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 = 3.1 kg and radius R = 0.24 m. The rods each have mass mr = 1.3 kg and length L = 0.77 m.

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?

So I already know the answers are .2569kgm^2, .08928 kgm^2, and 1.3738 kgm^2 (respectively)

But when doing the math why do you divide by 3, 2, etc?

EX (PROBLEM 1): I= ((1.3)(.77^2))/3 ( why do you divide by three where did the number come from))

Explanation / Answer

here,

md = 3.1 kg

R = 0.24 m

mr= 1.3 kg

L = 0.77 m

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

Ir = 1.3 * 0.77^2 /3

Ir = 0.26 kg.m^2

in this question the numbers 2 ,3 ....comes from the derivation of moment of inertia of different shapes

and using the two theorrms for moment of inertia

a) parallel axis theorm

b) perpendicular axis theorm

derivation for moment of inertia of rod

dI = dm *x^2

where dm = M L dx

dI = M L x^2dx

I = integration(dI)

Note: The lower limit is -h because the left side of the rod is -h units away from the axis of rotation. (We take right as positive)

Solving the integration, we have:

I=13M(L2–3Lh+3h2)

When the rotation axis is at one end of the rod (h = 0), we have:

I = M*L^2/3

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