Compare the weight of a body 100 km above and 100 km below the surface of the earth . Radius of the earth = 6400 km .

Answer

To compare the weight of a body 100 km above and 100 km below the surface of the Earth, we can use the formula for gravitational acceleration at different distances from the center of the Earth.

### Step 1: Determine the parameters
- Radius of the Earth (R) = 6400 km
- Height above the surface (h) = 100 km
- Depth below the surface (h) = 100 km

### Step 2: Calculate the gravitational acceleration above the surface
When a body is at a height \( h \) above the surface of the Earth, the gravitational acceleration \( g' \) can be calculated using the formula:
\[ g' = \frac{GM}{(R + h)^2} \]
where \( G \) is the universal gravitational constant and \( M \) is the mass of the Earth.

Substituting the values:
\[ g' = \frac{GM}{(6400 + 100)^2} = \frac{GM}{(6500)^2} \]

### Step 3: Calculate the gravitational acceleration below the surface
When a body is at a depth \( h \) below the surface of the Earth, the gravitational acceleration \( g'' \) can be calculated using the formula:
\[ g'' = \frac{GM}{(R - h)^2} \]
Substituting the values:
\[ g'' = \frac{GM}{(6400 - 100)^2} = \frac{GM}{(6300)^2} \]

### Step 4: Compare the two gravitational accelerations
To compare \( g' \) and \( g'' \), we can take the ratio:
\[ \frac{g'}{g''} = \frac{(R - h)^2}{(R + h)^2} = \frac{(6400 - 100)^2}{(6400 + 100)^2} = \frac{(6300)^2}{(6500)^2} \]

### Step 5: Simplify the ratio
Calculating the ratio:
\[ \frac{g'}{g''} = \frac{6300^2}{6500^2} = \left(\frac{6300}{6500}\right)^2 = \left(\frac{63}{65}\right)^2 \]

### Step 6: Relate the weights
Since weight \( W \) is proportional to gravitational acceleration \( g \):
\[ \frac{W'}{W''} = \frac{g'}{g''} = \left(\frac{63}{65}\right)^2 \]

### Conclusion
The weight of the body 100 km above the surface of the Earth is less than the weight of the body 100 km below the surface of the Earth by the factor of \( \left(\frac{63}{65}\right)^2 \).