Question

Problem 7 A solid cone of angle 28, base ro, and density pe is rotating with initial angular velocity a inside a conical seat, as shown in Fig. P1.53. The clearance h is filled with oil of viscosity μ Neglecting air drag, derive an analytical expression for the cone's angular velocity at) if there is no applied torque. cis oil 20 PL.53

Answer 1

Torques equation dT = r tau dA = r times mu r_0/h times 2 pi r dr/sin that integral dT = integral^r_1_0 2 pi w mu/h sin theta r^3 dr T = 2 pi mu w/h sin theta(r^4/4)^r_0_0 T = pi mu w r^4_0/2h sin theat (2) Torque required to slow drawn the one: T = I_0 dw/dt = 3/10 m r^2_0 dw/dt (3) equation (2) & (3) 3/10 m r^2_0 dw/dt = pi mu r^4_0/2h sin theta times 10/3 m r^2_0 dt (ln w)^w_w_0 = 5 pi mu r^2_0/33mh sin theta w/w_0 = exp[5pi mu r^2_0 t/3 mh sin theta] w = w_0 exp[5 pi mu r^2_0 t/3 mh sin theta]