[转]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?

 
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 

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.

we have



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



Cylinder has uniform density, where þ = constant




Volume of this cylinder is



Mass M is


since 


Moment of inertia for hollow cylinder is