Randall D. Knight — Rotation of a Rigid Body: Problem 85

Objects that rotate in air or water experience a torque due to drag. With quadratic drag, a drag torque that's negligible at low rpm quickly becomes significant as the rpm increases. Consider a square bar with cross section \(a \times a\) and length \(L\). It is rotating on an axle through its center at angular velocity \(\omega\) in a fluid of density \(\rho\). Assume that the drag coefficient \(C_d\) is constant along the length of the bar. Find an expression for the magnitude of the drag torque on the bar. Hint: Begin by considering the drag force on a small piece of the bar of length \(dr\) at distance \(r\) from the axle.