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A ceiling fan consists of a small cylindrical disk with 5 thin rods coming from the center. The disk has mass md = 2.8 kg and radius R = 0.26 m. The rods each have mass mr = 1.3 kg and length L = 0.74 m.

When the fan is turned on, it takes t = 3.7 s and a total of 14 revolutions to accelerate up to its full speed.

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 = 3.7 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

here,

mass of disk, md = 2.8 kg
radius of disk, r = 0.26 m

Moment of inertia of Disk, Id = 0.5*m*r^2
Id = 0.5*2.8*0.26^2
Id = 0.09464 Kg.m^2

mass of rods, mr = 1.3 kg
Length of rods, L = 0.74 m

Moment of inertia of rods, Ir = 1/3*mr*L^2
Ir = 1/3 * 1.3 * 0.74^2
Ir = 0.237 Kg.m^2

Net moment of inertia of system, I = Id + Ir
I = 0.09464 + 0.237
I = 0.332 Kg.m^2

time t = 3.7 s
theta = 14 rev = 14 * 6.283 = 87.962 rad

From second equation of motion, alpha = 2*theta/t^2
solving for angular acceleration, alpha = 2*87.962/(3.7)^2

alpha = 12.851 rad/s^2

Part 5:
Since, angular speed, w = alpha*t ( first eqn of rotational motion)

w = 12.851 * 3.7
w = 47.549 rad/s

Part 6:
Rotational energy, RKE = 0.5 * I * w^2
RKE = 0.5 * 0.332 * 47.549^2
RKE = 375.311 J

Part 7:
RKe = 375.311/2 = 187.655 J

w = sqrt(2*RKE/I)
w = sqrt(2*187.655/0.332)
w = 33.622 rad/s

Part 8:
since, alpha = w/t = 33.622/3.7 = 9.087 rad/s^2

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