moment of inertia

Moment of inertia is what mass becomes when you spin. For a rigid body rotating about a fixed axis, the moment of inertia is I = Σ m_i·r_i², summed over all the particles that make up the body (r_i is the perpendicular distance from each particle to the axis). For continuous bodies the sum becomes an integral I = ∫r² dm. It has units of kg·m².

Its role in rotational dynamics mirrors mass's role in linear dynamics. Angular momentum is L = I·ω (mirroring p = m·v); rotational kinetic energy is ½·I·ω² (mirroring ½·m·v²); torque and angular acceleration obey τ = I·α (mirroring F = m·a). The single difference is that I depends both on the mass and on how it is distributed relative to the axis — doubling the radius of a ring quadruples its moment of inertia, even at the same total mass.

For common shapes, I takes simple closed forms: a solid disk, ½·m·r²; a hollow hoop, m·r²; a solid sphere, (2/5)·m·r²; a thin rod about its centre, (1/12)·m·L². The same mass in different distributions can differ in I by a factor of three or more, which is why a hollow sphere rolls down a ramp slower than a solid one (same mass, larger I, more of the energy goes into rotation and less into translation) — and why figure skaters can change their angular velocity by drawing in their arms.