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 m_c, = 2.7 kg and radius R = 0.26 m. The rods each have mass m_r = 1.3 kg and length L = 0.89 m. What is the moment of inertia of each rod about the axis of rotation? What h the moment of inertia of the disk about the axis of rotation? What is the moment of inertia of the whole ceiling fan? When the fan is turned on, it takes t - 3.4 s and a total of 12 revolutions to accelerate up to its full speed. What is the magnitude of the angular acceleration? What is the final angular speed of the fan? What ts the final rotational energy of the fan? 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 = 3.4 s. What is the final angular speed of the fan? What is the magnitude of the angular acceleration while the fan slows down?

Explanation / Answer

a) Moment of inertia of rod about one end

I = 1/3 * M L2 = 1/3 * 1.3 * 0.892 = 0.343 kg.m2

b) I = 1/2 M R2 = 1/2 * 2.7 * 0.262 = 0.091 kg.m2

c) moment of inertia of the whole ceiling fan = 5 (1/3 * M L2 ) + (1/2 M R2) = 5 * 0.343 + 0.091 = 1.8 kg.m2

d) alpha = 4 pi N / t2 = 4 * pi * 12 / 3.42 = 13 rad/s2

e) wf = wi + alpha t = 0 + 13 * 3.4 = 44.35 rad/s

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6) Rotational energy = 1/2 I w2 = 1/2 * 1.8 * 44.352 = 1770 J

7) Energy second setting = 1770/2 = 885 = 0.5 * 1.8 * w2

w = sqrt [(885 * 2) / 1.8 ] = 31.36 rad/s

8) The magnitude of the angular acceleration while the fan slows down

= (44.35 - 31.36) / 3.4 = 3.82 rad/s2

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