If the earth suddenly shrinks (without changing mass) to half of its present radius, then acceleration due to gravity will 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 its present radius while keeping its mass constant, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula for Acceleration Due to Gravity**:
The acceleration due to gravity \( g \) 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 (Earth in this case),
- \( R \) is the radius of the planet.

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

3. **Substitute the New Radius into the Gravity Formula**:
We need to find the new acceleration due to gravity \( g' \) using the new radius:
\[ g' = \frac{GM}{(R')^2} \]
Substituting \( R' = \frac{R}{2} \) into the equation gives:
\[ g' = \frac{GM}{\left(\frac{R}{2}\right)^2} \]

4. **Simplify the Expression**:
Simplifying the denominator:
\[ g' = \frac{GM}{\frac{R^2}{4}} = \frac{GM \cdot 4}{R^2} = 4 \cdot \frac{GM}{R^2} \]
Thus, we can express \( g' \) in terms of the original \( g \):
\[ g' = 4g \]

5. **Conclusion**:
Therefore, if the Earth shrinks to half its radius while keeping its mass constant, the new acceleration due to gravity will be:
\[ g' = 4g \]
This means the acceleration due to gravity will increase by a factor of 4.

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

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