An infinite number of bodies,each of mass 2 , | vedclass

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(C) The gravitational potential $V$ at a point due to a point mass $m$ at a distance $r$ is given by $V = -\frac{Gm}{r}$.For a system of multiple mass

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$-4G$

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(C) The gravitational potential $V$ at a point due to a point mass $m$ at a distance $r$ is given by $V = -\frac{Gm}{r}$.For a system of multiple masses,the total potential is the algebraic sum of the potentials due to each mass.Here,each body has a mass $m = 2 \, kg$ and they are located at distances $r_1 = 1 \, m, r_2 = 2 \, m, r_3 = 4 \, m, r_4 = 8 \, m, \dots$ from the origin.The total gravitational potential $V$ at the origin is:$V = -\frac{G(2)}{1} - \frac{G(2)}{2} - \frac{G(2)}{4} - \frac{G(2)}{8} - \dots$$V = -2G \left( 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots \right)$The term in the bracket is an infinite geometric series with the first term $a = 1$ and common ratio $r = \frac{1}{2}$.The sum of an infinite geometric series is $S = \frac{a}{1 - r}$.$S = \frac{1}{1 - 1/2} = \frac{1}{1/2} = 2$.Therefore,$V = -2G(2) = -4G$.

An infinite number of bodies,each of mass $2 \, kg$,are situated on the $x-$axis at distances $1 \, m, 2 \, m, 4 \, m, 8 \, m, \dots$ respectively,from the origin. The resulting gravitational potential due to this system at the origin will be:

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