Question

A wheel is performing combined translational and rotational motion on a frictionless horizontal surface on a truck which is moving with a constant linear velocity $v=10\text{m/s}$. Centre of the wheel is also moving with a constant linear velocity $v=10\text{m/s}$ with respect to ground. The wheel is rotating with angular velocity $\omega =2\text{rad/s}$. Find velocity of point $A$ (Which is the topmost point of wheel as shown in the figure) with respect to ground. Radius of the wheel is $1\text{m}$.

• A. $10\hat{i}$
• B. $8\hat{i}$
• C. $6\hat{i}$
• D. zero

Answer 1

The correct option is B $8\hat{i}$

With respect to ground, the wheel is performing (translation + rotational) motion.
Translational velocity of wheel $\left(V_{\text{com}}\right)=$ velocity of truck.
i.e $V_{\text{com}}$ of wheel $=v=10\text{m/s}$
As we know,
$\overrightarrow{V_A}={\overrightarrow{V}}_{\text{com}}+\overrightarrow{\omega }\times {\overrightarrow{r}}_{AO}$
$O$ is the centre of wheel and $A$ is the highest point on the wheel.
$\overrightarrow{V_A}=10\hat{i}+2\left(\hat{k}\right)\times 1\left(\hat{j}\right)$
$\overrightarrow{V_A}=10\hat{i}+2(-\hat{i})[\because \hat{k}\times \hat{j}=-\hat{i}]$
$\overrightarrow{V_A}=8\hat{i}$