Calculate the ratio of weights of a body when it is taken to 80 km above the surface of earth and when taken to 80 km below the surface of earth.

Answer

To calculate the ratio of weights of a body when it is taken to 80 km above the surface of the Earth and when taken to 80 km below the surface of the Earth, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Weight and Acceleration due to Gravity**:
- The weight of a body (W) is given by the formula:
\[ W = m \cdot g \]
where \( m \) is the mass of the body and \( g \) is the acceleration due to gravity.

2. **Acceleration due to Gravity Above the Surface**:
- When the body is taken to a height \( h \) above the surface of the Earth, the acceleration due to gravity \( g_1 \) at that height is given by:
\[ g_1 = g \left(1 - \frac{2h}{R}\right) \]
where \( R \) is the radius of the Earth (approximately 6400 km) and \( h = 80 \) km.

3. **Substituting Values for \( g_1 \)**:
- Plugging in the values:
\[ g_1 = g \left(1 - \frac{2 \times 80}{6400}\right) = g \left(1 - \frac{160}{6400}\right) = g \left(1 - 0.025\right) = g \times 0.975 \]

4. **Acceleration due to Gravity Below the Surface**:
- When the body is taken to a depth \( d \) below the surface of the Earth, the acceleration due to gravity \( g_2 \) at that depth is given by:
\[ g_2 = g \left(1 - \frac{d}{R}\right) \]
where \( d = 80 \) km.

5. **Substituting Values for \( g_2 \)**:
- Plugging in the values:
\[ g_2 = g \left(1 - \frac{80}{6400}\right) = g \left(1 - 0.0125\right) = g \times 0.9875 \]

6. **Calculating the Ratio of Weights**:
- The ratio of the weights \( \frac{W_1}{W_2} \) when the body is at height and depth can be expressed as:
\[ \frac{W_1}{W_2} = \frac{m \cdot g_1}{m \cdot g_2} = \frac{g_1}{g_2} \]
- Substituting the expressions for \( g_1 \) and \( g_2 \):
\[ \frac{W_1}{W_2} = \frac{0.975g}{0.9875g} = \frac{0.975}{0.9875} \]

7. **Calculating the Final Ratio**:
- Performing the division:
\[ \frac{W_1}{W_2} = \frac{0.975}{0.9875} \approx 0.987 \]

### Final Answer:
The ratio of the weights of the body when taken to 80 km above the surface of the Earth and when taken to 80 km below the surface of the Earth is approximately:
\[ \frac{W_1}{W_2} \approx 0.987 \]