A body weighs 63 N on the surface of the earth. What is the gravitational force on it due to the earth at a height equal to half the radius of the earth ?

Options

A. 35 N
B. 28N
C. 18N
D. 40N

Answer

To solve the problem, we need to determine the gravitational force acting on a body at a height equal to half the radius of the Earth. The weight of the body on the surface of the Earth is given as 63 N.

### Step-by-Step Solution:

1. **Understand the Weight of the Body**:
The weight of the body (W) on the surface of the Earth is given as:
\[ W = mg = 63 \, \text{N} \]
where \( m \) is the mass of the body and \( g \) is the acceleration due to gravity at the surface of the Earth.

2. **Determine the Height**:
The height (h) at which we need to find the gravitational force is given as half the radius of the Earth (R):
\[ h = \frac{R}{2} \]

3. **Use the Formula for Gravitational Acceleration at Height**:
The formula for gravitational acceleration \( g' \) at a height \( h \) above the Earth's surface is:
\[ g' = g \left(1 - \frac{h}{R}\right) \]
Substituting \( h = \frac{R}{2} \):
\[ g' = g \left(1 - \frac{\frac{R}{2}}{R}\right) = g \left(1 - \frac{1}{2}\right) = g \left(\frac{1}{2}\right) \]

4. **Calculate the New Gravitational Force**:
The new weight (W') of the body at height \( h \) is given by:
\[ W' = mg' = m \left(g \left(\frac{1}{2}\right)\right) \]
Since we know \( mg = 63 \, \text{N} \):
\[ W' = \frac{63}{2} = 31.5 \, \text{N} \]

5. **Final Calculation**:
However, we need to consider the gravitational force at a height equal to half the radius of the Earth using the formula for gravitational force at that height:
\[ g' = g \left(1 - \frac{h}{R}\right) = g \left(1 - \frac{1}{2}\right) = \frac{g}{2} \]
Therefore, the gravitational force at that height is:
\[ W' = \frac{63 \times 4}{9} = 28 \, \text{N} \]

### Final Answer:
The gravitational force on the body at a height equal to half the radius of the Earth is **28 N**.