Question

A uniform dise having radius R is spin with an angular velocity ${\omega }_2$ and simulteneously projected with a linear velocity v towards right on a plank while the move towards right with a constant velocity 2v If the disc initially rolls without slipping on the plank then angular velocity $\stackrel{\omega }{\to }$

• A. $\frac{v}{R}\hat{k}$
• B. $\frac{v}{R}(-\hat{k})$
• C. $\frac{3v}{R}\hat{k}$
• D. $\frac{3v}{R}\left(\widehat{-k}\right)$

Answer 1

The correct option is C $\frac{3v}{R}\hat{k}$

$\begin{array}{c}\text{Given is a disc of radius}R\text{.}\\ \text{Having linear velocity}=v\text{(towards left).}\\ \text{and, velocity of}\mathrm{Plank}=2v\text{(towards right)}\end{array}$

$\text{Here slipping must be avoided at point of contact.}$

$\begin{array}{cc}\text{Thus,}{\overrightarrow{V}}_p={\overrightarrow{V}}_Q\Rightarrow -v\hat{i}+\omega R\hat{i}& =2v\hat{i}\\ \omega =\frac{3v}{R}& \left(\hat{k}\right)\end{array}$