Classical Mechanics
The moment of inertia (also called the mass moment of inertia or rotational inertia) is a physical quantity that measures how resistant a body is to changes in its rotational motion about a given axis. It plays the same role in rotational dynamics that mass plays in linear dynamics: for a point mass, it is the product of the mass and the square of the distance from the axis of rotation. For a continuous rigid body, it is the integral of that quantity over all mass elements. The SI unit of moment of inertia is the kilogram-meter squared (kg·m²).
The moment of inertia appears in the rotational analog of Newton's second law: torque equals moment of inertia times angular acceleration (τ = Iα). Thus, for a given torque, a larger moment of inertia produces a smaller angular acceleration, just as a larger mass produces a smaller linear acceleration for a given force. The physical meaning is that the moment of inertia measures how the mass is distributed relative to the axis of rotation; mass farther from the axis contributes more to the moment.
The value of the moment of inertia depends not only on the total mass but also on its distribution. For a continuous body, it is computed by integrating the mass element times the square of its perpendicular distance from the axis: I = ∫ r² dm. For a collection of point masses, the sum is I = Σ mᵢ rᵢ². For symmetric shapes, standard formulas exist; for example, a solid cylinder rotating about its central axis has I = (1/2)MR².
A common shortcut is the parallel axis theorem, which allows the moment of inertia about any axis to be calculated from the moment of inertia about a parallel axis through the center of mass. The theorem states I = I_cm + Md², where M is the total mass and d is the distance between the two axes.
For a rigid body, the moment of inertia depends on the orientation of the axis; the body has principal axes about which the product of inertia terms vanish, and the corresponding moments are called principal moments of inertia. For a uniformly dense sphere, all principal axes are equivalent, giving the same moment about any diameter. For a rod, the moment about an axis perpendicular to its length through the center is (1/12)ML², whereas about an end it is (1/3)ML².
The moment of inertia is central in engineering design, from flywheels that store rotational kinetic energy to the stability of spacecraft and the behavior of gyroscopes. In biomechanics, the moment of inertia of a limb about a joint affects the effort required to swing it.
In rotational collisions, the conservation of angular momentum is used along with the moment of inertia; for example, a figure skater pulls in her arms to reduce her moment of inertia and thus spin faster. The concept also appears in celestial mechanics, where the moments of inertia of planets and moons provide clues about their internal mass distribution, as measured by the moment of inertia factor (I/MR²).
The term 'moment of inertia' was introduced by Leonhard Euler in his 1765 book 'Theoria motus corporum solidorum seu rigidorum', although the concept had been earlier developed by Christiaan Huygens in his studies of pendulums and by Jakob Bernoulli.
There exist higher-order moments called the second moment of area (also known as the area moment of inertia) used in beam theory, which is distinct from the mass moment of inertia. The radius of gyration is a related length equal to the square root of (I/m). For irregular bodies, the moment of inertia can be measured experimentally by suspending the body and measuring the period of small oscillations. In general relativity, the moment of inertia of a rotating black hole does not follow the Newtonian formula but rather is proportional to the square of its mass, as described by the Kerr solution.
The concept of moment of inertia was first formulated by Huygens and later formalized by Euler.
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