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_r = 1.3 kg and length L = 0.71m. When the fan is turned on, it takes t = 3.5 s and a total of 13 revolutions to accelerate up to its full speed. 1. 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 = 3.5 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

moment of inertia Jr of each rod about the axis of rotation
Jr = mr*r^2/3 = 1.3*0.71^2/3 = 0.21844kg*m^2

moment of inertia Jd of the disk about the axis of rotation?
Jd = md*r^2/2 = 1.40*0.26^2 = 0.09464kg*m^2

moment of inertia J of the whole ceiling fan?
J = 5*Jr+Jd = 0.21844*5+0.09464 = 1.18684kg*m^2

When the fan is turned on, it takes t = 3.5 s and a total of 13 revolutions to accelerate up to its full speed. What is the magnitude of the angular acceleration ?
angle = 2*PI*n = /2*t^2
angular acceleration = 4*3.1416*13/3.5^2 = 13.32 rad/sec^2

1) final angular speed of the fan?
= *t = 13.32*3.5 = 46.65 rad/sec

2) final rotational energy of the fan?
Ek = J*^2/2 = 1.1868*46.65^2/2 = 4609 joule

3) 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 = 3.5 s.
What is the final angular speed 1 of the fan?
1 = (*2)/2 = 46.65*0,707 = 33.0rad/sec ...in such a way that 1^2 = ^2/2

4) What is the magnitude of the angular acceleration 1 while the fan slows down?
1 = (1-)/3.5 = (33.0-46.65)/3.5 = -3.89m/sec^2

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