A ceiling fan consists of a small cylindrical disk with 5 thin rods coming from
ID: 1452981 • 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 = 2.8 kg and radius R = 0.26 m. The rods each have mass mr = 1.3 kg and length L = 0.74 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 = 3.7 s and a total of 14 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.
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
A.
Moment of inertia of each rod about the axis of rotation
Iend = ML^2/3 = 1.3*0.74^2/3
Iend = 0.237 kg-m^2
B.
moment of inertia of the disk about the axis of rotation
Id = MR^2/2 = 2.8*0.26^2/2 = 0.0946 kg-m^2
C.
It = 5*Iend + Id = 5*0.237 + 0.0946 = 1.2796 kg-m^2
D.
theta = wi*t + 0.5*alpha*t^2
theta = 2*pi*N rad = 2*3.14*14
wi = 0
t = 3.7
alpha = 2*theta/t^2
alpha = 2*2*3.14*14/3.7^2 = 12.84 rad/sec^2
E.
wf = wi + alpha*t
wf = 0 + 12.84*3.7 = 47.51 rad/sec
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