A ceiling fan consists of a small cylindrical disk with 5 thin rods coming from
ID: 1377643 • Letter: A
Question
A ceiling fan consists of a small cylindrical disk with 5 thin rods coming from the center. The disk has mass md = 3.2 kg and radius R = 0.26 m. The rods each have mass mr = 1.4 kg and length L = 0.7 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 s and a total of 12 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 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?
rad/s2
Explanation / Answer
1) Irod = mr*L^2/3
= 1.4*0.7^2/3
= 0.2287 kg.m^2
2) Idisk = md*R^2/2
= 3.2*0.26^2/2
= 0.108 kg.m^2
3) I = Idisk + 5*Irod
= 0.108 + 5*0.2287
= 1.25 kg.m^2
4) theta = wo*t + 0.5*alfa*t^2
theta = 0 + 0.5*alfa*t^2
==> t =2*theta/t^2
= 2*12*2*pi/3^2
= 16.76 rad/s^2
5) w = wo + alfa*t
= 0 + 16.76*3
= 50.28 rad/s
6) KE(rotationla) = 0.5*I*w^2
= 0.5*1.25*50.28^2
= 1580 J
7) KE' = KE/2
w'^2 = w^2/2
w' = w/sqrt(2)
= 16.76/1.414
= 11.85 rad/s
8) |alfa| = |(w'-w)|/t
= |(11.85-16.76)|/3
= 1.64 rad/s^2
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