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# A smooth narrow tube in the form of an arc AB of a circle of center O and radius r is fixed so that A is vertically above O and OB is horizontal. Particles P and Q of mass m and 2m respectively. with a light in extensible string connecting them (length $\large\frac{\pi\;r}{2}$) are placed inside the tube with P at A and Q at B. and released from rest. Assuming that the string remains during motion, find speed of particles when P reaches B

$(a)\;\sqrt {(1+\pi) gr} \quad (b)\; \sqrt {\frac{2}{3}(1+\pi) gr} \quad (c)\;\sqrt {2 \pi gr} \quad (d)\; \sqrt {\frac{1}{3} \pi g r}$

and B will loose height $=\large\frac{\pi r}{2}$, length of string
$\therefore mgr+2mg \bigg(\large\frac{\pi r}{2}\bigg)=\frac{1}{2}$$(m+2m)v^2$
$v= \sqrt {\large\frac{2}{3} \normalsize (1+\pi) gr}$