If the radius of the earth were to shrink by 1% its mass remaining the same, the acceleration due to gravity on the earth's surface would

Options

A. Decrease by `2%`
B. Remain unchanged
C. Increase by `2%`
D. Increase by `1%`

Answer

To solve the problem, we need to determine how the acceleration due to gravity (g) changes when the radius of the Earth shrinks by 1%, while its mass remains constant.

### Step-by-Step Solution:

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

2. **Identifying the change in radius**:
If the radius of the Earth shrinks by 1%, the new radius \( R' \) can be expressed as:
\[ R' = R - 0.01R = 0.99R \]

3. **Calculating the new acceleration due to gravity**:
We can substitute the new radius into the formula for gravity:
\[ g' = \frac{GM}{(R')^2} = \frac{GM}{(0.99R)^2} \]
Simplifying this gives:
\[ g' = \frac{GM}{0.9801R^2} = \frac{g}{0.9801} \]

4. **Finding the relationship between new and old gravity**:
Since \( g = \frac{GM}{R^2} \), we can express \( g' \) in terms of \( g \):
\[ g' = g \cdot \frac{1}{0.9801} \]
This means that \( g' \) is greater than \( g \).

5. **Calculating the percentage increase in gravity**:
To find the percentage increase in gravity, we calculate:
\[ \text{Percentage Increase} = \left( \frac{g' - g}{g} \right) \times 100\% \]
Substituting \( g' = g \cdot \frac{1}{0.9801} \):
\[ \text{Percentage Increase} = \left( \frac{\frac{g}{0.9801} - g}{g} \right) \times 100\% \]
Simplifying this gives:
\[ \text{Percentage Increase} = \left( \frac{1 - 0.9801}{0.9801} \right) \times 100\% \approx 2.02\% \]

6. **Conclusion**:
Therefore, the acceleration due to gravity increases by approximately 2% when the radius of the Earth shrinks by 1%.

### Final Answer:
The acceleration due to gravity on the Earth's surface would **increase by approximately 2%**.