A dog is walking M in is rolling on a circular disc

Dear Student,

Here,angular momentum of the system is zero.
Moment of inertia of the dog is,
 $I=m{\left(\frac{R}{2}\right)}^2$  (since dog is moving in radius $\frac{R}{2}$)
and angular frequency ,${\omega }_d$= n
so angular momentum is,
$L=I{\omega }_d=m\frac{R^2}{4}n$
Now the moment of inertia of the disc is $\frac{1}{2}M{R}^2$.
Suppose the disc is rotating with a angular velocity is $\omega$. Therefore angular momentum is
$L_{disc}=\frac{1}{2}M{R}^2\omega$.
Now,
$L_{disc}=L$
$$\begin{gathered} \frac{1}{2}M{R}^2\omega =\frac{1}{4}m{R}^{2}n \\ \Rightarrow \\ M\omega =\frac{1}{2}\times mn \\ \Rightarrow \\ \omega \mathbf{=}\frac{\mathbf{m}\mathbf{n}}{\mathbf{2}\mathbf{M}}\mathbf{,}\mathbf{ }\mathit{O}\mathit{p}\mathit{t}\mathit{i}\mathit{o}\mathit{n}\mathbf{ }\mathbf{2} \end{gathered}$$

Regards.