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.

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

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

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

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

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

Analysis for Parallel Axis Theorem 1 2 3 4 2 = 3 = zero because of the definition of centre of mass