distance between 2 bodies is increased to 3 times of its original value.what is the effect on the gravitational force between them

Question

anonymous · Answer

You can take ratio of the gravitational force with the 2nd distance to that with the 1st one. In both cases masses m and M are the same, and G is constant:

$\large{\frac{F_{g2}}{F_{g1}} = \frac{G\frac{mM}{(r_2)^2}}{G\frac{mM}{(r_1)^2}}}= G\frac{mM}{(r_2)^2}* \frac{(r_1)^2}{GmM}= \left (\frac{r_1}{r_2}\right)^2$

Since $r_2$ is 3 times $r_1$, then $r_2 = 3r_1$:

$\large{\left (\frac{r_1}{r_2}\right)^2 = \left (\frac{r_1}{3r_1}\right)^2= \left (\frac{1}{3}\right)^2= \frac{1}{9}} $

By increasing the distance 3 times the gravitational force has just $\frac{1}{9}$ of its original strength.