CBSE Class 11-science Answered

Prove that the acceleration due to gravity decreses by twice when an object is taken to same height instead of depth.

Asked by saktipada1963 | 23 Sep, 2019, 10:23: AM

Expert Answer

Acceleration due to gravity on earth's surface,

begin mathsize 14px style g space equals space fraction numerator G space M over denominator R squared end fraction space.................... left parenthesis 1 right parenthesis end style

Acceleration due to gravity at a height h above earth's surface,

begin mathsize 14px style g subscript plus h end subscript space equals space fraction numerator G space M over denominator open parentheses R plus h close parentheses squared end fraction space.................... left parenthesis 2 right parenthesis end style

Acceleration due to gravity at a depth h below earth's surface,

begin mathsize 14px style g subscript negative h end subscript space equals space fraction numerator G space M over denominator R cubed end fraction space open parentheses R minus h close parentheses space space space space space.................... left parenthesis 3 right parenthesis end style

Eqn.(2) can be approximated as

begin mathsize 14px style g subscript plus h end subscript space equals space fraction numerator G space M over denominator open parentheses R plus h close parentheses squared end fraction space equals space fraction numerator G space M over denominator R squared end fraction 1 over open parentheses 1 plus begin display style h over R end style close parentheses squared space equals space fraction numerator G space M over denominator R squared end fraction open parentheses 1 plus h over R close parentheses to the power of negative 2 end exponent space almost equal to fraction numerator G space M over denominator R squared end fraction open parentheses 1 minus fraction numerator 2 h over denominator R end fraction close parentheses space space.............. left parenthesis 4 right parenthesis end style

Eqn.(3) is rewritten as

begin mathsize 14px style g subscript negative h end subscript space equals space fraction numerator G space M over denominator R squared end fraction space open parentheses 1 minus h over R close parentheses space space space space space.................... left parenthesis 5 right parenthesis end style

From eqn.(4) and (5), we see that decrease in acceleration due to gravity is twice at a height h above earth surface

in comparision with acceleration due to gravity at depth h below earth surface

Answered by Thiyagarajan K | 25 Sep, 2019, 12:56: PM

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