Question

How do you know that the distance between centre of mass and the centre of the circle is $\frac{2r}{\pi }$

Answer 1

Using parallel Axis Theorem
$IO=ICM+M{d}^2$
Where IO=Moment of inertia about centre $M{R}^2$
and ICM=Moment of inertia about centre of mass
$d=\frac{2R}{\pi }$ (distance between centre of mass and centre)
$ICM=IO-M{\left(\frac{2R}{\pi }\right)}^2$
$=M{R}^2\left[\frac{1-4}{{\pi }^2}\right]$