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_d =2.8 kg and radius R. = 0.26 m. The rods each have mass m_1. = 1.3 kg and length L = 0.72 m. 1) What is the moment of inertia of each rod about the axis of rotation? kg-m^2 2) What is the moment of inertia of the disk about the axis of rotation? Kg-m^2 What is the moment of inertia of the whole ceiling fan? kg-m^2 4) When the fan is turned on. it takes t = 3.6 s and a total of 14 revolutions to accelerate up to its full speed. What is the magnitude of the angular acceleration? rad/s^2 5) What is the final angular speed of the fan? rad/s J 6) What is the final rotational energy of the fan? J 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 = 3.6 s. What is the final angular speed of the fan? rad/s 8) What is the magnitude of the angular accel

Explanation / Answer

1) I_rod = mL²/3 = 1.3kg * (0.72m)²/3 = 0.22464 kg·m² 0.225 kg·m²

2) I_disk = ½mr² = ½ * 2.8kg * (0.26m)² = 0.09464 kg·m² 0.095 kg·m²

3) I = I_disk + 5*I_rod = 1.22 kg·m²

4) = ½t²
14rev * 2 rads/rev = ½ * * (3.6s)² = 13.58 rad/s²

5) = *t = 13.58rad/s² * 3.6s = 48.9 rad/s

6) KE = ½I² = ½ * 1.22kg·m² * (48.9 rad/s)² = 1458.64 J

7) w2 = sqrt[KE/I] = 34.58 rad/s

8) Alpha = (w2 - w)/t = 3.98 rad/s^2

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