Two asteroids exert a gravitational force, F, on each other. Some time later, the asteroids are now three times as far from each other as before. Which of the following represents the gravitational force at this distance?

F/3

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F/2

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F/6

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F/9

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Solution

The correct option is D F/9
Let the initial distance between the two asteroid is $r_{1}$ and after some time the distance between them is $r_{2} = 3 r_{1}$ .
Let, $F_{1} = \frac{G m_{1} m_{2}}{(r_{1})^{2}}$ and $F_{2} = \frac{G m_{1} m_{2}}{(r_{2})^{2}}$
$\frac{F_{1}}{F_{2}} = \frac{\frac{G m_{1} m_{2}}{(r_{1})^{2}}}{\frac{G m_{1} m_{2}}{(r_{2})^{2}}}$
$\frac{F_{1}}{F_{2}} = \frac{(r_{2})^{2}}{(r_{1})^{2}}$
$\frac{F_{1}}{F_{2}} = \frac{(3 r_{1})^{2}}{(r_{1})^{2}}$
$\frac{F_{1}}{F_{2}} = \frac{9 (r_{1})^{2}}{(r_{1})^{2}}$
$F_{2} = \frac{F_{1}}{9}$
$F_{2} = \frac{F}{9}$

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Two asteroids exert a gravitational force, F, on each other. Some time later, the asteroids are now three times as far from each other as before. Which of the following represents the gravitational force at this distance?

The correct option is D F/9Let the initial distance between the two asteroid is r1 and after some time the distance between them is r2=3r1. Let, F1=Gm1m2(r1)2 and F2=Gm1m2(r2)2 F1F2=Gm1m2(r1)2Gm1m2(r2)2 F1F2=(r2)2(r1)2 F1F2=(3r1)2(r1)2 F1F2=9(r1)2(r1)2 F2=F19 F2=F9