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Chapter 13: Q. 13 (page 353)

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

Expert verified

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

Step by step solution

01

Given information.

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

02

Calculation.

Let the new radius of earth be R.

The current free-fall acceleration is given by g=GMR2._____(2)

After shrinking, the free-fall acceleration will be : g'=3g=GMR'2.___(3)

03

Continuation of calculation.

Divide equation (3) by (2) and solve the equation for R'to get :

localid="1648482229762" 3g/g=GM/R'2GM/R2⇒3=R2R'2.⇒R'2=R23.⇒R'=R3.

04

Final answer.

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

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