Question

Suppose we could shrink the earth without changing its mass. At what fraction of its current radius would the free-fall acceleration at the surface be three times its present value?

Short answer

The radius of earth must reduce by $13$times its current value to increase the free-fall constant to three times its current value.

Step-by-step solution

Given information.

Free-fall acceleration is the acceleration experienced by a freely falling body towards a body with gravitational attraction.

Calculation.

Let the new radius of earth be $R.$

The current free-fall acceleration is given by $g=GM{R}_2.$_____(2)

After shrinking, the free-fall acceleration will be : $g^{\text{'}}=3g=GM{R^{\text{'}}}_2.$___(3)

Continuation of calculation.

Divide equation (3) by (2) and solve the equation for $R^{\text{'}}$to get :

localid="1648482229762" $$\begin{gathered} 3g/g=GM/{R^{\text{'}}}_{2}GM/R_2 \\ \Rightarrow 3=R_2{R^{\text{'}}}_2. \\ \Rightarrow {R^{\text{'}}}_2=R_{23}. \\ \Rightarrow R^{\text{'}}=R_3. \end{gathered}$$

Final answer.

The radius of earth must shrink to $13$times its current value to increase the free-fall constant to three times its current value.