Question

A disc of mass $M$ is attached to a horizontal massless spring of force constant $K$ so that it can roll without slipping along a horizontal surface. If the disc is pushed a little towards right and then released, its center of mass executes SHM with a period of

• A. $2\pi$ $\sqrt{\frac{M}{K}}$
• B. $2\pi$ $\sqrt{\frac{3M}{K}}$
• C. $2\pi$ $\sqrt{\frac{M}{2K}}$
• D. $2\pi$ $\sqrt{\frac{3M}{2K}}$

Answer 1

Answer: (D)

$2\pi$ $\sqrt{\frac{3M}{2K}}$

Spring exerts a force equal to $Kx$ on the center of mass of the disc.

Friction acts on opposite direction to cause rolling without slipping.

$Kx-f=M{a}_{cm}$

$fR=\frac{1}{2}M{R}^2\alpha$

Solving gives $f=\frac{Kx}{3}$

Hence restoring force acting on sphere=$kx-f=\frac{2Kx}{3}$

Hence $\omega =\sqrt{\frac{2K}{3M}}$

Hence time period of oscillation=$\frac{2\pi }{\omega }=2\pi \sqrt{\frac{3M}{2K}}$