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 how the acceleration due to gravity changes when the radius of the Earth decreases to 80% of its current value while keeping the mass of the Earth constant.

### Step-by-Step Solution:

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

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

3. **Substitute the new radius into the formula:**
We need to find the new acceleration due to gravity \( g' \) using the new radius:
\[ g' = \frac{G \cdot M}{(R')^2} = \frac{G \cdot M}{(0.8R)^2} \]
Simplifying this gives:
\[ g' = \frac{G \cdot M}{0.64R^2} \]

4. **Relate the new acceleration to the original acceleration:**
We can relate \( g' \) to the original acceleration \( g \):
\[ g' = \frac{g}{0.64} \]

5. **Calculate the new value of g':**
Given that the original acceleration due to gravity \( g \) is approximately \( 9.8 \, \text{m/s}^2 \):
\[ g' = \frac{9.8}{0.64} \approx 15.31 \, \text{m/s}^2 \]

6. **Determine the percentage increase in g:**
To find the percentage increase in the value of \( g \):
\[ \text{Percentage Increase} = \frac{g' - g}{g} \times 100 \]
Substituting the values:
\[ \text{Percentage Increase} = \frac{15.31 - 9.8}{9.8} \times 100 \approx 56\% \]

### Final Answer:
The new value of the acceleration due to gravity \( g' \) is approximately \( 15.31 \, \text{m/s}^2 \), and the percentage increase in the value of \( g \) is about \( 56\% \).