Please give detailed solution for the given question: If the radius of earth decreases without changing the mass, would pendulum clocks placed at the surface of earth GAIN time?

Dear student,
Let's take an example,
If we consider the Earth to be a perfect sphere, then the acceleration due to gravity at its surface is given by g equals G M over R squared.
Here, M is the mass of Earth; R is the radius of the Earth and G is universal gravitational constant.

If the radius of the earth is decreased by 1%, then the new radius becomes

R apostrophe equals R minus R over 100 equals 99 over 100 R rightwards double arrow R apostrophe equals 0.99 R

New acceleration due to gravity will be given by
g apostrophe equals G fraction numerator M over denominator R apostrophe squared end fraction equals G fraction numerator M over denominator left parenthesis 0.99 R right parenthesis squared end fraction rightwards double arrow g apostrophe equals 1.02 cross times open parentheses G M over R squared close parentheses equals 1.02 g
Hence, the value of the acceleration due to gravity increases when the radius is decreased.

Percentage increase in the acceleration due to gravity is given by
fraction numerator g apostrophe minus g over denominator g end fraction cross times 100 space space equals fraction numerator 0.02 g over denominator g end fraction cross times 100 space space equals 2 percent sign
​​​​​​​T1gso an increase in g will decrease the time period of pendulum.Regards

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