In some damping systems, a circular disk immersed in oil is used as a damper, as shown in Fig. P2-125. Show that the damping torque is proportional to angular speed in accordance with the relation \(\mathrm{T}_{\text {damping }}=C \omega\) where \(C=\) \(0.5 \pi \mu(1 / a+1 /b) R^{4}\). Assume linear velocity profiles on both sides of the disk and neglect the tip effects.

FIGURE P2-125

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a R b Disk Damping oil