If both the mass and radius of the earth, each decreases by 50%, the acceleration due to gravity would

Options

A. remain same
B. decrease by 50%
C. decrease by 100%
D. increase by 100%

Explanation

To solve the problem of how the acceleration due to gravity changes when both the mass and radius of the Earth decrease by 50%, 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 the Earth 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 Earth,
- \( R \) is the radius of the Earth.

### Step 2: Determine the new mass and radius
If both the mass and radius of the Earth decrease by 50%, we can express the new mass \( M' \) and new radius \( R' \) as follows:
- New mass \( M' = M - 0.5M = 0.5M \)
- New radius \( R' = R - 0.5R = 0.5R \)

### Step 3: Substitute the new values into the formula
Now, we can substitute \( M' \) and \( R' \) into the formula for \( g \):
\[ g' = \frac{G \cdot M'}{(R')^2} \]
Substituting the values we found:
\[ g' = \frac{G \cdot (0.5M)}{(0.5R)^2} \]

### Step 4: Simplify the expression
Now simplify the expression:
\[ g' = \frac{G \cdot (0.5M)}{(0.25R^2)} = \frac{0.5G \cdot M}{0.25R^2} = \frac{0.5}{0.25} \cdot \frac{G \cdot M}{R^2} = 2 \cdot g \]
Thus, the new acceleration due to gravity \( g' \) is twice the original acceleration due to gravity.

### Step 5: Conclusion
Therefore, if both the mass and radius of the Earth decrease by 50%, the acceleration due to gravity would increase by 100%.

### Final Answer
The acceleration due to gravity would be \( 2g \), which means it has increased by 100%.

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