Find the value of acceleration due to gravity in a mine at a depth of 80 km from the surface of the earth . Radius of the earth = 6400 km .

Answer

To find the value of acceleration due to gravity at a depth of 80 km from the surface of the Earth, we can use the formula for gravitational acceleration at a depth \(d\):

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

where:
- \(g'\) is the acceleration due to gravity at depth \(d\),
- \(g\) is the acceleration due to gravity at the surface of the Earth (approximately \(9.8 \, \text{m/s}^2\) or \(10 \, \text{m/s}^2\)),
- \(d\) is the depth (80 km in this case),
- \(R\) is the radius of the Earth (6400 km).

### Step-by-Step Solution:

1. **Identify the values**:
- Depth \(d = 80 \, \text{km}\)
- Radius of the Earth \(R = 6400 \, \text{km}\)
- Acceleration due to gravity at the surface \(g \approx 9.8 \, \text{m/s}^2\) (or \(10 \, \text{m/s}^2\))

2. **Convert the depth and radius to the same units**:
- Since both values are in kilometers, we can use them directly.

3. **Substitute the values into the formula**:
\[ g' = g \left(1 - \frac{d}{R}\right) \]
\[ g' = g \left(1 - \frac{80}{6400}\right) \]

4. **Calculate the fraction**:
\[ \frac{80}{6400} = \frac{1}{80} \]

5. **Substitute back into the equation**:
\[ g' = g \left(1 - \frac{1}{80}\right) \]

6. **Simplify the expression**:
\[ g' = g \left(\frac{80 - 1}{80}\right) = g \left(\frac{79}{80}\right) \]

7. **Final expression for \(g'\)**:
\[ g' = \frac{79}{80} g \]

8. **If we take \(g \approx 9.8 \, \text{m/s}^2\)**:
\[ g' = \frac{79}{80} \times 9.8 \approx 9.79 \, \text{m/s}^2 \]

### Conclusion:
The value of acceleration due to gravity at a depth of 80 km is approximately \(9.79 \, \text{m/s}^2\).