If radius of earth contracted by 0.1% , its mass remaining same then weight of the body at earth's surface will increase by

Options

A. `0.1%`
B. `0.2%`
C. `0.3%`
D. Remains same

Answer

To solve the problem of how the weight of a body at the Earth's surface will change if the radius of the Earth contracts by 0.1% while its mass remains the same, we can follow these steps:

### Step-by-Step Solution:

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

2. **Formula for Acceleration due to Gravity**: The acceleration due to gravity \( g \) can be expressed as:
\[ g = \frac{GM}{R^2} \]
where \( G \) is the universal gravitational constant, \( M \) is the mass of the Earth, and \( R \) is the radius of the Earth.

3. **Change in Radius**: If the radius of the Earth contracts by 0.1%, then the new radius \( R' \) can be calculated as:
\[ R' = R - 0.001R = 0.999R \]

4. **New Acceleration due to Gravity**: Substitute \( R' \) into the formula for \( g \):
\[ g' = \frac{GM}{(R')^2} = \frac{GM}{(0.999R)^2} = \frac{GM}{0.998001R^2} \]
Thus, the new acceleration due to gravity \( g' \) becomes:
\[ g' = \frac{g}{0.998001} \]

5. **Calculate the Change in Weight**: The new weight \( W' \) of the body is:
\[ W' = mg' = m \left(\frac{g}{0.998001}\right) = \frac{mg}{0.998001} = \frac{W}{0.998001} \]

6. **Percentage Change in Weight**: To find the percentage change in weight, we can calculate:
\[ \Delta W = W' - W = \left(\frac{W}{0.998001} - W\right) = W\left(\frac{1 - 0.998001}{0.998001}\right) = W\left(\frac{0.001001}{0.998001}\right) \]
The percentage change in weight is:
\[ \text{Percentage Change} = \left(\frac{\Delta W}{W}\right) \times 100 = \left(\frac{0.001001}{0.998001}\right) \times 100 \approx 0.1003\% \]

7. **Conclusion**: Therefore, the weight of the body at the Earth's surface will increase by approximately \( 0.1003\% \).