Calculate the depth below the surface of the earth where acceleration due to gravity becomes half of its value at the surface of the earth . Radius of the earth = 6400 km.

Answer

To find the depth below the surface of the Earth where the acceleration due to gravity becomes half of its value at the surface, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We need to find the depth \( d \) below the Earth's surface where the acceleration due to gravity \( g' \) is half of the surface gravity \( g \).

2. **Know the Values**: The radius of the Earth \( R \) is given as 6400 km. The acceleration due to gravity at the surface is denoted as \( g \).

3. **Set Up the Equation**: The relationship between the acceleration due to gravity at a depth \( d \) and at the surface is given by:
\[ g' = g \left(1 - \frac{d}{R}\right) \]
Since we want \( g' \) to be half of \( g \), we can write:
\[ g' = \frac{g}{2} \]

4. **Substituting into the Equation**: Substitute \( g' \) into the equation:
\[ \frac{g}{2} = g \left(1 - \frac{d}{6400}\right) \]

5. **Cancel \( g \)**: Since \( g \) is not zero, we can divide both sides by \( g \):
\[ \frac{1}{2} = 1 - \frac{d}{6400} \]

6. **Rearranging the Equation**: Rearranging gives:
\[ \frac{d}{6400} = 1 - \frac{1}{2} \]
\[ \frac{d}{6400} = \frac{1}{2} \]

7. **Solving for \( d \)**: Multiply both sides by 6400 to find \( d \):
\[ d = \frac{1}{2} \times 6400 = 3200 \text{ km} \]

### Final Answer:
The depth below the surface of the Earth where the acceleration due to gravity becomes half of its value at the surface is **3200 km**.

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