Question

A uniform thin cylindrical disk of mass $M$ and radius $R$ is attached to two identical massless springs of spring constant $k$ which are fixed to the wall as shown in the figure. The springs are attached to the axle of the disk symmetrically on either side at a distance $d$ from its centre. The axle is massless and both the springs and the axle are in horizontal plane. The unstretched length of each spring is $L$. The disk is initially at its equilibrium position with its centre of mass $(CM)$ at a distance $L$ from the wall. The disk rolls without slipping with velocity $\overrightarrow{ V }_{0}= V _{0} \hat{ i }$. The coefficient of friction is $\mu .$

The maximum value of $V _{0}$ for which the disk will roll without slipping is

• A. $\mu g \sqrt{\frac{M}{k}}$
• B. $\mu g \sqrt{\frac{ M }{2 k }}$
• C. $\mu g \sqrt{\frac{3 M}{k}}$
• D. $\mu g \sqrt{\frac{5 M}{2 k}}$

Answer 1

Answer: (C)

$2 kx - f _{\max }= ma$
$2 kx \cdot r = I _{ p } \alpha$
$f _{\max }=\mu mg$
$\Rightarrow x =\frac{3}{2} \frac{\mu mg }{ k }$
$\Rightarrow \frac{1}{2}(2 k ) x ^{2}=\frac{1}{2} I _{ p } \omega^{2}$
$\Rightarrow \gamma=\mu g \sqrt{\frac{3 m }{ k }}$