Prove that acceleration due to gravity decreases with the depth and height.
Answers
Let A be the point on the surface of the Earth and B be the point at an altitude h.
Let M be the mass of the Earth and R be the radius of the Earth.
Consider the Earth as a spherical shaped body.
The acceleration due to gravity at point A on the surface of the Earth is,
begin mathsize 14px style straight g equals GM over straight R squared space space space... space left parenthesis 1 right parenthesis end style
Let the body be placed at B at a height h from the surface of the Earth.
The acceleration due to gravity at B is,
begin mathsize 14px style straight g apostrophe equals fraction numerator GM over denominator left parenthesis straight R plus straight h right parenthesis squared end fraction space space space... space left parenthesis 2 right parenthesis end style
Dividing eqn (1) by (2), we get,
begin mathsize 14px style fraction numerator straight g over denominator straight g apostrophe end fraction equals fraction numerator GM divided by straight R squared over denominator left square bracket GM divided by left parenthesis straight R plus straight h right parenthesis squared right square bracket end fraction fraction numerator straight g over denominator straight g apostrophe end fraction equals fraction numerator GM cross times left parenthesis straight R plus straight h right parenthesis squared over denominator GM cross times straight R squared end fraction fraction numerator straight g over denominator straight g apostrophe end fraction equals fraction numerator left parenthesis straight R plus straight h right parenthesis squared over denominator straight R squared end fraction fraction numerator straight g over denominator straight g apostrophe end fraction equals open parentheses straight R over straight R plus straight h over straight R close parentheses squared fraction numerator straight g over denominator straight g apostrophe end fraction equals open parentheses 1 plus straight h over straight R close parentheses squared straight g apostrophe equals straight g open parentheses 1 plus straight h over straight R close parentheses to the power of negative 2 end exponent Now comma space expanding space by space using space binomial space theorem comma space we space get comma straight g apostrophe equals straight g open parentheses 1 minus fraction numerator 2 straight h over denominator straight R end fraction close parentheses end style
The value of acceleration due to gravity decreases with increase in height above the surface of the Earth.
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