1. The acceleration due to gravity at the surface of the earth is

$g = \frac{GM}{R^2}$

where:

• $G$ is the gravitational constant,
• $M$ is the mass of the earth,
• $R$ is the radius of the earth.

2. Given that the mass remains the same, only the radius changes.

If the radius shrinks by $2\%$, then

$R' = 0.98R$

3. The new acceleration due to gravity will be

$g' = \frac{GM}{(R')^2} = \frac{GM}{(0.98R)^2} = \frac{g}{(0.98)^2}$

Now,

$(0.98)^2 = 0.9604$

So,

$g' = \frac{g}{0.9604} \approx 1.0412\, g$

4. Thus, the fractional increase is

$\frac{g' - g}{g} = 1.0412 - 1 = 0.0412$

$= 4.12\%$

So the acceleration due to gravity increases by approximately $4\%$.

5. Checking options:

• A: decrease by $2\%$ $\rightarrow$ incorrect
• B: decrease by $4\%$ $\rightarrow$ incorrect
• C: increase by $2\%$ $\rightarrow$ incorrect
• D: increase by $4\%$ $\rightarrow$ correct

Therefore, the correct option is D.