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Moment of inertia quantifies the resistance of a physical object to angular acceleration. Moment of inertia is to rotational motion as mass is to linear motion.
In general, an object's moment of inertia depends on its shape and distribution of mass within that shape: the greater the concentration of material away from the centroid, the larger the moment of inertia. It also varies depending upon the axis of rotation specified; values relative to the object's centroid are typically taken as baseline values. See the list of moments of inertia for specific examples. The parallel axis theorem can be used to determine moments of inertia relative to displaced axes of rotation.
Rotational versions of Newton's Second Law and the formulas for momentum and kinetic energy, use the moment of inertia of an object (with torque, angular velocity and angular acceleration replacing force, velocity and acceleration, respectively).
Moment of inertia is often represented by the letter .
A rigid body can be considered an infinite number of infinitely small particles, each with mass . If each particle is a distance from a particular axis of rotation, then the moment of inertia of the rigid body about that axis is given by:
Mathematical derivation
Continuous mass distributions require an infinite sum over all the point mass moments which make up the whole. This is accomplished by integrating all the masses over all three-dimensional space involved:
is defined by the spatial density distribution ρ.
Area moment of intertia
Bending stress within a beam depends upon the moment of inertia of the cross-sectional area of the beam. This area moment of inertia is derived by replacing the term for mass, m, in the above formula, with a term for area, A, and integrating over two-dimensional space.
See also: list of moments of inertia, torque
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