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

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=$\;r \alpha$

$T+f=\large\frac{ma}{2}$

$T-f=ma$

$2T=\large\frac{3ma}{2} \; => T=\large\frac{3ma}{4}$

$2K_{1}x=\large\frac{3ma}{4}$

$w=x\;(\large\frac{8K_{1}}{3m})$

$w=\sqrt{\large\frac{8K_{1}}{3m}}$

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