Moment of Inertia · aiwiki.page

Moment of inertia is a measure of the distribution of mass about a specified axis. In classical mechanics, it plays a role in rotational motion analogous to that of mass in translational motion: for a given torque about a fixed axis, a body with a greater moment of inertia undergoes less angular acceleration. Unlike mass, moment of inertia depends on the location and orientation of the chosen axis as well as on the body's mass distribution. It is usually denoted by $I$ . (openstax.org)

Definition and physical meaning

For a collection of point masses, the moment of inertia about an axis is

I=\sum_i m_i r_{\perp i}^{\,2},

where $m_i$ is the mass of the $i$ th particle and $r_{\perp i}$ is its perpendicular distance from the axis, not necessarily its distance from a particular point. For a continuous mass distribution, the sum becomes an integral:

I=\int r_\perp^2\,dm.

If the volume density is $\rho(\mathbf r)$ , then $dm=\rho(\mathbf r)\,dV$ , so

I=\int_V \rho(\mathbf r)r_\perp^2\,dV.

The definition applies whether or not the object is actually rotating. (openstax.org)

The squared-distance factor gives mass farther from the axis a disproportionately large contribution. Moving a point mass to twice its original distance from the axis multiplies its contribution by four. Consequently, objects with equal mass and equal outer radius can have different moments of inertia: a thin hoop has a greater moment of inertia about its central axis than a uniform solid disk. (openstax.org)

In the International System of Units, moment of inertia is measured in $\mathrm{kg\,m^2}$ , and its physical dimension is mass times length squared. Contributions from separate components add, provided that all are evaluated about the same axis. (openstax.org)

Role in rotational dynamics

For a rigid body rotating about a fixed axis with constant moment of inertia,

\tau_{\mathrm{ext},\parallel}=I\alpha,

where $\tau_{\mathrm{ext},\parallel}$ is the component of net external torque along the axis and $\alpha$ is angular acceleration. This is the rotational counterpart of Newton's second law, $F=ma$ . The scalar equation concerns the torque component along the rotation axis; it is not a general vector equation for arbitrary three-dimensional motion. (openstax.org)

For rotation about that axis, the rotational kinetic energy is

K_{\mathrm{rot}}=\frac12 I\omega^2,

where $\omega$ is angular velocity. This follows by adding the kinetic energies of the body's particles, whose tangential speeds are $v_i=\omega r_{\perp i}$ . (openstax.org)

The component of angular momentum along the axis is

L_\parallel=I\omega.

If the external torque about the axis vanishes, this component is conserved. A figure skater who draws their arms inward reduces their moment of inertia and increases their angular speed, approximately satisfying

I_{\mathrm{initial}}\omega_{\mathrm{initial}} = I_{\mathrm{final}}\omega_{\mathrm{final}}.

This example involves a changing configuration rather than a single rigid body with constant $I$ . (ocw.mit.edu)

For general rigid-body motion, kinetic energy separates into translation of the center of mass and rotation about it:

K=\frac12 Mv_{\mathrm{CM}}^2 +\frac12\boldsymbol{\omega}^{T} \mathbf I_{\mathrm{CM}}\boldsymbol{\omega}.

Here $\mathbf I_{\mathrm{CM}}$ is the inertia tensor evaluated at the center of mass. (mitp-content-server.mit.edu)

Calculation and standard examples

Calculations begin by specifying the axis and expressing each mass element's perpendicular distance from it. Symmetry can simplify the integral, but a formula derived for one axis cannot automatically be used for another. (openstax.org)

Common results are listed below. Extended bodies are assumed to have uniform density; rods and shells are idealized as thin. (openstax.org)

Body	Axis	Moment of inertia
Point mass $M$	At perpendicular distance $R$	$MR^2$
Thin circular hoop, radius $R$	Through center, perpendicular to its plane	$MR^2$
Solid disk or cylinder, radius $R$	Central symmetry axis	$\tfrac12 MR^2$
Solid sphere, radius $R$	Any diameter	$\tfrac25 MR^2$
Thin spherical shell, radius $R$	Any diameter	$\tfrac23 MR^2$
Thin rod, length $\ell$	Through midpoint, perpendicular to rod	$\tfrac1{12}M\ell^2$
Thin rod, length $\ell$	Through one end, perpendicular to rod	$\tfrac13 M\ell^2$

For example, a uniform rod has linear mass density $M/\ell$ . Taking the origin at its midpoint gives

I=\frac{M}{\ell} \int_{-\ell/2}^{\ell/2}x^2\,dx =\frac1{12}M\ell^2.

The integration limits change when the axis passes through an end, producing the different result shown in the table. (openstax.org)

Axis theorems

The parallel-axis theorem relates the moment of inertia about an axis through the center of mass to that about a parallel axis:

I=I_{\mathrm{CM}}+Md^2,

where $d$ is the perpendicular separation between the axes. One of the two axes must pass through the center of mass; the formula is not a direct rule for shifting between any two arbitrary parallel axes. It also shows that, among parallel axes with a given direction, the center-of-mass axis has the smallest moment of inertia. (openstax.org)

For a thin planar mass distribution in the $xy$ -plane, three mutually perpendicular axes meeting at one point obey the perpendicular-axis theorem:

I_z=I_x+I_y.

This follows because $I_z=\int(x^2+y^2)\,dm$ , whereas $I_x=\int y^2\,dm$ and $I_y=\int x^2\,dm$ . The restriction to a planar distribution is essential; the equation does not hold for a general three-dimensional body. (ocw.mit.edu)

Inertia tensor and principal axes

A single scalar describes inertia about one axis. General three-dimensional rotation requires the inertia tensor, a second-order tensor represented by a symmetric $3\times3$ matrix:

I_{ij} =\int\left(r^2\delta_{ij}-x_i x_j\right)\,dm,

with coordinates measured from a specified origin. Thus, for example,

I_{xx}=\int(y^2+z^2)\,dm, \qquad I_{xy}=-\int xy\,dm.

For an axis through that origin with unit direction vector $\mathbf n$ , its scalar moment of inertia is

I_{\mathbf n}=\mathbf n^T\mathbf I\mathbf n.

(ocw.mit.edu)

About the center of mass, angular momentum and rotational energy satisfy

\mathbf L=\mathbf I\boldsymbol{\omega}, \qquad K_{\mathrm{rot}} =\frac12\boldsymbol{\omega}^{T}\mathbf I\boldsymbol{\omega}.

Angular momentum therefore need not be parallel to angular velocity. (mitp-content-server.mit.edu)

The tensor can be diagonalized using mutually perpendicular principal axes of inertia. Its eigenvalues are the principal moments $I_1,I_2,I_3$ , and its eigenvectors give the principal directions. In this coordinate system,

K_{\mathrm{rot}} =\frac12\left(I_1\omega_1^2+ I_2\omega_2^2+I_3\omega_3^2\right).

Symmetry often identifies these axes directly. Repeated principal moments allow more than one choice of principal directions; for a uniform sphere, any orthogonal set of axes through its center is principal. (mitp-content-server.mit.edu)

Applications and distinctions

Moment of inertia enters calculations of rotating machinery, wheels, and flywheels. For a fixed angular speed, increasing $I$ increases the rotational energy stored in a flywheel; for a fixed applied axial torque, increasing $I$ decreases angular acceleration. These are different consequences of the same mass-distribution property. (openstax.org)

In engineering, mass moment of inertia must be distinguished from the second moment of area, also commonly called “area moment of inertia.” The latter is a geometric quantity, such as

I_{A,x}=\int_A y^2\,dA,

with units of $\mathrm{m^4}$ , rather than $\mathrm{kg\,m^2}$ . It describes the distribution of cross-sectional area and is not a body's rotational inertia. (engineeringstatics.org)

Finally, moment of inertia is not a measure of friction or a force opposing rotation. It is a property of mass distribution. Its definition remains usable for a changing configuration, but treating it as a constant in rigid-body equations requires that the relevant distribution relative to the axis remain unchanged. (openstax.org)

References

4 Moment of Inertia and Rotational Kinetic Energy — University Physics Volume 1openstax.org
5 Calculating Moments of Inertia — University Physics Volume 1openstax.org
3 Dynamics of Rotational Motion: Rotational Inertia — College Physics 2eopenstax.org
Ch. 10 Summary — University Physics Volume 1openstax.org
2 Conservation of Momentum — Physicsopenstax.org
09(F14) Chapter 2: Rigid Body Dynamicsocw.mit.edu
Structure and Interpretation of Classical Mechanics: Chapter 2mitp-content-server.mit.edu
Statics: Mass Moment of Inertiaengineeringstatics.org
Statics: Integral Properties of Shapesengineeringstatics.org