A body of weight 72 N moves from the surface of earth at a height half of the radius of earth , then geavitational force exerted on it will be

Options

A. 36 N
B. 32 N
C. 144 N
D. 50 N

Answer

To find the gravitational force exerted on a body of weight 72 N when it is moved to a height equal to half the radius of the Earth, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Information:**
- The weight of the body on the surface of the Earth is \( W = 72 \, \text{N} \).
- The height \( h \) at which the body is moved is \( h = \frac{R}{2} \), where \( R \) is the radius of the Earth.

2. **Calculate the Mass of the Body:**
- The weight of the body is given by the formula:
\[ W = m \cdot g \]
where \( g \) is the acceleration due to gravity at the surface of the Earth.
- Rearranging this gives:
\[ m = \frac{W}{g} \]
- We know that \( g \approx 9.8 \, \text{m/s}^2 \) (standard value). Thus:
\[ m = \frac{72}{9.8} \approx 7.35 \, \text{kg} \]

3. **Determine the New Gravitational Force at Height \( h \):**
- The formula for gravitational force at a distance \( r \) from the center of the Earth is given by:
\[ F = \frac{G M m}{r^2} \]
where \( G \) is the gravitational constant and \( M \) is the mass of the Earth.
- At height \( h = \frac{R}{2} \), the distance from the center of the Earth becomes:
\[ r = R + h = R + \frac{R}{2} = \frac{3R}{2} \]

4. **Substituting into the Gravitational Force Formula:**
- Now, substituting \( r \) into the gravitational force formula:
\[ F = \frac{G M m}{\left(\frac{3R}{2}\right)^2} \]
- Simplifying this gives:
\[ F = \frac{G M m}{\frac{9R^2}{4}} = \frac{4 G M m}{9 R^2} \]

5. **Relate the New Force to the Weight on the Surface:**
- We know that the weight on the surface of the Earth is:
\[ W = \frac{G M m}{R^2} \]
- Therefore, we can express the new force \( F \) in terms of the weight \( W \):
\[ F = \frac{4}{9} W \]

6. **Calculate the New Gravitational Force:**
- Substituting \( W = 72 \, \text{N} \):
\[ F = \frac{4}{9} \times 72 = \frac{288}{9} = 32 \, \text{N} \]

### Final Answer:
The gravitational force exerted on the body at a height equal to half the radius of the Earth is **32 N**.

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