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Question

(C) The gravitational potential $V$ at a point due to a mass element $dm$ at a distance $r$ is given by $dV = -\frac{G dm}{r}$.For an arc of mass $M$

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$(V_O)_1 = (V_O)_2 = (V_O)_3$

Vedclass · Answer

(C) The gravitational potential $V$ at a point due to a mass element $dm$ at a distance $r$ is given by $dV = -\frac{G dm}{r}$.For an arc of mass $M$ and radius $R$,every mass element $dm$ of the arc is at the same distance $R$ from the center $O$.Therefore,the gravitational potential at the center $O$ is the integral of the potential due to all mass elements: $V_O = \int dV = \int -\frac{G dm}{R} = -\frac{G}{R} \int dm = -\frac{GM}{R}$.Since the potential depends only on the total mass $M$ and the distance $R$,and both are identical for all three arcs,the gravitational potential at the center is the same for all three cases.Thus,$(V_O)_1 = (V_O)_2 = (V_O)_3$.

Compare the gravitational potential at the center $O$ for the given arcs,each having mass $M$ and radius $R$.

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