Physics - Gravitation Question with Solution | TestHub

The gravitational potential $V$ due to a point mass $m$ at a distance $r$ is given by $V=-\frac{Gm}{r}$.

For multiple masses, the total potential is the sum of individual potentials.

Here, masses $m$ are at distances $r_n=2^{n-1}$ for $n=1,2,3,\dots$.

So, $r_1=1$ m, $r_2=2$ m, $r_3=4$ m, etc.

The total gravitational potential at $(0,0)$ is:

$$V_{total}=\sum _{n=1}^{\infty }-\frac{Gm}{r_n}=-Gm\left(\frac{1}{1}+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\dots \right)$$

This is a geometric series with first term $a=1$ and common ratio $r=\frac{1}{2}$.

The sum of an infinite geometric series is $S=\frac{a}{1-r}$ for $|r|<1$.

$$S=\frac{1}{1-1/2}=\frac{1}{1/2}=2$$

Therefore, $V_{total}=-Gm(2)=-2Gm$.

Given $V_{total}=-YGm$, comparing this, we find $Y=2$.