Question

if the radius of the earth shrink by 2%,mass remaing same, then how would the value of accelertion due to gravity change???



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Answer 1

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Answer 2

$\huge\bold{Solution}$

◆Let M, R be the mass and radius of earth.The acceleration due to gravity on the surface of Earth is given by

$g = \frac{GM}{R {}^{2} } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: .........(1)$

●Differentiating it, we have

$dg = - \frac{2GM}{R {}^{3} }dR \: \: \: \: \: \: \: \: \: \: .......(2)$

●Dividing (2) by (1), we have,

$\frac{dg}{g} = - \frac{2dR}{R}$

●When radius of Earth shrinks by 2% then,

$\frac{dR}{R} = - \frac{2}{100}$

●From (1) & (2) , the percentage increase in the value of "g" is =

$\frac{dg}{g} \times 100 = - \frac{2dR}{T} \times 100 = - 2( \frac{ - 2}{100} \times 100) \\ \\ = 4\%$

◆Hence the value of acceleration due to gravity change by 4%.