Question

If the radius of the earth shrinks by $2.0\%$, mass remaining constant, then how would the value of acceleration due to gravity change. ?

Answer 1

The value of acceleration due to gravity is

$g_1=\frac{GM}{R_1^2}............\left(1\right)$

When radius gets shrink by 2% then, new radius becomes, $R_2=98$% = 0.98, then value of acceleration due to gravity is

$g_2=\frac{GM}{R_2^2}=\frac{GM}{0.98R_1}..........\left(2\right)$

Taking ratio of equations (1) and (2)

$\frac{g_1}{g_2}={\left(\frac{R_2}{R_1}\right)}^2={\left(\frac{0.98R_1}{R_1}\right)}^2=0.9604$

$\frac{g_1}{g_2}=0.9604$

$g_2=1.04g_1$

Hence, g increases by 4%.