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 suddenly shrinks to half of its present radius while keeping its mass constant, we can follow these steps:

### Step 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:
\[ g = \frac{G \cdot M}{R^2} \]
where:
- \( G \) is the universal gravitational constant,
- \( M \) is the mass of the planet,
- \( R \) is the radius of the planet.

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

### Step 3: Substitute the new radius into the formula
Now, we need to find the new acceleration due to gravity \( g' \) using the new radius:
\[ g' = \frac{G \cdot M}{(R')^2} \]
Substituting \( R' = \frac{R}{2} \):
\[ g' = \frac{G \cdot M}{\left(\frac{R}{2}\right)^2} \]

### Step 4: Simplify the equation
Now, simplify the expression:
\[ g' = \frac{G \cdot M}{\frac{R^2}{4}} \]
This can be rewritten as:
\[ g' = \frac{G \cdot M \cdot 4}{R^2} \]
Thus, we have:
\[ g' = 4 \cdot \frac{G \cdot M}{R^2} \]
Since \( \frac{G \cdot M}{R^2} = g \), we can substitute:
\[ g' = 4g \]

### Step 5: Conclusion
Therefore, if the Earth shrinks to half of its present radius without changing its mass, the acceleration due to gravity will become:
\[ g' = 4g \]

### Final Answer
The acceleration due to gravity will be four times its original value.

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