Homework H$\mathrm{.}4.$R

Given: A particle P $,$ having a mass of $m,$ is free to slide on a smooth, lightweight bar. The bar is free to rotate in a horizontal plane about a vertical shaft passing through O$.$ A spring, having a stiffness $k$ and unstretched length $R_0,$ is connected between P and O $.$ The spring is compressed to half of its unstretched length and released when the bar has a rotational speed of ${}_1,$ as shown in the figure below left.

Find: For the position when $R=R_0($shown in the figure below right$)$:

$($a$)$ Determine the rotation rate ${}_2$ of the bar;

$($b$)$ Determine the value of $R^{}.$

HORIZONTAL PLANE

Leave your final answers in terms of $m,k,R_0$ and ${}_1.$