A simple ring
A uniform ring of mass M and radius r has negligible thickness. Each element of the ring has mass m and thus inertia mr². The whole ring must have mass M and inertia Mr².
1 A hollow cylinder spinning on its longitudinal axis is made up of many rings. If the mass of the whole cylinder is M and its radius is r, what is the inertia?
2 Consider a uniform disk of mass M and radius r spinning about an axis perpendicular to the disk and through its centre. The axis is perpendicular to the plane of the disk. Each element of width dr is approximately a ring, and has mass of m (kg/m²) so its moment of inertia is 2πr (dx) m ρ². Sum these for all x (integrate) to get an expression; recognise that M=πr²m and therefore the inertia is ½ Mr².
3 Find the moment of inertia of a uniform hollow cone of base radius r and height h and mass M when rotated about its axis. I think ½ Mr² is an obvious result.
4 Find the moment of inertia of a uniform solid cylinder of base radius r and length h and mass M when rotated about its axis. I think ½ Mr² is another obvious result.
5 Show that the moment of inertia of a uniform solid cone of base radius r and height h and mass M is, when rotated about its axis, 3/10 Mr².
6 Consider a hollow sphere. Draw a diagram to help you show that the elemental moment of inertia is (2πmr² cos ø dø) (rcosø)² and use this to show that the moment of inertia of a hollow sphere is ⅔ Mr².
