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?

Explanation:

The acceleration due to gravity on the surface of the Earth is given by:

where

G is the gravitational constant

M is the mass of the Earth

R is the radius of the Earth

Here we want to find the new Earth radius R' for which the gravitational acceleration at the surface, g', would be 3 times the current value of g:

So we would have

Solving the equation for R', we find

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