Question

Find the moment of inertia of a uniformly dense hollow cylinder rotating about the y- axis. [Clearly define all model elements and show all work for full credit]

Check your answer with the following limiting conditions:

- When a0 (solid disk/cylinder)

- When a ≈ b (thin hoop/cylinder)

Answer 1

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

When a = 0

when a = b