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Find moment of inertia of a uniform hollow cylinder

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Find moment of inertia of a uniform hollow cylinder?
 
Classical Mechanics - Moment of inertia of a uniform hollow cylinder Classical Mechanics - Moment of inertia of a uniform hollow cylinder
Find the moment of inertia of a uniform hollow cylinder of inner radius b, outer radius a, and mass M?
 
Hollow Cylinder
We know that the moment of inertia for hoop with radius R is mR2. We can divide cylinder into thin concentric hoops of thickness dR.
Density = Mass per unit volume
Density = dm / dV
Density
where:
þ; - Density
dm – Mass of a ring or radius R
dV - Volume of a ring or radius R
Lets assume height of the cylinder is h.
volume
we have
density
mass

We can obtain moment of inertia by integrating over all these hoops
moment of inertia for hoop
inertia

Cylinder has uniform density, where þ = constant
inertia
Image
Image

Volume of this cylinder is
volume
volume

Mass M is
mass
Image
since Image
Image

Moment of inertia for hollow cylinder is
moment of inertia for hollow cylinder
 
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