In the Hartnell governor shown in Fig. 2.71, the stiffness of the spring is \(10^{4} \mathrm{~N} / \mathrm{m}\) and the weight of each ball is \(25 \mathrm{~N}\). The length of the ball arm is \(20 \mathrm{~cm}\), and that of the sleeve arm is \(12 \mathrm{~cm}\). The distance between the axis of rotation and the pivot of the bell crank lever is \(16 \mathrm{~cm}\). The spring is compressed by \(1 \mathrm{~cm}\) when the ball arm is vertical. Find

(a) the speed of the governor at which the ball arm remains vertical and

(b) the natural frequency of vibration for small displacements about the vertical position of the ball arms.

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Spring -Ball 12 cm- 16 cm FIGURE 2.71 Hartnell governor. + 20 cm