A body hanging from a massless spring stretches it by 3 cm on earth's surface. At a place 800 km above the earth's surface, the same body will stretch the spring by
(Radius of Earth = 6400 km)

Options

A. `((34)/(27))cm`
B. `((64)/(27))cm`
C. `((27)/(64))cm`
D. `((27)/(34))cm`

Answer

To solve the problem, we need to determine how much a spring stretches when a body is placed on it at a height of 800 km above the Earth's surface.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- A body stretches a spring by 3 cm on the Earth's surface.
- We need to find the stretch of the spring (let's call it \( x' \)) when the body is at a height of 800 km above the Earth's surface.

2. **Using the Formula for Gravitational Acceleration**:
- The acceleration due to gravity \( g \) at a distance \( r \) from the center of the Earth is given by:
\[ g = \frac{GM}{r^2} \]
- At a height \( h \) above the Earth's surface, the distance from the center of the Earth becomes \( R + h \), where \( R \) is the radius of the Earth.

3. **Finding the New Gravitational Acceleration**:
- The gravitational acceleration at height \( h \) (800 km) is:
\[ g' = \frac{GM}{(R + h)^2} \]
- We can express \( g' \) in terms of \( g \):
\[ g' = g \left( \frac{R^2}{(R + h)^2} \right) \]

4. **Calculating the Ratio of Gravitational Accelerations**:
- The ratio of the gravitational accelerations at height \( h \) and at the surface is:
\[ \frac{g'}{g} = \left( \frac{R}{R + h} \right)^2 \]
- Substituting \( R = 6400 \) km and \( h = 800 \) km:
\[ \frac{g'}{g} = \left( \frac{6400}{6400 + 800} \right)^2 = \left( \frac{6400}{7200} \right)^2 = \left( \frac{64}{72} \right)^2 = \left( \frac{8}{9} \right)^2 = \frac{64}{81} \]

5. **Relating the Stretch of the Spring**:
- The stretch of the spring is proportional to the gravitational acceleration:
\[ \frac{x'}{x} = \frac{g'}{g} \]
- Therefore:
\[ x' = x \cdot \frac{g'}{g} = 3 \, \text{cm} \cdot \frac{64}{81} \]

6. **Calculating \( x' \)**:
- Now, substituting the value of \( x \):
\[ x' = 3 \cdot \frac{64}{81} = \frac{192}{81} \, \text{cm} = \frac{64}{27} \, \text{cm} \]

### Final Answer:
The stretch of the spring at a height of 800 km above the Earth's surface is \( \frac{64}{27} \) cm.