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 = 2.6 kg and radius R = 0.24 m. The rods each have mass mr = 1.4 kg and length L = 0.77 m.

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

kg-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?

kg-m2

4) When the fan is turned on, it takes t = 3.7 s and a total of 13 revolutions to accelerate up to its full speed.

What is the magnitude of the angular acceleration?

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 = 3.7 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

here,

mass of disk , md = 2.6 kg

R = 0.24 m

mr = 1.4 kg

L = 0.77 m

a)

the moment of inertia of each rod about the axis of rotation , Ir = mr * l^2 /3

Ir = 1.4 * 0.77^2 /3

Ir = 0.28 kg.m^2

b)

the moment of inertia of the disk about the axis of rotation , Id = 0.5 * md * R^2

Id = 0.5 * 2.6 * 0.24^2

Id = 0.07488 kg.m^2

c)

the moment of inertia of the whole ceiling fan , I = Id + 4 * Ir

I = 0.07488 + 4 * 0.28 kg.m^2

I = 1.19 kg.m^2

d)

t = 3.7 s

theta = 13 * 2pi = 81.64 rad

theta = w0 * t + 0.5 * alpha * t^2

81.64 = 0 + 0.5 * alpha * 3.7^2

alpha = 11.9 rad/s^2

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