A body weight 45 N on the surface of the earth. What is the gravitational force acting on it due to the earth at a height equal to half the radius of the earth ?

Options

A. 40 N
B. 25 N
C. 20 N
D. 15 N

Explanation

To solve the problem, we need to find the gravitational force acting on a body at a height equal to half the radius of the Earth, given that its weight on the surface of the Earth is 45 N.

### Step-by-Step Solution:

1. **Understand the Weight on the Surface of the Earth**:
The weight \( W \) of the body on the surface of the Earth is given by:
\[ W = mg \]
where \( m \) is the mass of the body and \( g \) is the acceleration due to gravity at the surface of the Earth.

2. **Calculate the Mass of the Body**:
From the weight given:
\[ W = 45 \, \text{N} \]
We can express the mass \( m \) as:
\[ m = \frac{W}{g} \]
Here, \( g \) is approximately \( 9.8 \, \text{m/s}^2 \) (the acceleration due to gravity at the surface of the Earth).

3. **Determine the Height**:
We need to find the gravitational force at a height equal to half the radius of the Earth. Let \( R_e \) be the radius of the Earth. Therefore, the height \( h \) is:
\[ h = \frac{R_e}{2} \]

4. **Use the Formula for Gravitational Acceleration at Height**:
The acceleration due to gravity at a height \( h \) above the Earth's surface is given by:
\[ g' = g \left( \frac{R_e}{R_e + h} \right)^2 \]
Substituting \( h = \frac{R_e}{2} \):
\[ g' = g \left( \frac{R_e}{R_e + \frac{R_e}{2}} \right)^2 = g \left( \frac{R_e}{\frac{3R_e}{2}} \right)^2 = g \left( \frac{2}{3} \right)^2 = g \cdot \frac{4}{9} \]

5. **Calculate the New Weight**:
The new weight \( W' \) at height \( h \) is given by:
\[ W' = mg' = m \left( \frac{4g}{9} \right) \]
Substituting \( m = \frac{W}{g} \):
\[ W' = \frac{W}{g} \cdot \frac{4g}{9} = \frac{4W}{9} \]
Now substituting \( W = 45 \, \text{N} \):
\[ W' = \frac{4 \times 45}{9} = \frac{180}{9} = 20 \, \text{N} \]

### Final Answer:
The gravitational force acting on the body at a height equal to half the radius of the Earth is **20 N**.