Question

if the radius of the earth shrinks by 1.5% (mass remaining the same) .then the value of acceleration due to gravity change by..

Answer 1

An object of mass m on the surface of the Earth (with its current dimensions) is attracted by the entire mass M of the Earth. Newton’s Universal Law of Gravitation states that the force between two masses separated by R is

GMm/R^2

Thus, mg = GMm/R^2

This gives the gravitational acceleration on the surface of the Earth:

g = GM/R^2

Now suppose the radius R of the Earth shrinks by 1.5%, retaining the same mass.

Then the new radius = R - 1.5%R = 0.985 R

Then the modified g

g* = GM/(0.985R)^2

Thus g* /g = [GM/(0.985R)^2]/ [ GM/R^2] = 1/0.985^2 = 1.0307

Fractional change in g

(g*-g) /g = g* /g - 1 ~ 0.03

About a 3% increase.