A ceiling fan consists of a small cylindrical disk with 5 thin rods coming from

Question

A ceiling fan consists of a small cylindrical disk with 5 thin rods coming from the center. The disk has mass 3.2 kg and radius R = 0.25 m. The rods each have mass 1.4 kg and length L = 0.85 m.

When the fan is turned on, it takes t = 3.7 s and a total of 16 revolutions to accelerate up to its full speed.What is the magnitude of the angular acceleration?

i know i could use the constant angular acceleration formula: theta=1/2*(angular acceleration)*(time^2) and get the right answer 14.68677 but i was wondering why cant i use this constant acceleration formula: angular velocity=(angular acceleration)*(time)???
i tried using the second formula and i got exactly half the answer (7.343 ), why is that???
i mean we have the time and we could find the angular velocity using this formula: angular velocity=2pi/period, so why dont i get the right answer using this formula?

Explanation / Answer

See everything is correct except the last formulae(angular velocity=2pi/period). See this is valid only for constant angular velocity problems only. Period is the time taken to complete one revolution and it changes according to the angular velocity. Here, the period changes as angular velocity change because of the angular acceleration. So, you better use the formula (angular velocity)^2=2*(angular acceleration)*(theta). From here, we can find the angular velocity and then you substitute this into the formula.

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