Moment of Inertia of a Toilet Paper Roll, Thick Tube, or Annulus

This is an explanation of how to find the moment of inertia (rotational inertia) about the usual axis of rotation of a toilet paper roll, annulus, thick ring, or tube with thick walls. For systems of discrete masses, the moment of inertia is Σ r^2 m. For continuous distributions, this becomes ∫ r^2 dm. For many students, the most challenging part of finding moments of inertia is coming up with the relationship for dm, an infinitesimal chunk of mass. This can be found as dm = (mass/area) dA, in which dA is a tiny chunk of area. In this case, we’ll use dm = [mass/area] 2πr dr.