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 centre of mass of the disk undergoes simple harmonic motion with angular frequency $\omega$ equal to
  1. $\sqrt{\frac{k}{M}}$
  2. $\sqrt{\frac{2 k}{M}}$
  3. $\sqrt{\frac{2 k}{3 M}}$
  4. $\sqrt{\frac{4 k}{3 M}}$

Solution

$ \begin{aligned} & F_{\text {net }}=-\left(\frac{4 k}{3}\right) \cdot x \\ & \therefore a=\frac{F_{\text {net }}}{M}=-\left(\frac{4 k}{3 M}\right) x=-\omega^2 x \\ & \therefore \omega=\sqrt{\frac{4 k}{3 M}} \end{aligned} $ $\therefore$ correct option is (d). `

Asked in: JEE Advanced 2008 (Paper 2)

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