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

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Aicharya · Accepted Answer

$ 	ext { } \begin{aligned} W=\Delta U & =U_f-U_i=U_{\infty}-U_p \ & =-U_P \ & =-m V_P \ & =-V_P \quad(	ext { as } m=1) \end{aligned} $ Potential at point $P$ will be obtained by integration as given below Let $d M$ be the mass of small ring as shown $ \begin{aligned} d M & =\frac{M}{\pi(4 R)^2-\pi(3 R)^2}(2 \pi r) d r \ & =\frac{2 M r d r}{7 R^2} \ d V_P & =-\frac{G \cdot d M}{\sqrt{16 R^2+r^2}} \ & =-\frac{2 G M}{7 R^2} \int_{3 R}^{4 R} \frac{r}{\sqrt{16 R^2+r^2}} \cdot d r \ & =-\frac{2 G M}{7 R}(4 \sqrt{2}-5) \ 	herefore \quad W & =+\frac{2 G M}{7 R}(4 \sqrt{2}-5) \end{aligned} $ $	herefore$ correct option is (a)

A thin uniform annular disc (see figure) of mass $M$ has outer radius $4 R$ and inner radius $3 R$. The work…

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