Question

A uniform disc is spun with an angular velocity $\vec{\omega}$ and simultaneously projected with a linear velocity $v$ towards left on a plank, while the plank moves towards right with a constant velocity $2\, v$. If the disc rolls without sliding on the plank just after its spinning, the magnitude of $\vec{\omega}$ (in radian/s) is___.
(Take $v=3\, m / s , R=1\, m$ ).

Answer 1

For pure rolling of the disc on the plank,

$\vec{v}_{P}=\vec{v}_{Q}$ where $\vec{v}_{Q}=2 v \hat{i}$
This gives $\vec{v}_{P}=2 \vec{v}$
Since, $\vec{v}_{P}=\vec{v}_{P C}+\vec{v}_{C}$
Substituting $\vec{v}_{P}=2 v \hat{i}$ and $\vec{v}_{C}=-v \hat{i}$
We have $\vec{v}_{P C}=3 v \hat{i}$
Then using the formula $\vec{v}_{P C}=\vec{\omega}_{P C} \times \vec{r}_{P C}$
Where $\vec{r}_{P C}=-R \hat{j}$
We can find $\vec{\omega}_{P C}(=\vec{\omega})$
$=\frac{3 v}{R} \hat{k}=\frac{3 \times 3}{1} \hat{k}=9 \hat{k}\, rad / \sec$