A ceiling fan consists of a small. uniform, cylindrical disk with 5 thin rods co
ID: 1461215 • Letter: A
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 = 0.24 m. The rods each have mass mr = 1.4 kg and length L = 0.73 m.
1) What is the moment of inertia of each rod about the axis of rotation?
In kg*m2
2) What is the moment of inertia of the disk about the axis of rotation?
In kg*m2
3) What is the moment of inertia of the whole ceiling fan?
In kg*m2
4) When the fan is turned on, it takes t = 3.5 s and a total of 17 revolutions to accelerate up to its full speed.
What is the magnitude of the angular acceleration?
In rad/s2
5) What is the final angular speed of the fan?
In rad/s
6) What is the final rotational energy of the fan?
In Joules
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 = 3.5 s.
What is the final angular speed of the fan?
In rad/s
8) What is the magnitude of the angular acceleration while the fan slows down?
In rad/s2
Explanation / Answer
1) Moment of ineria of each rod, Ir = mrL2/3 = 1.4 * 0.732 / 3 = 0.25 kg-m2
2) Moment of inertia of disk, Id = mdR2/2 = 2.5 * 0.242 /2 = 0.07 kg-m2
3) Moment of inertia of whole fan, If = Id + 5Ir = 0.07 + (5 * 0.25) = 1.32 kg-m2
4) Total angular distance, = 2 * 17 = 106.8 rad
Now, = t2/2
=> angular acceleration, = 2/t2 = (2 * 106.8) / 3.52 = 17.44 rad/s2
5) Final angular speed, = t = 17.44 * 3.5 = 61.04 rad/s2
6) Final rotational energy, KErot = If2/2 = 1.32 * 61.042 / 2 = 2459.1 J
7) New rotational energy = If12/2 = 2459.1/2 = 1229.55 J
So, new angular speed, 1 = [2 * 1229.55 / 1.32]1/2 = 43.16 rad/s
8) Magnitude of angular acceleration, 1 = ( - 1) / t = (61.04 - 43.16) / 3.5 = 5.11 rad/s2
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