Question

A wheel of radius $1\text{m}$ is performing combined translational and rotational motion on a frictionless horizontal surface with an angular velocity $4\text{rad/s}$ and translational velocity of its COM is $V_0=10\text{m/s}$ as shown in figure. Find the magnitude of velocity of point $P$, if the line $OP$ makes angle ${30}^{\circ }$ with the horizontal.

• A. $10\text{m/s}$
• B. $4\text{m/s}$
• C. $2\sqrt{39}\text{m/s}$
• D. $4\sqrt{39}\text{m/s}$

Answer 1

The correct option is C $2\sqrt{39}\text{m/s}$


Given,
Radius of wheel $R=1\text{m}$
Angular velocity of wheel $\omega =4\text{rad/s}$
Let velocity at point $P=V_P$
Then, $\overrightarrow{V_P}=\overrightarrow{V_0}+\overrightarrow{R}\times \overrightarrow{\omega }$
Here, $V_0=10\text{m/s}$
and $R\omega =1\times 4=4\text{m/s}$
So, $\left|V_P\right|=\sqrt{\left(V_0^2\right)+(R\omega {)}^2+2V_0(R\omega )cos\theta }$
Here $\theta$ is the angle between $V_0$ & $\omega R$ which is $90-30={60}^{\circ }$, so $\cos {60}^{\circ }=1/2$
$\left|V_P\right|=\sqrt{(10{)}^2+(4{)}^2+2\times 10\times 4\times 1/2}$
$\left|V_P\right|=2\sqrt{39}\text{m/s}$