Question

A cylinder of radius R is rolling with velocity of centre of mass $v=\frac{\omega R}{2}$ towards right. Keeping the centre of the cylinder as the origin, Find the coordinates of the point with zero net velocity.

• A. $\frac{R}{2},0$
• B. $0,\frac{-R}{2}$
• C. $\frac{R}{\sqrt{2}},\frac{R}{\sqrt{2}}$
• D. $0,\frac{R}{\sqrt{2}}$

Answer 1

The correct option is B

$0,\frac{-R}{2}$

Let point p which has net zero velocity be at an angle $\theta$ as shown, and at a distance r from the centre.

Now ${\overrightarrow{v}}_{p,0}={\overrightarrow{v}}_p-{\overrightarrow{v}}_0$

$\Rightarrow$ Velocity of p with respect to ground,${\overrightarrow{v}}_p={\overrightarrow{v}}_{p,0}+{\overrightarrow{v}}_0$

$\Rightarrow {\overrightarrow{v}}_p=(-\omega rsin\theta +V)\hat{i}+\omega rcos\theta \hat{j}$

=0(zero net velocity)

$\Rightarrow {\omega }_{r}cos\theta =0\Rightarrow \theta ={90}^{\circ },andv={\omega }_{r}sin\theta$,

$\Rightarrow butv=\frac{\omega R}{2}\therefore r=\frac{R}{2}$

$\Rightarrow coordinatesofpare(0,-\frac{R}{2})$