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.

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

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

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

4)When the fan is turned on, it takes t = 3.1 s and a total of 16 revolutions to accelerate up to its full speed.
What is the magnitude of the angular acceleration?

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

6)What is the final rotational energy of the fan?

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 = 3.1 s.

What is the final angular speed of the fan?

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

Explanation / Answer

1) I_rod = mr*L^2/3

= 1.4*0.78^2/3

= 0.284 kg.m^2

2) I_disk = 0.5*md*R^2

= 0.5*2.9*0.22^2

= 0.0702 kg.m^2

3) I_fan = 5*I_rod + I_disk

= 5*0.284 + 0.702

= 2.122 kg.m62

4) angular displacement, theta = 16 revolutios

= 16*2*pi rad

Apply, theta = 0.5*alfa*t^2

==> alfa = 2*theta/t^2

= 2*16*2*pi/3.1^2

= 20.92 rad/s^2

5) w2 = w1 + alfa*t

= 0 + 20.92*3.1

= 64.852 rad/s

6) KEf = 0.5*I*w2^2

= 0.5*2.122*20.92^2

= 464.34 J

7) new angular speed, w' = w/sqrt(2)

= 64.85/1.414

= 45.86 rad/s

8) alfa = (w - w')/t

= (64.85 - 45.86)/3.1

= 6.126 rad/s^2

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