CALC A child is pushing a merry-go-round. The angle through which the merry-go-round has turned varies with time according to ?(t) = ?t + ?t3, where ? = 0.400 rad/s and ? = 0.0120 rad/s3. (a) Calculate the angular velocity of the merry-go-round as a function of time. (b) What is the initial value of the angular velocity? (c) Calculate the instantaneous value of the angular velocity ?z at t = 5.00 s and the average angular velocity ?av-z for the time interval t = 0 to t = 5.00 s. Show that ?av-z is not equal to the average of the instantaneous angular velocities at t = 0 and t = 5.00 s, and explain.

Solution 5E Angular velocity is the rate of change of angular displacement. Therefore, angular velocity = AngularTimeplaceme…..(1) Given that, (t) =t + t , where = 0.400 rad/s and = 0.0120 rad/s 3 (a) We can calculate the angular velocity as a function of time. This is done as follows. 3 Differentiating (t) =t + t wrt time, d(t) dt = + 3t 2 (t) = + 3t …..(2) This is the required expression for angular velocity as a function of time. (b) The initial value of angular velocity is given at time t = 0. From equation 2, at time t=0 2 (0) = + 3 × 0 (0) = 0.400 rad/s This is the required value of initial angular velocity. (c) At time t = 5.00 s, Angular velocity is (5) = 0.400 rad/s + 3 × 0.0120 rad/s × 5 s 3 2 2 (5) = 1.30 rad/s This is the required instantaneous angular velocity at t = 5.00 rad/s 3 We have, (t) =t + t At t= 0 s, (0) = 0 At t = 5.00 s 3 3 3 (5) = 5 s × 0.400 rad/s + 0.0120 rad/s × 5 s (5) = 3.5 rad (5)(0) The a verage angular velocity av.z= 5.000 av.z= (3.5 rad 0)/5.00 s = 0.7 rad/s Now, total instantaneous angular velocity is = (0) + (5) = 0 + 1.30 rad/s = 1.30 rad/s Average of this instantaneous angular velocity is = (1.30 rad/s)/2 = 0.65 rad/s = 0.65 rad/s Now, 0.65 rad/s is different from 0.7 rad/s. By definition average angular velocity is the total displacement divided by the total time taken to get displaced. Therefore, calculation for the same yields a different result than the average value between the two points.