A ceiling fan consists of a small. uniform, cylindrical disk with 5 thin rods co

Question

A ceiling fan consists of a small. uniform, cylindrical disk with 5 thin rods coming from the center. The disk has mass md= 2.5 kg and radius R = 0.23 m. The rods each have mass mr = 1.3 kg and length L = 0.88 m.

1) 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 = 3.2 s and a total of 15 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 = 3.2 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

For Rod, formula is (ML^2 /24)
Moment of Inertia for each rod: . [{1.3*(0.88^2)*(1/24)}= 0.041 kg m^2
Moment of Inertia of the disk: .(0.5*2.5*0.22^2= 0.064 kg m^2
Moment of Inertia for the whole fan: [0.041*5+0.064] = 0.269 kg m^2

When the fan is turned on, it takes t = 3.2 s and a total of 15 revolutions to accelerate up to its full speed. What is the magnitude of angular acceleration?
15*2*pi= 0.5*alpha*3.2^2 or angular acceleration alpha = 60*pi/(3.2^2) = 18.39 rad/s^2
Also:
The final angular speed of the fan? = 18.39*3.2 = 58.87 rad/s

The final rotational energy of the fan = 0.5*I*58.87^2 = 466.1 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 = 3.2 s

Energy in the second setting = 466.1/2 = 233 = 0.5*.269*omega^2 or
reduced angular velocity, omega = sq rt[(233*2)/.269] = 41.6 rad/s

The magnitude of the angular acceleration while the fan slows down = (58.87-41.6)/3.2 = 5.39rad/s^2

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