Paragraph: A uniform thin cylindrical disk of mass $M$ and radius $R$ is attached to two identical massless…
Paragraph:
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 a horizontal plane. The unstretched length of each spring is $L$. The disk is initially at its equilibrium position with its centre of mass $(C M)$ at a distance Lfrom the wall. The disk rolls without slipping with velocity $\mathbf{v}_0=v_0 \hat{\mathbf{i}}$ The coefficient of friction is $\mu$. Question:
The maximum value of $v_0$ for which the disk will roll without slipping is
$\mu g \sqrt{\frac{M}{k}}$
$\mu g \sqrt{\frac{M}{2 k}}$
$\mu g \sqrt{\frac{3 M}{k}}$
$\mu g \sqrt{\frac{5 M}{2 k}}$
Solution
In case of pure rolling, mechanical energy will remain conserved.
$
\begin{array}{rlrl}
& \therefore & \frac{1}{2} M v_0^2+\frac{1}{2}\left(\frac{1}{2} M R^2\right)\left(\frac{v_0}{R}\right)^2 & =2\left[\frac{1}{2} k x_{\max }^2\right] \\
& \therefore & x_{\max } & =\sqrt{\frac{3 M}{4 k}} v_0 \\
& \text { As } & & =\frac{2 k x}{3} \\
& f_{\max } & =\mu M g=\frac{2 k x_{\max }}{3}=\frac{2 k}{3} \sqrt{\frac{3 M}{4 k}} v_0 \\
& v_0 & =\mu g \sqrt{\frac{3 M}{k}}
\end{array}
$
$\therefore$ correct option is (c).
;
