Moment of Inertia: Point Mass

This is the building block from which every other moment of inertia is assembled: a compact mass $m$ at distance $r$ from the axis contributes $I = mr^2$. It comes directly from the energy. That mass moves at $v = \omega r$, so its kinetic energy is $\tfrac{1}{2}m(\omega r)^2 = \tfrac{1}{2}(mr^2)\omega^2$, and the bracketed quantity is what has to be called $I$ if $\tfrac{1}{2}I\omega^2$ is going to work. The $r^2$ is not chosen, it is inherited from the square in kinetic energy.

A 2 kg mass whirling on a 1.5 m tether has $I = 2 \times 1.5^2 = 4.5$ kg·m². Move it to 3 m and $I$ becomes 18 kg·m² — the mass has not changed, the distance has doubled, and the resistance to being spun up has quadrupled. That square is the most important fact in rotational mechanics, and it is why a diver tucks, why a tightrope walker's pole is long rather than heavy, and why putting material at the rim of a flywheel is worth far more than putting it near the hub.

Every extended body's moment of inertia is this formula summed over all its mass, $I = \sum m_i r_i^2$, or integrated for a continuous body. That integration is exactly where the coefficients on the other pages come from: the $\tfrac{1}{2}$ of a solid disk, the $\tfrac{2}{5}$ of a solid sphere, the $\tfrac{1}{12}$ of a rod about its centre. Each is a statement about how the mass of that shape is distributed relative to the axis, and nothing more.

The $r$ is the perpendicular distance to the axis — a line — not the distance to a point. For a mass sitting 3 m along the axis and 0.5 m out from it, the moment of inertia about that axis is $m \times 0.5^2$, not $m \times 3.04^2$; the along-axis position is irrelevant, because that part of the displacement is not being swung round. This trips people up as soon as the geometry stops being flat. The related habit worth building is always naming the axis before quoting an $I$: the same point mass has a completely different moment of inertia about a different line, and unlike mass, which is one number for the object, $I$ is one number per axis. When several masses share an axis, their moments of inertia simply add, which makes building up a real assembly a matter of arithmetic rather than calculus — and the parallel-axis theorem, $I = I_{cm} + md^2$, is just this formula treating a whole body as though it were a point at its own centre of mass.