= Moment of inertia
{title2=$I$}
{wiki}
= Moments of inertia
{synonym}
The <moment of inertia> about a specified axis measures how a <mass> distribution resists changes of <angular velocity> about that axis. For <point masses> $m_i$ at perpendicular distances $r_{\perp i}$, or a continuous distribution with <mass density> $\rho$,
$$
I=\sum_i m_i r_{\perp i}^2,
\qquad I=\int r_\perp^2\,dm=\int r_\perp^2\rho\,dV.
$$
Its <SI> unit is $\mathrm{kg\,m^2}$. The distance is to the entire axis, not to one point on it: moving a <point mass> along the axis leaves its contribution unchanged. The <moment of inertia> depends on both the distribution and the chosen axis; it is not determined by total <mass> alone. About an axis fixed in an <inertial frame>, a <rigid body> with larger <moment of inertia> has smaller <angular acceleration> under the same net axial <torque>, and greater <rotational kinetic energy> at the same <angular speed>.<The Feynman Lectures on Physics, Volume I by Richard Feynman and others/Chapter 19: Center of Mass; Moment of Inertia>{ref}
For a simple model, put four equal <point masses> of total <mass> $M$ on light rigid spokes at radius $a$. Their <moment of inertia> is $Ma^2$. Doubling $a$ quadruples the <moment of inertia> without changing $M$. The illustrated placements $a=R/3$ and $a=R$ give $MR^2/9$ and $MR^2$. Actual spokes and finite-sized weights contribute their own <moments of inertia>; the <point mass> model neglects them. Repositioning weights during rotation also breaks the fixed-distribution assumption, so it requires a separate <work> and <angular momentum> calculation.
\Image[/rotational-inertia-energy-mass-distribution.png]
{title=Same total <mass>, different radial placement and <moment of inertia>}
{description=Original model diagrams, not experimental data. The upper pair has four equal <point masses> on negligible-mass spokes, at $R/3$ and $R$ from a perpendicular central axis. The lower pair compares a uniform disc with an ideal thin ring of the same <mass> $M$ and outer radius $R$. The displayed <moments of inertia> determine <rotational kinetic energy> at a shared <angular speed>. The visible thickness of the ring and size of the point markers are only drawing conventions.}
The <moment of inertia of a uniform disc> follows by dividing it into concentric <annuli>: $dm=(2M/R^2)r\,dr$ and $I=\int_0^R r^2(2M/R^2)r\,dr=MR^2/2$. For an ideal thin ring every <mass> element is at $R$, so $I=MR^2$. A ring of finite radial thickness instead uses the <moment of inertia of a uniform annulus>, $I=M(R^2+b^2)/2$, with inner radius $b$. A <moment of inertia> is thus a compact summary of radial placement rather than a complete description of shape. For several joined pieces, their <moments of inertia> add only when each is calculated about the same axis.<University Physics Volume 1 by William Moebs and others/10.5 Calculating Moments of Inertia>[§10.5]{ref}
A <physical pendulum> makes the <moment of inertia> measurable through timing. Martín Monteiro, Cecilia Stari, Cecilia Cabeza and Arturo C. Martí used a metal bar with suspension holes $1.0\,\mathrm{cm}$ apart and an attached smartphone. They report bar length $1.199\,\mathrm m$, bar <mass> $0.2518\,\mathrm{kg}$ and phone <mass> $0.1311\,\mathrm{kg}$. Changing the suspension hole changes the pivot axis. Their readouts were phone <gyroscope> <angular velocity> records, analyzed by <autocorrelation> or a fit to a <sinusoid>, and an independent photogate; the logged records for the <sinusoid> fits lasted $10\,\mathrm s$. They observed agreement between the three timing methods and the geometry-based model, including a minimum of <oscillation period> as the suspension point varied. These are reported apparatus details and observations, not measurements made for this article.<Experimental analysis of a physical pendulum with variable suspension point by Martín Monteiro and others>{ref}
To infer a <moment of inertia> from such a measurement, determine the total <mass> $M_{\mathrm{tot}}$ including every attached part, locate the assembly's <centre of mass> $G$, and measure its distance $h>0$ below the horizontal pivot $O$. Record successive crossings in the same direction, or the elapsed <time> $t_N$ for $N$ complete <oscillations>, giving <oscillation period> $P=t_N/N$. The <physical pendulum> model with a fixed pivot, a <uniform gravitational field>, negligible <friction> and the <small-angle approximation> gives
$$
I_O\ddot\theta=-M_{\mathrm{tot}}gh\sin\theta,
\qquad P\simeq2\pi\sqrt{\frac{I_O}{M_{\mathrm{tot}}gh}},
\qquad I_{O,\mathrm{inferred}}=M_{\mathrm{tot}}gh\left(\frac{P}{2\pi}\right)^2.
$$
The crossing times, <masses> and geometric distances are the measured inputs; $I_{O,\mathrm{inferred}}$ is an inference conditional on this model, while an <integral> over the assumed shape supplies an independent model prediction. The <parallel axis theorem> converts the inferred pivot <moment of inertia> to one about a parallel axis through $G$. The controls and interpretation of that comparison are described there. An axial measurement determines one scalar <moment of inertia>, not the complete <inertia tensor> needed for general rotation.
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