Question

A disc of radius $r$ has linear velocity $V$ and angular velocity $\omega$ as shown in figure. If point $B$ is at a distance $\frac{r}{2}$ from centre, then find the velocity of point $A$ w.r.t point $B$ $\left({\overrightarrow{V}}_{AB}\right)$.

• A. $\omega r$
• B. $\frac{\omega r}{3}$
• C. ${\omega }^{2}r$
• D. $\frac{\omega r}{2}$

Answer 1

The correct option is D $\frac{\omega r}{2}$

As the disc rolls on the surface the
${\overrightarrow{V}}_A={\overrightarrow{V}}_{COM}+(\overrightarrow{\omega }\times {\overrightarrow{r}}_{AO})\dots \dots \left(i\right)$
Similarly,
${\overrightarrow{V}}_B={\overrightarrow{V}}_{COM}+(\overrightarrow{\omega }\times {\overrightarrow{r}}_{BO})\dots \dots \left(ii\right)$
${\overrightarrow{V}}_{AB}={\overrightarrow{V}}_A-{\overrightarrow{V}}_B$
So, using equation $\left(i\right)$ and $\left(ii\right)$, we get
${\overrightarrow{V}}_{AB}=\overrightarrow{\omega }\times ({\overrightarrow{r}}_{AO}-{\overrightarrow{r}}_{BO})$
$\Rightarrow {\overrightarrow{V}}_{AB}=\overrightarrow{\omega }\times {\overrightarrow{r}}_{AB}=\frac{\omega r}{2}$