Question

Consider the general motion of a wheel of radius $2\text{m}$ which can be viewed as pure translation with the velocity $\overrightarrow{v}$ and pure rotation about $\text{O}$ with angular velocity $5\text{rad/s}$. Which of the following is incorrect?

• A. ${\overrightarrow{V}}_{AO}=-10\hat{i}$
• B. ${\overrightarrow{V}}_{BO}=-10\hat{j}$
• C. ${\overrightarrow{V}}_{CO}=10\hat{i}$
• D. ${\overrightarrow{V}}_{DO}=-10\hat{j}$

Answer 1

The correct option is D ${\overrightarrow{V}}_{DO}=-10\hat{j}$

As we know,
${\overrightarrow{V}}_{PQ}=\overrightarrow{\omega }\times {\overrightarrow{r}}_{PQ}$
So here, ${\overrightarrow{V}}_{AO}=\omega \times {\overrightarrow{r}}_{AO}$
$=5(-\hat{k})\times 2(-\hat{j})=-10\hat{i}$


As we can see, the direction of velocity w.r.t $\text{O}$ will be in negative $x$ direction.
Similarly
${\overrightarrow{V}}_{BO}=\overrightarrow{\omega }\times {\overrightarrow{r}}_{BO}=5(-\hat{k})\times 2\left(\hat{i}\right)=(-5\times 2)(\hat{k}\times \hat{i})$
$=10\text{m/s}(-\hat{j})=-10\hat{j}$
${\overrightarrow{V}}_{CO}=\omega \times {\overrightarrow{r}}_{CO}=5(-\hat{k})\times 2\left(\hat{j}\right)=-10(\hat{k}\times \hat{j})$
$=10\left(\hat{i}\right)(\because \hat{k}\times \hat{j}=-\hat{i})$
${\overrightarrow{V}}_{DO}=\omega \times {\overrightarrow{r}}_{DO}=5(-\hat{k})\times 2(-\hat{i})=10\times (\hat{k}\times \hat{i})$
$=10\left(\hat{j}\right)$
$[\because \hat{k}\times \hat{i}=\hat{j}]$
Hence, option $D$ is incorrect.