If W is the weight of a body on the surface of the earth, its weight at a height equal to radius of the earth would be

Options

A. W/2
B. 2 W
C. W/4
D. 4 W

Explanation

To solve the problem, we need to determine the weight of a body at a height equal to the radius of the Earth. Let's break this down step by step.

### Step 1: Understand the weight on the surface of the Earth
The weight \( W \) of a body on the surface of the Earth 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 at the surface of the Earth.

### Step 2: Define the height and distance from the center of the Earth
The radius of the Earth is denoted as \( r \). If the body is raised to a height equal to the radius of the Earth, the total distance from the center of the Earth becomes:
\[ \text{Distance from center} = r + r = 2r \]

### Step 3: Calculate the acceleration due to gravity at height \( 2r \)
The acceleration due to gravity at a distance \( d \) from the center of the Earth is given by:
\[ g' = \frac{G \cdot M}{d^2} \]
where \( G \) is the gravitational constant and \( M \) is the mass of the Earth. At a distance of \( 2r \):
\[ g' = \frac{G \cdot M}{(2r)^2} = \frac{G \cdot M}{4r^2} \]

### Step 4: Express the new weight at height \( 2r \)
The weight \( W_1 \) of the body at this new height can be calculated using the new acceleration due to gravity \( g' \):
\[ W_1 = m \cdot g' = m \cdot \frac{G \cdot M}{4r^2} \]

### Step 5: Relate the new weight to the original weight
Since the original weight \( W \) is given by:
\[ W = m \cdot g = m \cdot \frac{G \cdot M}{r^2} \]
We can express \( W_1 \) in terms of \( W \):
\[ W_1 = \frac{m \cdot \frac{G \cdot M}{4r^2}}{m \cdot \frac{G \cdot M}{r^2}} \cdot W = \frac{W}{4} \]

### Conclusion
Thus, the weight of the body at a height equal to the radius of the Earth is:
\[ W_1 = \frac{W}{4} \]

### Final Answer
The weight of the body at a height equal to the radius of the Earth is \( \frac{W}{4} \).

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