If the radius of earth were to decrease by 1%, its mass remaining the same, the acceleration due to gravity on the surface of the earth will

Options

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

Explanation

To solve the problem, we need to understand how the acceleration due to gravity (g) changes when the radius of the Earth decreases while keeping its mass constant.

### Step-by-Step Solution:

1. **Understand the Formula for Acceleration Due to Gravity**:
The formula for the acceleration due to gravity at the surface of a planet is given by:
\[ g = \frac{GM}{R^2} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( R \) is the radius of the Earth.

2. **Identify the Changes**:
- The mass \( M \) of the Earth remains constant.
- The radius \( R \) decreases by 1%. This means:
\[ \Delta R = -0.01R \]

3. **Relate Changes in g to Changes in R**:
Since \( g \) is inversely proportional to the square of the radius, we can express the relative change in \( g \) as:
\[ \frac{\Delta g}{g} = -2 \frac{\Delta R}{R} \]
Here, \( \Delta g \) is the change in acceleration due to gravity, and \( \Delta R \) is the change in radius.

4. **Substitute the Change in Radius**:
Since \( \Delta R = -0.01R \), we can substitute this into the equation:
\[ \frac{\Delta g}{g} = -2 \left(-0.01\right) = 0.02 \]

5. **Convert to Percentage Change**:
To find the percentage change in \( g \):
\[ \frac{\Delta g}{g} \times 100 = 0.02 \times 100 = 2\% \]
This indicates that the acceleration due to gravity increases by 2%.

6. **Conclusion**:
Therefore, if the radius of the Earth decreases by 1% while its mass remains the same, the acceleration due to gravity on the surface of the Earth will increase by 2%.

### Final Answer:
The acceleration due to gravity on the surface of the Earth will increase by 2%.