Question

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

• A. decrease by $2$%
• B. increase by $2$%
• C. increase by $4$%
• D. decrease by $4$%

Answer 1

The correct option is C increase by $4$%

$g=\frac{GM}{R^2}$, if $R$ decreases, then $g$ increases
Taking log on both sides, we have
$\log g=\log G+\log M-2\log R$
Differentiating it we get
$\frac{dg}{g}=0+0-\frac{2dR}{R}$
$=-2\times \left(\frac{-2}{100}\right)=\frac{4}{100}$
$=\frac{4}{100}\times 100$
$=4$%