Question

A circular disc of mass $m$ and radius $R$ is set into motion on a horizontal floor with a linear speed v in the forward direction and an angular speed $\omega =v/R$ in clockwise direction as shown in figure. The magnitude of the total angular momentum of the disc about bottommost point $O$ of the disc is $\alpha mvR$. Find $\alpha$.
Diagram

Answer 1

Draw a diagram of given problem


Calculate the total angular momentum

Given,
Mass $=m$
Angular speed $=\omega =\frac{v}{R}$
Angular momentum $=\alpha mvR$
Therefore, for first diagram,
$\overrightarrow{L}={\overrightarrow{L}}_{cm}+m({\overrightarrow{r}}_0\times {\overrightarrow{v}}_0)$
${\overrightarrow{L}}_{cm}=I_{\omega }$
$=\left(\frac{1}{2}m{R}^2\right)\left(\frac{v}{R}\right)$
$=\frac{1}{2}mvR$
And,
$m({\overrightarrow{r}}_0\times {\overrightarrow{v}}_0)=mRv$
Hence,
$\left|\overrightarrow{L}\right|=\frac{1}{2}mvR+mvR$
$\left|\overrightarrow{L}\right|=\frac{3}{2}mvR$

Final answer$:1.5$