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# A disc of mass 'm' and radius 'r' is present on a rough surface . Find the angular frequency (w) of the S.H.M . Consider pure rolling of the disc

$(a)\;\sqrt{\large\frac{8K_{1}}{3m}}\qquad(b)\;\sqrt{\large\frac{4 K_{1}}{3m}}\qquad(c)\;\sqrt{\large\frac{K_{1}}{3m}}\qquad(d)\;None\;of\;these$

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Answer : (c) $\;\sqrt{\large\frac{K_{1}}{3m}}$
Explanation :
Let the disc roll on the surface and the centre be moved by a distance x .
Therefore , The top most point of the disc will move by a distance of 2x .
Therefore , Considering free body diagram of the systems
$f$= (friction)
$T=2 K_{1}x$
$T-f=ma$
Considering torque about centre
$Tr+fr=I\;\alpha$
$(T+f)\;r=\large\frac{mr^2}{2} \times \alpha$
As the disc rolls a=$\;r \alpha$
$T+f=\large\frac{ma}{2}$
$T-f=ma$
$2T=\large\frac{3ma}{2} \; => T=\large\frac{3ma}{4}$
$2K_{1}x=\large\frac{3ma}{4}$
$w=x\;(\large\frac{8K_{1}}{3m})$
$w=\sqrt{\large\frac{8K_{1}}{3m}}$
answered Mar 10, 2014 by
edited Mar 10, 2014 by yamini.v

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