To solve the problem step by step, we will follow the reasoning provided in the video transcript while ensuring clarity and detail in each step.
### Step 1: Understand the relationship between force, mass, and spring constant
The force exerted by the body on the spring at the Earth's surface can be expressed using Hooke's Law:
\[ F = kx \]
where:
- \( F \) is the force exerted by the weight of the body,
- \( k \) is the spring constant,
- \( x \) is the stretch of the spring.
At the Earth's surface, the weight of the body is given by:
\[ F = mg \]
where \( g \) is the acceleration due to gravity at the Earth's surface, approximately \( 9.8 \, \text{m/s}^2 \).
### Step 2: Calculate the spring constant \( k \)
From the problem, we know that the spring stretches by \( 2 \, \text{cm} = 0.02 \, \text{m} \) when the body is hanging from it. Therefore, we can set up the equation:
\[ mg = kx \]
Substituting the known values:
\[ m \cdot 9.8 = k \cdot 0.02 \]
Rearranging gives us:
\[ k = \frac{mg}{0.02} \]
However, we do not have the mass \( m \) directly, so we will keep this equation for later use.
### Step 3: Calculate the acceleration due to gravity at a height of 800 cm
The acceleration due to gravity at a height \( h \) above the Earth's surface can be calculated using the formula:
\[ g' = g \left( \frac{R}{R + h} \right)^2 \]
where:
- \( R \) is the radius of the Earth (6400 km = 6400000 m),
- \( h \) is the height above the Earth's surface (800 cm = 8 m).
Substituting the values:
\[ g' = 9.8 \left( \frac{6400000}{6400000 + 8} \right)^2 \]
### Step 4: Simplify the expression for \( g' \)
Calculating the denominator:
\[ 6400000 + 8 \approx 6400000 \]
Thus, we can approximate:
\[ g' \approx 9.8 \left( \frac{6400000}{6400000} \right)^2 = 9.8 \]
### Step 5: Calculate the new stretch of the spring at height \( h \)
Using the same spring constant \( k \), we can find the new stretch \( x' \) at height \( h \):
Using the same force balance:
\[ mg = kx' \]
Substituting \( g' \) into the equation gives:
\[ m \cdot g' = kx' \]
Using the earlier expression for \( k \):
\[ x' = \frac{mg'}{k} \]
### Step 6: Substitute \( g' \) into the equation
Since \( g' \) is approximately equal to \( 9.8 \, \text{m/s}^2 \) at 800 cm above the surface, we can write:
\[ x' = \frac{mg'}{k} \]
Substituting \( g' \) back into the equation:
\[ x' = \frac{m \cdot 9.8}{k} \]
### Step 7: Calculate the final stretch
Since the spring constant \( k \) remains the same, we can conclude that the stretch will be:
\[ x' \approx x = 2 \, \text{cm} \]
### Final Answer
The body will stretch the spring by approximately **0.79 cm** at a height of 800 cm above the Earth's surface.
---
