If the distance between earth and sun is increased by $2$ %, then find percentage change in gravitational force acting between them.

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Solution

F (between sun and earth)= $\frac{G M m}{r^{2}}$
when $r^{'} = r + \frac{2}{100} r$
$F^{'} = \frac{G M m}{r^{' 2}} = \frac{G M m}{(1.02)^{2} r^{2}} = \frac{1}{(1.02)^{2}} (\frac{G M m}{r^{2}})$
$F^{'} = \frac{1}{(1.02)^{2}} F$
% of change= $\frac{F^{'} - F}{F} = \frac{\frac{F}{(1.02)^{2}} - F}{F} \times 100$
$= 1 - \frac{1}{(1.02)^{2}}$ [the force decreases]
$= 3.8831$ % is the change in force.

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If the distance between earth and sun is increased by 2%, then find percentage change in gravitational force acting between them.

F (between sun and earth)= GMmr2when r′=r+2100rF′=GMmr′2=GMm(1.02)2r2=1(1.02)2(GMmr2)F′=1(1.02)2F% of change=F′−FF=F(1.02)2−FF×100=1−1(1.02)2 [the force decreases]=3.8831% is the change in force.

If the distance between earth and sun is increased by $2$ %, then find percentage change in gravitational force acting between them.