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 = 2.7 kg and radius R = 0.25 m. The rods each have mass mr = 1.2 kg and length L = 0.85 m.
1. When the fan is turned on, it takes t = 3.2 s and a total of 16 revolutions to accelerate up to its full speed.What is the magnitude of the angular acceleration?
2. What is the final angular speed of the fan?
3.What is the final rotational energy of the fan?
4.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 = 3.2 s.What is the final angular speed of the fan?
5.What is the magnitude of the angular acceleration while the fan slows down?

Explanation / Answer

1) = ½t²
16rev * 2 rads/rev = ½ * * (3.2s)² = 19.63 rad/s²

2) f = *t = 19.63rad/s² * 3.2s = 62.82 rad/s

3) I_rod = mL²/3 = 1.2kg * (0.85m)²/3 = 0.289 kg·m²

I_disk = ½mr² = ½ * 2.7kg * (0.25m)² = 0.084375 kg·m²

I = I_disk + 5*I_rod = 1.529375 kg·m²
KE = ½I² = ½ * 1.529375 kg·m² * (62.82 rad/s)² = 3017.73 J

4) KE(rot2) = KE(rot1)

w2 = sqrt[KE(rot1)/I] = sqrt[3017.73/1.529375] = 44.42 rad/s

5) the angular acceleration = (w2 - wf)/dt = (44.42 - 62.82)/3.2 = 5.75 rad/s^2

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