If mass of earth decreases by 25% and its radius increases by 50% , then acceleration due to gravity at its surface decreases by nearly

Options

A. `89%`
B. `67%`
C. `33%`
D. `11%`

Answer

To solve the problem, we need to calculate the change in acceleration due to gravity (g) when the mass of the Earth decreases by 25% and its radius increases by 50%.

### Step-by-Step Solution:

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

2. **Calculate the New Mass of the Earth**:
If the mass of the Earth decreases by 25%, the new mass \( M' \) can be calculated as:
\[ M' = M - 0.25M = 0.75M = \frac{3M}{4} \]

3. **Calculate the New Radius of the Earth**:
If the radius of the Earth increases by 50%, the new radius \( R' \) can be calculated as:
\[ R' = R + 0.5R = 1.5R = \frac{3R}{2} \]

4. **Calculate the New Acceleration due to Gravity**:
Substitute \( M' \) and \( R' \) into the formula for \( g \):
\[ g' = \frac{G \cdot M'}{(R')^2} = \frac{G \cdot \frac{3M}{4}}{\left(\frac{3R}{2}\right)^2} \]
Simplifying the denominator:
\[ (R')^2 = \left(\frac{3R}{2}\right)^2 = \frac{9R^2}{4} \]
Now substituting this back into the equation for \( g' \):
\[ g' = \frac{G \cdot \frac{3M}{4}}{\frac{9R^2}{4}} = \frac{G \cdot 3M}{4} \cdot \frac{4}{9R^2} = \frac{3GM}{9R^2} = \frac{1}{3} \cdot \frac{GM}{R^2} = \frac{g}{3} \]

5. **Calculate the Change in Acceleration due to Gravity**:
The change in acceleration due to gravity can be calculated as:
\[ \Delta g = g - g' = g - \frac{g}{3} = \frac{3g}{3} - \frac{g}{3} = \frac{2g}{3} \]

6. **Calculate the Percentage Decrease**:
The percentage decrease in acceleration due to gravity is given by:
\[ \text{Percentage Decrease} = \frac{\Delta g}{g} \times 100 = \frac{\frac{2g}{3}}{g} \times 100 = \frac{2}{3} \times 100 = 66.67\% \]

### Final Answer:
The acceleration due to gravity at the surface decreases by nearly **67%**.