Gravitational field intensity

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Question

If the radius of the earth were to shrink by 2% its mass remaining the same, by how much percentage would the acceleration due to gravity on the earth's surface would increase?

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Solution

$g = \frac{GM}{R^{2}}$ (if R increases, g decreases)
Taking logarithms on both the sides, we get
$\log g = \log G + \log M - 2 \log R$
Differentiating, we get $\frac{dg}{g} = 0 + 0 - \frac{2 dR}{R}$
$\frac{dg}{g} = - 2 (\frac{- 2}{100}) = \frac{4}{100}$
$% increase in g = \frac{d g}{g} \times 100 = \frac{4}{100} \times 100 = 4 %$

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Gravitational field intensity

Remember concepts with our Masterclasses.

If the radius of the earth were to shrink by 2% its mass remaining the same, by how much percentage would the acceleration due to gravity on the earth's surface would increase?

g=GMR2 (if R increases, g decreases)Taking logarithms on both the sides, we getlog⁡g=log⁡G+log⁡M−2log⁡RDifferentiating, we get dgg=0+0−2dRRdgg=−2−2100=4100% increase in g=dgg×100=4100×100=4%

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