Moment of Inertia of a Rigid Hoop

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

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

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

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

6. y’ y dx x O x L Testing the Parallel Axis Theorem 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 1 2 3 4 2 = 3 = zero because of the definition of centre of mass