$4\mathrm{.}22$ A rigid and uniform bar undergoing rotational motions in the vertical plane is

restrained by a torsional spring of stiffness $k_t$ at the pivot point $O,$ as shown in

Figure E$4\mathrm{.}22.$ The bar has a mass $m$ and a length $l.$ Determine what $k_t$ should be$,$ so

that small oscillations about the upright position of the bar $($i$.$e$.,=0),$ are stable.

igure E$4\mathrm{.}22.$