Question

If the radius of the earth shrinks by 4% and there is no change in its mass. By how much does the value of `g’ change?

• A. 2%
• B. 4%
• C. 16%
• D. 8%

Answer 1

The correct option is D

8%

Let us assume that,
$M_e$ = mass of the earth
$R_1$ and $R_2$ are the radii of the earth before and after shrink respectively.
$g_1$ and $g_2$ are the acceleration due to gravity before and after shrink respectively.
We know,
$g_1=\frac{G{M}_e}{(R_1{)}^2}$ & $g_2=\frac{G{M}_e}{(R_2{)}^2}$
Given,
$R_2=\left(\frac{96}{100}\right)R_1=0.96R_1\therefore \frac{g_2}{g_1}=\frac{G{M}_e}{(R_2{)}^2}\times \frac{(R_1{)}^2}{G{M}_e}=\frac{(R_1{)}^2}{(0.96R_1{)}^2}=\frac{1}{(0.96{)}^2}=1.08$
So, increase in % = $\left[\frac{(1.08-1)}{1}\right]\times 100$ = 8%
So, value of "g" changes by 8%.