When the radius of earth is reduced by 1% without changing the mass, then the acceleration due to gravity will

Options

A. increase by 2%
B. decrease by 1.5%
C. increase by 1%
D. decrease by 1%

Answer

To solve the problem, we need to understand how the acceleration due to gravity (g) is affected by changes in the radius of the Earth (R) while keeping the mass (M) constant. The formula for acceleration due to gravity at the surface of a planet is given by:

\[ g = \frac{GM}{R^2} \]

where:
- \( G \) is the universal gravitational constant,
- \( M \) is the mass of the Earth,
- \( R \) is the radius of the Earth.

### Step-by-step Solution:

1. **Understand the Initial Conditions**:
- Let the initial radius of the Earth be \( R \).
- The initial acceleration due to gravity is \( g = \frac{GM}{R^2} \).

2. **Determine the New Radius**:
- If the radius is reduced by 1%, the new radius \( R' \) can be calculated as:
\[ R' = R - 0.01R = 0.99R \]

3. **Calculate the New Acceleration due to Gravity**:
- Using the new radius, the new acceleration due to gravity \( g' \) can be calculated as:
\[ g' = \frac{GM}{(R')^2} = \frac{GM}{(0.99R)^2} \]

4. **Simplify the Expression**:
- Substitute \( R' \) into the equation:
\[ g' = \frac{GM}{(0.99R)^2} = \frac{GM}{0.9801R^2} \]
- Since \( g = \frac{GM}{R^2} \), we can express \( g' \) in terms of \( g \):
\[ g' = \frac{g}{0.9801} \]

5. **Calculate the Factor Increase**:
- To find the new value of \( g' \):
\[ g' \approx 1.0204g \]
- This indicates that the new acceleration due to gravity is approximately 2.04% greater than the original acceleration due to gravity.

### Final Answer:
When the radius of the Earth is reduced by 1% without changing the mass, the acceleration due to gravity will increase by approximately 2.04%.