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 = 3.4 kg and radius R = 0.22 m. The rods each have mass mr = 1.3 kg and length L = 0.81 m.

1)What is the moment of inertia of each rod about the axis of rotation? ans: 0.28

2)What is the moment of inertia of the disk about the axis of rotation?  ans: 0.082

3) What is the moment of inertia of the whole ceiling fan? asn : 1.53

4) When the fan is turned on, it takes t = 3.7 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 = 3.7 s.

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

Explanation / Answer

Here ,

1)

Moment of inertia of each rod , I = 1/3 * mr * l^2

Moment of inertia of each rod 1/3 * 1.3 * 0.81^2

Moment of inertia of each rod = 0.284 Kg.m^2

the Moment of inertia of each rod is 0.284 Kg.m^2

2)

for the disk

moment of inertia of disk = 0.5 * m * r^2

moment of inertia of disk = 0.5 * 3.4 * 0.22^2

moment of inertia of disk = 0.082 Kg.m^2
3)

for the whole fan

total moment of inertia = moment of inertia of rod + moment of inertia of disk

total moment of inertia = 0.082 + 0.284

total moment of inertia = 0.366 Kg.m^2

the total moment of inertia is 0.366 Kg.m^2

4)

angle = 12 revs = 12 * 2pi radians

now , let the angular acceelration is a

Using second equaiton of motion

theta =wi * t + 0.5 * a * t^2

12 * 2pi =0 * 3.7 + 0.5 * a * 3.7^2

solvng for a

a = 11 rad/s^2

the angular accelration of the fan os 11 rad/s^2

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