Two celestial bodies are separated by some distance. If the mass of any one of the point like bodies is doubled while the mass of other is halved then how far should they be taken so that the gravitational force between them becomes one-fourth?

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Solution

We know that,
Gravitational For $θ$ $(F_{4}) = \frac{G m_{1} m_{2}}{d^{2}}$
If $F_{4} = \frac{F_{4}}{4}, m_{1} = 2 m$ and $m_{2} = \frac{m_{2}}{2}$
Then ,
$F_{4} = G \frac{m_{1}^{'} m_{2}^{'}}{(d^{2})} = G \frac{m_{1} m_{2}}{(d^{'})^{2}}$
As $F_{4}^{'} = F_{4} / 4$
Then, $\frac{G m_{1} m_{2}}{(d^{'})^{2}} = G \frac{m_{1} m_{2}}{4 (d)^{2}}$
$\Rightarrow d^{'} = 2 d$
Their separated should be doubled

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Two celestial bodies are separated by some distance. If the mass of any one of the point like bodies is doubled while the mass of other is halved then how far should they be taken so that the gravitational force between them becomes one-fourth?

We know that,Gravitational For θ (F4)=Gm1m2d2If F4=F44,m1=2m and m2=m22Then , F4=Gm′1m′2(d2)=Gm1m2(d′)2As F′4=F4/4Then, Gm1m2(d′)2=Gm1m24(d)2⇒d′=2dTheir separated should be doubled