Introduction to Classical Mechanics · Problem 9.5
Angular momentum, Part II (General L) — Morin
Physics — Mechanics · nice-cylinder
Problem
What must the ratio of height to radius of a cylinder be so that every axis is a principal axis (with the CM as the origin)?Intuitive explanation
A cylinder has two natural types of symmetry axes: one along its length and infinitely many perpendicular to it through the center. For every axis to be principal, the moment of inertia must be identical in all directions—meaning the inertia tensor becomes a scalar multiple of the identity. This happens only when the two independent moments of inertia (parallel and perpendicular to the axis) are equal. Find the height-to-radius ratio that makes these match.
Intuitive explanation
A cylinder has two natural types of symmetry axes: one along its length and infinitely many perpendicular to it through the center. For every axis to be principal, the moment of inertia must be identical in all directions—meaning the inertia tensor becomes a scalar multiple of the identity. This happens only when the two independent moments of inertia (parallel and perpendicular to the axis) are equal. Find the height-to-radius ratio that makes these match.