A body weights 72 N on the surface of the earth. What is the gravitational force acting on it due to the earth at a height to half the radius of the earth from the surface ?

Options

A. 16 N
B. 32 N
C. 8 N
D. 48 N

Explanation

To solve the problem step by step, we need to determine the gravitational force acting on a body at a height that is half the radius of the Earth from the surface. The weight of the body on the surface of the Earth is given as 72 N.

### Step 1: Understand the given data
- Weight of the body on the surface of the Earth, \( W = 72 \, \text{N} \)
- The height \( h \) from the surface is given as half the radius of the Earth, so \( h = \frac{R}{2} \), where \( R \) is the radius of the Earth.

### Step 2: Find the gravitational force at height \( h \)
The gravitational force \( F \) at a distance \( r \) from the center of the Earth can be expressed as:
\[ F = \frac{G \cdot M \cdot m}{r^2} \]
where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of the Earth,
- \( m \) is the mass of the body,
- \( r \) is the distance from the center of the Earth.

At the surface of the Earth, the distance \( r \) is equal to the radius of the Earth \( R \):
\[ W = \frac{G \cdot M \cdot m}{R^2} \]

### Step 3: Calculate the gravitational force at height \( h \)
At height \( h = \frac{R}{2} \), the distance from the center of the Earth becomes:
\[ r = R + h = R + \frac{R}{2} = \frac{3R}{2} \]

Now, we can express the gravitational force at this height:
\[ F_h = \frac{G \cdot M \cdot m}{\left(\frac{3R}{2}\right)^2} \]
\[ F_h = \frac{G \cdot M \cdot m}{\frac{9R^2}{4}} = \frac{4G \cdot M \cdot m}{9R^2} \]

### Step 4: Relate the forces
From the weight at the surface:
\[ W = \frac{G \cdot M \cdot m}{R^2} \]
We can express \( m \) in terms of \( W \):
\[ m = \frac{W \cdot R^2}{G \cdot M} \]

### Step 5: Substitute \( m \) into the equation for \( F_h \)
Substituting \( m \) into the equation for \( F_h \):
\[ F_h = \frac{4}{9} \cdot W \]
Substituting \( W = 72 \, \text{N} \):
\[ F_h = \frac{4}{9} \cdot 72 = \frac{288}{9} = 32 \, \text{N} \]

### Conclusion
The gravitational force acting on the body at a height of half the radius of the Earth from the surface is \( 32 \, \text{N} \).