1. Moment of Inertia of a Rigid Hoop
Find the moment of inertia of the hoop
of mass M and radius R about the z-axis. x y R O z
Mass element, dm
In this case, the radius is fixed for
all the mass elements:
NB This is also the moment of inertia for
a cylindrical tube of length L and infinitesimal
wall thickness.

2. Rigid Beam Here we can calculate the moment of inertia of the
beam about (a) its centre (y) and (b) about one end (y’). O dx x y L y’ y:
y’:
Mass per unit length, l = M/L,
so dm = l dx

3. Solid Cylinder z Our volume element here is dV = 2pLr dr
And our mass element dm = r dV = 2prLr dr dr r R

4. y dm (x,y) y’ r CM (xcm, ycm) D ycm xcm x’ x Parallel Axis Theorem
CM = centre of mass r D
CM (xcm, ycm)
ycm
xcm x x’ y’ y
dm (x,y)
mass element

5. PAT (cont) z CM
Axis through CM
Rotation
Axis (z) O D CM

6. Testing the Parallel Axis Theoremdx x y L y’
We know from before that Iy =ML2/12
and Iy’ = ML2/3
Now checking with PAT: D
D = L/2

7. Analysis for Parallel Axis Theorem
2 = 3 = zero because of the definition of centre of mass