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 = 2.5 kg and radius R = 0.23 m. The rods each have mass mr = 1.3 kg and length L = 0.89 m.

1)

What is the moment of inertia of each rod about the axis of rotation?

kg-m2

2)

What is the moment of inertia of the disk about the axis of rotation?

kg-m2

3)

What is the moment of inertia of the whole ceiling fan?

kg-m2

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?

rad/s2

5)

What is the final angular speed of the fan?

rad/s

6)

What is the final rotational energy of the fan?

J

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?

rad/s

8)

What is the magnitude of the angular acceleration while the fan slows down?

Explanation / Answer

Answer:  When the fan is turned on, it takes t = 3.3 s and a total of 18 revolutions to accelerate up to its full speed. What is the magnitude of angular acceleration?
18*2*pi= 0.5*alpha*3.3^2 or angular acceleration alpha = 72*pi/(3.3^2) = 20.77 rad/s^2

What I've found:
Moment of Inertia for each rod: . [{1.3*(0.87^2)*(1/24)}= 0.041, your answer is wrong formula is (ML^2 /24)
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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