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 = 3 kg and radius R = 0.23 m. The rods each have mass mr = 1.2 kg and length L = 0.89 m.

1)What is the moment of inertia of each rod about the axis of rotation?

answer 0.317kg-m2

2)What is the moment of inertia of the disk about the axis of rotation?

kg-m2

3)What is the moment of inertia of the whole ceiling fan?

answer 1.65 kg-m2

4)When the fan is turned on, it takes t = 3.6 s and a total of 15 revolutions to accelerate up to its full speed.

What is the magnitude of the angular acceleration?

answer 14.5 rad/s2

5)What is the final angular speed of the fan?

rad/s

6)

What is the final rotational energy of the fan?  

J

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 = 3.6 s.

What is the final angular speed of the fan?

rad/s

8)What is the magnitude of the angular acceleration while the fan slows down?

rad/s2

Explanation / Answer

1) the moment of inertia of each rod about the axis of rotation is

1/3*Mr*L^2= 0.31684 kg m^2

2) the moment of inertia of the disk about the axis of rotation is

(1/2) Md*R^2=0.07935 kg m^2

3) the moment of inertia of the whole ceiling fan is

5* 1/3*Mr*L^2+ (1/2) Md*R^2 = 1.66355kg m^2

4) the magnitude of the angular acceleration is

alpha =2*theta/t^2

=2*15*2*3.14/3.6^2

=14.53703 rad/s^2

5) the final angular speed of the fan is

Wf = alpha *t

= 52.333 rad/s......

6) the final rotational energy of the fan is

KE = 0.5*I*Wf^2

= 0.5*1.66355* 52.333^2

= 2.278017866*10^3 J...............

7) the final angular speed of the fan is

KEf = KE/2

= 1.13900893*10^3 J

KE f = 0.5*I*W^2

W = 37 rad/s

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

alpha = 37 - 52.333/3.6

= -4.2591 rad/s^2

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