Q$4-(25$ pts$.)$ Consider a rigid body that is made of seven identical circular rings. Each ring has a radius $R$ and mass $M$ with uniform mass density. This rigid body is free to rotate about an axis perpendicular to the plane of figure$/$paper and passing through point $P.$ It is initially held at rest in the vertical position as shown in the figure. It is then given a negligible push and set into counterclockwise rotation at $t=0.$ The gravitational acceleration $g>0$ is downwards as usual.

$($a$)(5$ pts$)$ Using integration $($sketch it$),$ calculate the moment of inertia $I_{\text{r}\text{i}\text{n}\text{g}}$ of a single circular ring about a perpendicular axis through its center of mass.

$($b$)(10$ pts$)$ Find the moment of inertia $I_P$ of this rigid body with respect to the pivot P$.$ Express it in terms of $I_{\text{r}\text{i}\text{n}\text{g}},M$ and $R.$

$($c$)(10$ pts$)$ Find the linear velocity of the center of mass of this rigid body when its angular displacement $($from its initial position with respect to the rotation axis$)$ becomes $5\frac{}{3}.$ Express it in terms of $I_P,g,M$ and $R.$