The system shown is released from rest with the lightweight slender bar \(O A\) in the vertical position shown. The torsional spring at \(O\) is undeflected in the initial position and exerts a restoring moment of magnitude \(k_{T} \theta\) on the bar, where \(\theta\) is the counterclockwise angular deflection of the bar. The string \(S\) is attached to point \(C\) of the bar and slips without friction through a vertical hole in the support surface. For the values \(m_{A}=2 \mathrm{~kg}\), \(m_{B}=4 \mathrm{~kg}, L=0.5 \mathrm{~m}\), and \(k_{T}=13 \mathrm{~N} \cdot \mathrm{m} / \mathrm{rad}:\)

(a) Determine the speed \(v_{A}\) of particle \(A\) when \(\theta\) reaches \(90^{\circ}\).

(b) \(\operatorname{Plot} v_{A}\) as a function of \(\theta\) over the range \(0 \leq \theta \leq 90^{\circ}\). Identify the maximum value of \(v_{A}\) and the value of \(\theta\) at which this maximum occurs.

B mB S MA A L kT 2 L 2