Show that if the acceleration due to gravity on any planet is n² times the acceleration due to gravity on the earth, then time period of simple pendulum on that planet is 1/n times the time period on simple pendulum on the earth.

Solution

Time period of oscillation is given by,

straight T equals 2 straight pi square root of straight ell over straight g end root

Let, g and g' be the acceleration due to gravity on the earth and planet respectively, and
T and T’ be their respective time periods.
Then,

T equals 2 pi square root of straight ell over g end root space space space space space a n d space T apostrophe equals space T equals 2 pi square root of fraction numerator straight ell over denominator g apostrophe end fraction end root space therefore space space space fraction numerator T apostrophe over denominator T end fraction equals square root of fraction numerator g over denominator g apostrophe end fraction end root equals square root of fraction numerator g over denominator n squared g end fraction end root equals 1 over n rightwards double arrow space space space space T apostrophe equals 1 over n T

Hence, proved.

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