A lathe, initially at rest, accelerates at 0.60 rad/s² for 10 s, then runs at a constant angular velocity for 20 s, and finally decelerates uniformly for 10 s to come to a complete stop. What is its average angular velocity? A)4.5 rad/s B)3.5 rad/s C)5.0 rad/s D)3.0 rad/s E)4.0 rad/s

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Explanation:The average angular velocity is the total angular displacement divided by the total time. The total angular displacement is the sum of the angular displacements during each of the three stages:$$	heta_1 = \frac{1}{2} \alpha_1 t_1^2 = \frac{1}{2} (0.60 	ext{ rad/s}^2) (10 	ext{ s})^2 = 30 	ext{ rad}$$$$	heta_2 = \omega_2 t_2 = (0.60 	ext{ rad/s}^2) (10 	ext{ s}) (20 	ext{ s}) = 120 	ext{ rad}$$$$	heta_3 = \frac{1}{2} \alpha_3 t_3^2 = \frac{1}{2} (-0.60 	ext{ rad/s}^2) (10 	ext{ s})^2 = -30 	ext{ rad}$$The total angular displacement is:$$	heta = 	heta_1 + 	heta_2 + 	heta_3 = 30 	ext{ rad} + 120 	ext{ rad} - 30 	ext{ rad} = 120 	ext{ rad}$$The total time is:$$t = t_1 + t_2 + t_3 = 10 	ext{ s} + 20 	ext{ s} + 10 	ext{ s} = 40 	ext{ s}$$Therefore, the average angular velocity is:$$\omega_{avg} = \frac{	heta}{t} = \frac{120 	ext{ rad}}{40 	ext{ s}} = \boxed{3.0 	ext{ rad/s}}$$

A Lathe, Initially at Rest, Accelerates at 0