

1. **Understand the formula for acceleration due to gravity below the surface of the Earth**:
The acceleration due to gravity at a depth \( d \) below the surface is given by:
\[ g' = g \left(1 - \frac{d}{R}\right) \]
where \( g \) is the acceleration due to gravity at the surface, \( d \) is the depth, and \( R \) is the radius of the Earth.

2. **Set up the equation for 70% of surface gravity**:
We need to find the depth \( d \) where \( g' = 0.7g \). Therefore, we can substitute \( g' \) in the formula:
\[ 0.7g = g \left(1 - \frac{d}{R}\right) \]

3. **Cancel \( g \) from both sides**:
Since \( g \) is not zero, we can divide both sides by \( g \):
\[ 0.7 = 1 - \frac{d}{R} \]

4. **Rearrange the equation**:
Rearranging gives:
\[ \frac{d}{R} = 1 - 0.7 \]
\[ \frac{d}{R} = 0.3 \]

5. **Solve for \( d \)**:
Now, multiply both sides by \( R \):
\[ d = 0.3R \]

6. **Substitute the value of \( R \)**:
Given \( R = 6400 \) km, we substitute:
\[ d = 0.3 \times 6400 \text{ km} \]
\[ d = 1920 \text{ km} \]

### Final Answer
The depth below the surface of the Earth where the acceleration due to gravity becomes 70% of its value at the surface is **1920 km**.