If a body weight 100 N on the surface of the earth, then how much it would weigh at a depth of R/2 bellow the surface of the earth of radius R?

Options

A. 50N
B. 75 N
C. 100 N
D. 25 N

Explanation

To solve the problem of how much a body weighing 100 N on the surface of the Earth would weigh at a depth of \( R/2 \) below the surface, we can follow these steps:

### Step 1: Understand the weight of the body on the surface
The weight \( W \) of the 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. Given that \( W = 100 \, \text{N} \), we can express the mass as:
\[ m = \frac{W}{g} = \frac{100}{g} \]

### Step 2: Determine the gravitational acceleration at a depth
When a body is at a depth \( d \) below the surface of the Earth, the gravitational acceleration \( g' \) at that depth is given by the formula:
\[ g' = g \left(1 - \frac{d}{R}\right) \]
where \( R \) is the radius of the Earth. In this case, the depth \( d \) is \( R/2 \):
\[ g' = g \left(1 - \frac{R/2}{R}\right) = g \left(1 - \frac{1}{2}\right) = g \cdot \frac{1}{2} = \frac{g}{2} \]

### Step 3: Calculate the weight at the depth
Now, we can calculate the weight of the body at the depth \( R/2 \):
\[ W' = m \cdot g' = m \cdot \frac{g}{2} \]
Substituting \( m \) from Step 1:
\[ W' = \left(\frac{100}{g}\right) \cdot \frac{g}{2} = \frac{100}{2} = 50 \, \text{N} \]

### Conclusion
Thus, the weight of the body at a depth of \( R/2 \) below the surface of the Earth is:
\[ \boxed{50 \, \text{N}} \]