If the radius of the earth suddenly decreases to `80%` of its present value, the mass of the earth remaining the same, the value of the acceleration due to gravity will

Options

A. remain unchanged
B. become `9.8/0.64ms^(-2)`
C. increase by `36%`
D. increase by about `56%`

Explanation

To solve the problem, we need to determine the new value of the acceleration due to gravity (g') when the radius of the Earth decreases to 80% of its present value while keeping the mass of the Earth (M) constant.

### Step-by-Step Solution:

1. **Understand 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,
- \( R \) is the radius of the Earth.

2. **Determine the new radius:**
If the radius of the Earth decreases to 80% of its present value, the new radius \( R' \) can be calculated as:
\[ R' = 0.8R \]

3. **Substitute the new radius into the formula:**
The new acceleration due to gravity \( g' \) at the surface with the new radius will be:
\[ g' = \frac{GM}{(R')^2} = \frac{GM}{(0.8R)^2} \]

4. **Simplify the expression:**
\[ g' = \frac{GM}{0.64R^2} = \frac{1}{0.64} \cdot \frac{GM}{R^2} = \frac{g}{0.64} \]

5. **Calculate the new value of g':**
Since \( g = \frac{GM}{R^2} \), we can express \( g' \) as:
\[ g' = \frac{g}{0.64} \approx 1.5625g \]
If we take \( g \approx 9.8 \, \text{m/s}^2 \):
\[ g' \approx 1.5625 \times 9.8 \approx 15.3125 \, \text{m/s}^2 \]

6. **Determine the percentage change in g:**
To find the percentage change in g, we can use the formula:
\[ \text{Percentage Change} = \left(\frac{g' - g}{g}\right) \times 100 \]
Substituting the values:
\[ \text{Percentage Change} = \left(\frac{15.3125 - 9.8}{9.8}\right) \times 100 \approx \left(\frac{5.5125}{9.8}\right) \times 100 \approx 56.24\% \]

### Final Answer:
The new value of the acceleration due to gravity will be approximately \( 1.5625g \) or \( 15.3125 \, \text{m/s}^2 \), which indicates an increase of approximately \( 56.24\% \).