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# Figure shows a rough track, a portion of which is in the form of a cylinder of radius R. with what minimum linear speed should a sphere of radius r be set rolling on the horizontal part so that it completes goes round the circle on the cylindrical part?

$\begin {array} {1 1} (a)\;\sqrt {\frac{27}{7}(R-r)} & \quad (b)\;\sqrt{\frac{27}{5}g(R-r)} \\ (c)\;\sqrt {\frac{7}{5}g(R+r)} & \quad (d)\;\sqrt {\frac{17}{5}g(R-r} \end {array}$

At the point $P, N=0$
$mg= \large\frac{mv^2p}{(R-v)}$
$V^2_p =(R-r) g$
$\large\frac{1}{2}$$mV_A^2 \bigg[1+ \large\frac{k^2}{R^2}\bigg]=\large\frac{1}{2}$$mV_p^2+mg2( R-r)$
$\qquad= \large\frac{1} {2}$$m \bigg[1+\large\frac{k^2}{R^2}\bigg]$$V_p^2+2 mg (R-r)$
$\qquad= \large\frac{1}{2}$$m \bigg[1+\large\frac{2}{5}\bigg]$$ V_p^2+2mg (R-r)$
$\qquad= \large\frac{7}{10} $$m (R-r)g+2mg(R-v) \frac{-1}{2}mV_A^2 \bigg[\frac{7}{5}\bigg]= \large\frac{27}{10}$$ m(R-r)g$
$V_A= \sqrt {\large\frac{27}{7} \normalsize (R-r)g}$
edited Jun 24, 2014

+1 vote