$2.$ The mechanical flyball governor operates with a vertical shaft O$-$O$.$ As the shaft's rotation speed is increased, the rotational radius of the balls $($each of mass $\backslash ($ m $\backslash ))$ tends to increase, causing the body $"$ A $"$ of mass $\backslash ($ M $\backslash )$ to be lifted up by the collar B$.$ Suppose the system begins at rest for a given $\backslash (\backslash$beta$=\backslash$beta$\_\{0\}\backslash )$ and a torque $\backslash (\backslash$tau $\backslash )$ is then applied to the shaft and held fixed. When the body A is "fully elevated" to some given $\backslash (\backslash$beta$=\backslash$beta$\_\{$f$\}\backslash ),$ it is observed to be spinning with angular speed $\backslash (\backslash$omega$\_\{$f$\}\backslash ).$ Through what total angle $\backslash (\backslash$theta $\backslash )$ does the shaft O$-$O rotate at the time when A becomes fully elevated? Neglect the mass of the collar B$,$ and assume a constant linear density $\backslash (\backslash$rho $\backslash )$ with units $[$m$/$L$]$ for the arms. See the figure. Hint: The inertia about a vertical axis for a slanted bar can be deduced from the inertia tensor for a bar, which we know from lecture.