Gravitation Ques 5
- 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
(2010)
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Answer:
Correct Answer: 5.( a )
Solution:
- $\begin{aligned} & W=\Delta U=U_f-U_i=U_{\infty}-U_p=-U_p=-m V_p \\ &=-V_p \quad \\ &\text { (as } m=1)\end{aligned}$
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 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}} d r \\ & =-\frac{2 G M}{7 R}(4 \sqrt{2}-5) \\ \therefore \quad W & =+\frac{2 G M}{7 R}(4 \sqrt{2}-5) \end{aligned} $
BETA
