The weight of an object on the surface of the Earth is 40 N. Its weight at a height equal to the radius of the Earth is

Options

A. 40 N
B. 20 N
C. 10 N
D. 30 N

Answer

To solve the problem, we need to determine the weight of an object at a height equal to the radius of the Earth, given that its weight on the surface of the Earth is 40 N.

### Step-by-step Solution:

1. **Understand the Weight on the Surface:**
- The weight \( W \) of the object on the surface of the Earth is given as \( W = 40 \, \text{N} \).

2. **Use the Formula for Weight Variation with Height:**
- The weight of an object at a height \( h \) above the Earth's surface can be calculated using the formula:
\[ g' = \frac{g}{(1 + \frac{h}{R})^2} \]
where \( g' \) is the acceleration due to gravity at height \( h \), \( g \) is the acceleration due to gravity on the surface, and \( R \) is the radius of the Earth.

3. **Set the Height Equal to the Radius of the Earth:**
- In this case, we are asked to find the weight at a height equal to the radius of the Earth, so we set \( h = R \).

4. **Substitute \( h = R \) into the Formula:**
- Substitute \( h = R \) into the formula:
\[ g' = \frac{g}{(1 + \frac{R}{R})^2} = \frac{g}{(1 + 1)^2} = \frac{g}{2^2} = \frac{g}{4} \]

5. **Relate Weight to Mass and Gravity:**
- The weight of the object at height \( h \) can be expressed as:
\[ W' = mg' = m \left(\frac{g}{4}\right) \]
- Since the weight on the surface is \( W = mg \), we can express \( m \) as:
\[ m = \frac{W}{g} \]
- Substitute this into the equation for \( W' \):
\[ W' = \frac{W}{g} \cdot \frac{g}{4} = \frac{W}{4} \]

6. **Calculate the New Weight:**
- Now, substituting the given weight \( W = 40 \, \text{N} \):
\[ W' = \frac{40 \, \text{N}}{4} = 10 \, \text{N} \]

### Final Answer:
The weight of the object at a height equal to the radius of the Earth is \( 10 \, \text{N} \).

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