Vector Calculus | Part IA, 2004

Let S1S_{1} be the 3 -dimensional sphere of radius 1 centred at (0,0,0),S2(0,0,0), S_{2} be the sphere of radius 12\frac{1}{2} centred at (12,0,0)\left(\frac{1}{2}, 0,0\right) and S3S_{3} be the sphere of radius 14\frac{1}{4} centred at (14,0,0)\left(\frac{-1}{4}, 0,0\right). The eccentrically shaped planet Zog is composed of rock of uniform density ρ\rho occupying the region within S1S_{1} and outside S2S_{2} and S3S_{3}. The regions inside S2S_{2} and S3S_{3} are empty. Give an expression for Zog's gravitational potential at a general coordinate x\mathbf{x} that is outside S1S_{1}. Is there a point in the interior of S3S_{3} where a test particle would remain stably at rest? Justify your answer.

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