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

Options

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

Answer

To solve the problem of how the acceleration due to gravity changes if the Earth shrinks to half its radius without changing its mass, 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 acceleration due to gravity,
- \( G \) is the universal gravitational constant,
- \( M \) is the mass of the Earth,
- \( R \) is the radius of the Earth.

2. **Identify the Changes**:
According to the problem, the mass \( M \) of the Earth remains constant, but the radius \( R \) is reduced to half its original value. Therefore, the new radius \( R' \) is:
\[ R' = \frac{R}{2} \]

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

4. **Simplify the Expression**:
Now, substituting \( R' \) into the equation:
\[ g' = \frac{GM}{\left(\frac{R}{2}\right)^2} = \frac{GM}{\frac{R^2}{4}} = \frac{GM \cdot 4}{R^2} = 4 \cdot \frac{GM}{R^2} \]

5. **Relate it Back to the Original Gravity**:
We know that \( g = \frac{GM}{R^2} \). Therefore:
\[ g' = 4g \]

6. **Conclusion**:
The new acceleration due to gravity \( g' \) when the Earth shrinks to half its radius is four times the original acceleration due to gravity:
\[ g' = 4g \]

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

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