Question

Find the incorrect option for the general rotation of a wheel of radius $r$ which can be viewed as pure translation with velocity $\overrightarrow{V_0}$ and pure rotation about $\text{O}$ with $\overrightarrow{\omega }$.

• A. $\overrightarrow{V_A}=\overrightarrow{V_0}+\overrightarrow{\omega }\times {\overrightarrow{r}}_{AO}$
• B. $\overrightarrow{V_B}=\overrightarrow{V_0}+\overrightarrow{\omega }\times {\overrightarrow{r}}_{CO}$
• C. ${\overrightarrow{V}}_{AO}=\overrightarrow{\omega }\times {\overrightarrow{r}}_{AO}$
• D. ${\overrightarrow{V}}_{AO}=\overrightarrow{V_0}+\overrightarrow{\omega }\times {\overrightarrow{r}}_{AO}$

Answer 1

Answer: (B), (D)

${\overrightarrow{V}}_{AO}=\overrightarrow{V_0}+\overrightarrow{\omega }\times {\overrightarrow{r}}_{AO}$

As we know,
$\overrightarrow{V_A}=\overrightarrow{V_O}+\overrightarrow{\omega }\times {\overrightarrow{r}}_{AO}$
$=\overrightarrow{V_0}+\overrightarrow{\omega }\times {\overrightarrow{r}}_{AO}$
and $\overrightarrow{V_B}$ will be
$\overrightarrow{V_B}=\overrightarrow{V_0}+\overrightarrow{\omega }\times {\overrightarrow{r}}_{BO}$
(From the equation of pure rolling, since, $\text{A},\text{B}$ and $\text{O}$ are moving with the same translation velocity $\overrightarrow{V_0}$)

$\Rightarrow {\overrightarrow{V}}_{AO}={\overrightarrow{V}}_A-{\overrightarrow{V}}_O$
$=\overrightarrow{\omega }\times {\overrightarrow{r}}_{AO}$
Hence, B & D are incorrect.