If the radius of the earth decreases by `10%`, the mass remaining unchanged, what will happen to the acceleration due to gravity?

Options

A. Decreases by `19%`
B. Increases by `19%`
C. Decreases by more than `19%`
D. Increases by more than `19%`

Explanation

To solve the problem, we need to determine how the acceleration due to gravity (g) changes when the radius of the Earth decreases by 10%, while the mass (M) remains unchanged.

### Step-by-Step Solution:

1. **Understanding the Formula for Acceleration Due to Gravity**:
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 gravitational constant,
- \( M \) is the mass of the Earth,
- \( R \) is the radius of the Earth.

2. **Determine the New Radius**:
If the radius of the Earth decreases by 10%, the new radius \( R' \) can be calculated as:
\[ R' = R - 0.1R = 0.9R \]

3. **Substituting the New Radius into the Gravity Formula**:
We can now express the new acceleration due to gravity \( g' \) using the new radius:
\[ g' = \frac{GM}{(R')^2} = \frac{GM}{(0.9R)^2} \]
Simplifying this gives:
\[ g' = \frac{GM}{0.81R^2} = \frac{g}{0.81} \]

4. **Calculating the Ratio of New Gravity to Old Gravity**:
To find the relationship between the new and old values of gravity, we can write:
\[ \frac{g'}{g} = \frac{1}{0.81} \]
This means:
\[ g' = \frac{100}{81}g \]

5. **Finding the Percentage Change in Gravity**:
The percentage change in gravity can be calculated as:
\[ \text{Percentage Change} = \left(\frac{g' - g}{g}\right) \times 100 \]
Substituting \( g' \):
\[ \text{Percentage Change} = \left(\frac{\frac{100}{81}g - g}{g}\right) \times 100 \]
Simplifying this gives:
\[ = \left(\frac{100 - 81}{81}\right) \times 100 = \frac{19}{81} \times 100 \]

6. **Calculating the Final Value**:
Now, calculating \( \frac{19}{81} \times 100 \):
\[ \approx 23.46\% \]
Therefore, the acceleration due to gravity increases by approximately 23%.

### Final Answer:
The acceleration due to gravity increases by approximately 23%.