If the radius of the earth shrinks by 1.5% ( mass remaining same) , then the value of acceleration due to gravity changes by

Options

A. 0.02
B. `-2 %`
C. ` 3 % `
D. `- 3 % `

Answer

To solve the problem of how the acceleration due to gravity changes when the radius of the Earth shrinks by 1.5% while keeping the mass constant, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula for Acceleration due to Gravity (g)**:
The acceleration due to gravity is given by the formula:
\[ g = \frac{G \cdot M}{R^2} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( R \) is the radius of the Earth.

2. **Determine the New Radius**:
If the radius of the Earth shrinks by 1.5%, the new radius \( R' \) can be calculated as:
\[ R' = R \times (1 - 0.015) = R \times 0.985 \]

3. **Calculate the New Acceleration due to Gravity (g')**:
The new acceleration due to gravity \( g' \) can be expressed as:
\[ g' = \frac{G \cdot M}{(R')^2} \]
Substituting \( R' \) into the equation:
\[ g' = \frac{G \cdot M}{(0.985R)^2} = \frac{G \cdot M}{0.970225R^2} \]

4. **Relate g' to g**:
Since \( g = \frac{G \cdot M}{R^2} \), we can express \( g' \) in terms of \( g \):
\[ g' = g \cdot \frac{1}{0.970225} \]
Therefore, we can find the ratio:
\[ g' = g \cdot (1.0306) \]

5. **Calculate the Percentage Change in g**:
The percentage change in the acceleration due to gravity can be calculated using the formula:
\[ \text{Percentage Change} = \frac{g' - g}{g} \times 100 \]
Substituting \( g' \):
\[ \text{Percentage Change} = \frac{(1.0306g - g)}{g} \times 100 = (1.0306 - 1) \times 100 = 0.0306 \times 100 = 3.06\% \]

6. **Conclusion**:
The change in the acceleration due to gravity when the radius of the Earth shrinks by 1.5% is approximately **3%**.