If the radius of the earth were to shrink by 1%, the density remaining constant, the acceleration due to gravity on the surface of the earth will be :

Question

Accepted Answer

When the radius of the Earth shrinks by 1% while maintaining constant density, the acceleration due to gravity on the surface of the Earth will be affected. Let's explore why Option 4, "less than g," is the correct answer.

### Explanation for Option 4: Less than g

The acceleration due to gravity on the surface of a planet is given by the formula:

$$ g = \frac{G \cdot M}{R^2} $$

where:
- $ G $ is the gravitational constant,
- $ M $ is the mass of the Earth,
- $ R $ is the radius of the Earth.

If the radius $ R $ decreases by 1%, the new radius $ R' $ becomes $ 0.99R $. The volume of the Earth, which is proportional to $ R^3 $, will decrease, but since the density remains constant, the mass $ M $ of the Earth will also decrease proportionally to the volume. Therefore, the new mass $ M' $ is given by:

$$ M' = \text{Density} \times \text{Volume} = \text{Density} \times \left(\frac{4}{3} \pi (0.99R)^3\right) $$

This results in a decrease in mass because the volume decreases. Substituting the new mass and radius into the formula for gravity:

$$ g' = \frac{G \cdot M'}{(0.99R)^2} $$

Since $ M' < M $ due to the decrease in volume, and the denominator $(0.99R)^2$ is smaller than $ R^2 $, the overall effect is a decrease in the value of $ g' $ compared to the original $ g $. Thus, the acceleration due to gravity will be less than the original $ g $.

### Explanation for Why Other Options are Incorrect

1. **Option 1: Infinite**
   - The acceleration due to gravity cannot become infinite because both the mass and radius of the Earth remain finite. The formula for gravity does not allow for an infinite result under these conditions.

2. **Option 2: g**
   - If the radius decreases while maintaining constant density, the mass decreases, leading to a decrease in gravity. Therefore, the acceleration due to gravity cannot remain the same as $ g $.

3. **Option 3: More than g**
   - For gravity to increase, the mass would need to increase or the radius would need to decrease significantly without a corresponding decrease in mass. Since the mass decreases due to the reduction in volume, gravity cannot be more than $ g $.

In conclusion, the correct answer is Option 4: "less than g," because the decrease in Earth's radius, while maintaining constant density, results in a decrease in mass, leading to a reduction in the acceleration due to gravity.