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 1%
B. remain unchanged
C. increase by 1%
D. increase by 2%

Explanation

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 the 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:
\[ g = \frac{G \cdot M}{R^2} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( R \) is the radius of the Earth.

2. **Initial Conditions**:
Let the initial radius of the Earth be \( R_1 \) and the initial acceleration due to gravity be \( g_1 \):
\[ g_1 = \frac{G \cdot M}{R_1^2} \]

3. **New Conditions After Radius Shrinks**:
If the radius shrinks by 1%, the new radius \( R_2 \) can be expressed as:
\[ R_2 = R_1 - 0.01 R_1 = 0.99 R_1 \]

4. **New Acceleration due to Gravity**:
The new acceleration due to gravity \( g_2 \) with the new radius \( R_2 \) is:
\[ g_2 = \frac{G \cdot M}{R_2^2} \]
Substituting \( R_2 \):
\[ g_2 = \frac{G \cdot M}{(0.99 R_1)^2} = \frac{G \cdot M}{0.9801 R_1^2} \]

5. **Relating \( g_2 \) to \( g_1 \)**:
Now, we can express \( g_2 \) in terms of \( g_1 \):
\[ g_2 = \frac{G \cdot M}{0.9801 R_1^2} = \frac{g_1}{0.9801} \]

6. **Calculating the Ratio of \( g_2 \) to \( g_1 \)**:
To find the percentage change, we calculate:
\[ \frac{g_2}{g_1} = \frac{1}{0.9801} \approx 1.0204 \]

7. **Finding the Percentage Increase**:
The percentage increase in \( g \) is:
\[ \text{Percentage Increase} = \left( \frac{g_2 - g_1}{g_1} \right) \times 100\% = \left( 1.0204 - 1 \right) \times 100\% \approx 2.04\% \]

8. **Final Result**:
Therefore, the acceleration due to gravity on the Earth's surface would increase by approximately 2%.

### Conclusion:
The acceleration due to gravity on the Earth's surface would increase by about 2% if the radius of the Earth were to shrink by 1%, while its mass remains the same.