Question

A ceiling fan consists of a small cylindrical disk with 5 thin rods coming from the center. The disk has mass md = 3.1 kg and radius R = 0.24 m. The rods each have mass mr = 1.3 kg and length L = 0.75 m.

1) SOLVED ALREADY BACKGROUND INFO

moment of inertia of each rod about the axis of rotation = .24kg-m²

^{2) SOLVED ALREADY BACKGROUND INFO} moment of inertia of the disk about the axis of rotation = .09kg-m²

What is the moment of inertia of each rod 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 12 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?

Answer 1

1.

Moment of inertia of rod is given by,

I_rod = (mr*L^2)/3

Given, mr = mass of rod = 1.3 kg

L = length of rod = 0.75 m

I_rod = 1.3*0.75^2/3

I_rod = 0.24 kg*m^2

2.

Moment of inertia for disk is given by,

I_disk = 0.5*md*R^2

given, md = mass of disk = 3.1 kg

R = radius of disk = 0.24 m

So, I_disk = 0.5*3.1*0.24^2

I_disk = 0.09 kg*m^2

3.

Moment of inertia of whole fan is given by,

I_fan = 5*I_rod + I_disk

I_fan = 5*0.24 + 0.09

I_fan = 1.29 kg*m^2

4.

By second rotational kinematics law,

= wi*t + 0.5**t^2

given, wi = initial angular speed = 0

t = time = 3.1 sec.

= angular acceleration = ??

= revolutions = 12 rev = 12*2*pi rad

12*2*pi = 0*3.1 + 0.5**3.3^2

= 2*12*2*pi/3.1^2

= 15.7 rad/s^2

5.

By first rotational kinematics law,

wf - wi = *t

here, wf = final angular speed = ??

wf = 0 + 15.7*3.1

wf = 48.7 rad/sec.