3.II.11C

Let $S_{1}$ be the 3 -dimensional sphere of radius 1 centred at $(0,0,0), S_{2}$ be the sphere of radius $\frac{1}{2}$ centred at $\left(\frac{1}{2}, 0,0\right)$ and $S_{3}$ be the sphere of radius $\frac{1}{4}$ centred at $\left(\frac{-1}{4}, 0,0\right)$. The eccentrically shaped planet Zog is composed of rock of uniform density $\rho$ occupying the region within $S_{1}$ and outside $S_{2}$ and $S_{3}$. The regions inside $S_{2}$ and $S_{3}$ are empty. Give an expression for Zog's gravitational potential at a general coordinate $\mathbf{x}$ that is outside $S_{1}$. Is there a point in the interior of $S_{3}$ where a test particle would remain stably at rest? Justify your answer.

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