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 md=2.6 kg and radius R=0.24 m the rods each has mass mr=1.4 kg and length L=0.75m1). what is the moment of inertia of each rod about the axis of rotation? 2). what is the moment of inertia of the disk about the axis of rotation? 3).what is the moment of inertia of the whole ceiling fan? 4).when the fan is turned on it takes t=3.6 s and a total of 12 revolutions to accelerate up to its full speed what is the magnitude of the angular acceleration? 5). what is the final angular speed of the fan? 6). what is the final rotational energy of the fan? 7).now the fan is turned to a lower setting where it ends with half of its rotational energy as before the time it takes to slow to this new speed is also t=3.6 s what is the final angular speed of the fan? 8). what is the magnitude of the angular acceleration while the fan slows down?

Explanation / Answer

When the fan is turned on, it takes t = 3.6 s and a total of 12 revolutions to accelerate up to its full speed. What is the magnitude of angular acceleration?
12*2*pi= 0.5*alpha*3.6^2 or angular acceleration alpha = 48*pi/(3.6^2) = 11.635 rad/s^2

What I've found:
Moment of Inertia for each rod: . [{1.4*(0.75^2)*(1/24)}= 0.0328
Moment of Inertia of the disk: .08228(0.5*3.2*0.22^2= 0.08228, you are right!
Moment of Inertia for the whole fan: [0.041*5+0.08228] = 1.03 kg m^2

Also:
The final angular speed of the fan? = 20.77*3.3 = 68.54 rad/s

The final rotational energy of the fan = 0.5*I*68.54^2 = 2414 J

Now the fan is turned to a lower setting where it ends with half of its rotational energy as before. The time it takes to slow to this new speed is also t = 3.3 s. What is the final angular speed of the fan

Energy in second setting = 2414/2 = 1207 = 0.5*1.02*omega^2 or
reduced angular velocity, omega = sq rt[(1207*2)/1.02] = 48.65 rad/s

The magnitude of the angular acceleration while the fan slows down = (68.54-48.65)/3.3 = 6.03 rad/s^2

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