Question

A uniform plank of mass $m$ free to move in horizontal direction only is placed on the top a cylinder of mass $2m$ and radius $R$.The plank is attached to a fixed wall by means of a light spring of spring constant $K$. There is no slipping between the cylinder and the plank and between cylinder and ground. The time period of small oscillations of system is

• A. $2\pi \sqrt{\frac{7m}{4k}}$
• B. $2\pi \sqrt{\frac{3m}{k}}$
• C. $\sqrt{\frac{k}{3m}}$
• D. $\frac{1}{2\pi }\sqrt{\frac{k}{3m}}$

Answer 1

The correct option is A $2\pi \sqrt{\frac{7m}{4k}}$

Rotate the sphere by an angle $\theta$ as shown.

Elongation in spring$=2Rsin\theta =2R\theta$

Net torque about point $\theta$

$\tau =2R\times 2R\theta \times K\tau =4R^{2}K\times \theta$

Compare with $\tau =4R^{2}K$

Moment of inertia of the system is

$I=\frac{2m{R}^2}{2}+M\times {\left(2R\right)}^2+2m{R}^{2}I=7m{R}^{2}T=2\pi \sqrt{\frac{I}{C}}T=2\pi \sqrt{\frac{7m{R}^2}{4R^{2}K}}T=2\pi \sqrt{\frac{7m}{4K}}$