quantity

moment of inertia

A measure of an object's resistance to rotational acceleration about an axis, depending on mass distribution.
8 related articles · 6 formulas

Description

Moment of inertia is the rotational analog of mass. It quantifies how difficult it is to change an object's rotational speed about a given axis. Unlike ordinary mass, which is a fixed property of an object, moment of inertia depends on how that mass is distributed relative to the axis of rotation: the farther the mass is from the axis, the larger the moment of inertia, and the harder it is to spin or stop the object. This is why a figure skater spins faster when pulling their arms in — they are reducing their moment of inertia.

History

The concept was formalized by Leonhard Euler in the 18th century, building on earlier work by Christiaan Huygens on compound pendulums. The term 'moment of inertia' itself was introduced by Euler in 1730.

How it works

Mathematically, for a point mass m at distance r from the axis, the moment of inertia is I = mr². For a continuous object, you sum (integrate) this over all mass elements. The total moment of inertia determines the angular acceleration α produced by a given torque τ via τ = Iα, mirroring Newton's second law F = ma. The concept extends to complex systems like spinning stars, gyroscopes, and even magnetic domain walls, where the effective mass of a wall is derived from its moment of inertia under precession.

In practice

Used in engineering to design flywheels for energy storage, in robotics to control arm movements, and in aerospace for satellite attitude control.

Fun fact

The moment of inertia of a neutron star is so large that its spin period can be measured with the precision of a few nanoseconds, making it one of the most accurate clocks in the universe.

Concept graph →

Formulas

$$h_0 = \frac{4\pi^2 G}{c^4} \frac{\varepsilon I_{zz} f_{\rm GW}^2}{d}$$
The amplitude of gravitational waves from a rotating ellipsoid.
What is in the formula
h_0variablegravitational wave strain amplitude — strain
εvariableellipticity — ellipticity
I_zzvariablemoment of inertia about rotation axis — moment of inertia kilogram metre squared
f_GWvariablegravitational wave frequency — frequency hertz
dvariabledistance to source — distance metre
GconstantNewton's constant of universal gravitation — gravitational constant = 6.67430e-11 cubic metre per kilogram per second squared
cconstantSpeed of light in vacuum — speed of light = 299792458 metre per second
$$\varepsilon = \frac{|I_{xx} - I_{yy}|}{I_{zz}}$$
Ellipticity measures deviation from axial symmetry.
What is in the formula
εvariableellipticity — ellipticity
I_xxvariablemoment of inertia about x-axis — moment of inertia kilogram metre squared
I_yyvariablemoment of inertia about y-axis — moment of inertia kilogram metre squared
I_zzvariablemoment of inertia about z-axis — moment of inertia kilogram metre squared
$$L_{\text{sd}} = -I \Omega \dot{\Omega}$$
Rotational energy loss rate equals moment of inertia times angular velocity times its derivative.
What is in the formula
Ivariablemoment of inertia — moment of inertia kilogram metre squared
Ωvariableangular velocity — angular velocity radian per second
Ω̇variabletime derivative of angular velocity — angular acceleration radian per second squared
$$I = \int r^2 dm$$
Moment of inertia measures mass distribution about axis.
What is in the formula
IvariableMoment of inertia about the axis — moment of inertia kilogram metre squared
rvariablePerpendicular distance from the axis to mass element — distance metre
dmvariableInfinitesimal mass element — mass kilogram
∫operatorIntegration over the entire body — integral operator
$$E_{\text{rot}} = \frac{1}{2} I \Omega^2$$
Energy due to rotation depends on moment of inertia and angular velocity.
What is in the formula
E_rotvariablerotational kinetic energy — energy joule
Ivariablemoment of inertia — moment of inertia kilogram metre squared
Ωvariableangular velocity — angular velocity radian per second
$$I = k M R^2$$
Moment of inertia for spherical objects with mass distribution.
What is in the formula
IvariableMoment of inertia — moment of inertia kilogram metre squared
kvariableDimensionless shape factor
MvariableMass of the sphere — mass kilogram
RvariableRadius of the sphere — radius metre

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