Calculating Moment of Inertia

Mass tells you how hard a body is to push. Moment of inertia tells you how hard it is to spin, and it depends on where the mass sits. Here is how to calculate it.

Collections of particles#

For separate particles, use the definition directly: $I = \sum m_i r_i^2$, where each $r_i$ is the perpendicular distance from the axis. Two heavy balls on a light rod, at distances r₁ and r₂ from an axis, give $I = m_1 r_1^2 + m_2 r_2^2$; the rod adds nothing.

Continuous bodies#

For a smooth distribution of matter, slice the body into elements of mass dm chosen so that every point of an element is practically the same distance r from the axis. Then

The integration limits must make the elements cover the whole body exactly once.

Uniform rod about its perpendicular bisector

Rod of mass M, length L, origin at its middle. An element between x and x + dx has mass $(M/L)\,dx$ and distance |x|:

Rectangular plate about a central line parallel to one edge

Plate of sides l and b, axis through the centre parallel to the side of length b. Slice it into strips parallel to the axis: a strip at distance x of width dx has mass $\frac{M}{lb}\,b\,dx = \frac Ml\,dx$, exactly like the rod. So $I = Ml^2/12$; the side b parallel to the axis doesn’t matter.

Ring about its axis

Every bit of a thin ring of radius R is at distance R from its axis: $I = \int R^2\,dm = MR^2$. A thin-walled hollow cylinder gives the same result for the same reason.

Disc about its axis

Slice the disc into thin rings of radius x and width dx. Area $2\pi x\,dx$, mass $dm = \dfrac{M}{\pi R^2}2\pi x\,dx = \dfrac{2M}{R^2}x\,dx$, and each ring contributes $x^2\,dm$:

A solid cylinder is a stack of such discs, so it too has $I = MR^2/2$ about its axis, whatever its length.

Hollow sphere about a diameter

Cut the shell into rings by planes perpendicular to the axis. The ring at polar angle θ has radius $R\sin\theta$ and width $R\,d\theta$, so its mass is $\dfrac{M}{4\pi R^2}(2\pi R\sin\theta)(R\,d\theta) = \tfrac M2\sin\theta\,d\theta$:

Solid sphere about a diameter

Build it from thin hollow shells of radius x and thickness dx. Each has mass $\dfrac{3M}{4\pi R^3}4\pi x^2\,dx = \dfrac{3M}{R^3}x^2\,dx$ and moment of inertia $\tfrac23(dm)x^2$: