Question

A uniform disc of mass m and radius R is pivoted at its centre O with its plane vertical as shown in figure, A circular portion of disc of radius $\frac{R}{2}$ is removed from it. The time period of small oscillations of remaining portion about O is

• A. $3\pi \sqrt{\frac{R}{g}}$
• B. $\pi \sqrt{\frac{13R}{g}}$
• C. $2\pi \sqrt{\frac{39R}{16g}}$
• D. $2\pi \sqrt{\frac{7R}{6g}}$

Answer 1

The correct option is B $\pi \sqrt{\frac{13R}{g}}$

$T=2\pi \sqrt{\frac{I}{mg{y}_{cm}}}I=\frac{M{R}^2}{2}-[\frac{1}{2}\frac{M}{4}{\left(\frac{R}{2}\right)}^2+\frac{M}{4}{\left(\frac{R}{2}\right)}^2]=\frac{13M{R}^2}{32}\Rightarrow Y_{cm}=\frac{R}{6}andm=\frac{3M}{4}T=2\pi \sqrt{\frac{13}{4}\frac{R}{g}}$