If the earth suddenly shrinks (without changing and half of its present radius , the acceleration due to gravity be

Options

A. g/2
B. 4g
C. g/4
D. 2g

Explanation

To solve the problem of how the acceleration due to gravity changes if the Earth shrinks to half of its present radius while keeping its mass the same, we can follow these steps:

### Step-by-Step Solution:

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

2. **Identify the new radius**:
If the Earth shrinks to half of its present radius, the new radius \( R' \) will be:
\[ R' = \frac{R}{2} \]

3. **Substitute the new radius into the formula**:
The new acceleration due to gravity \( g' \) at the surface of the Earth after it shrinks can be calculated by substituting \( R' \) into the formula:
\[ g' = \frac{GM}{(R')^2} = \frac{GM}{\left(\frac{R}{2}\right)^2} \]

4. **Simplify the equation**:
Simplifying the expression for \( g' \):
\[ g' = \frac{GM}{\left(\frac{R^2}{4}\right)} = \frac{GM \cdot 4}{R^2} = 4 \cdot \frac{GM}{R^2} \]
Therefore, we can express \( g' \) in terms of the original acceleration due to gravity \( g \):
\[ g' = 4g \]

5. **Conclusion**:
The new acceleration due to gravity when the Earth shrinks to half its radius while maintaining the same mass will be four times the original acceleration due to gravity.

### Final Answer:
The acceleration due to gravity will be \( 4g \).

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