A thin uniform annular disc (see figure) of mass M has outer radius 4R and inner radius 3R. The work required to take a unit mass from point P on its axis to infinity is

Engineering

Physics

Gravitational Potential Energy

Question

$\frac{2 GM}{7 R} (4 \sqrt{2} - 5)$

$\frac{2 GM}{5 R} (\sqrt{2} - 1)$

$\frac{GM}{4 R}$

$- \frac{2 GM}{7 R} (4 \sqrt{2} - 5)$

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$W = + \int dv = + \int \frac{Gdm}{r^{2}}$

$= - \int_{3 R}^{4 R} \frac{G \times \frac{M}{π (16 R^{2} - 9 R^{2})} \times 2 πxdx}{\sqrt{x^{2} + 16 R^{2}}}$

$= \frac{2 GM}{7 R^{2}} [4 \sqrt{2} - 5]$