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Moment of Inertia

The second moment, or moment of inertia, of a point mass m about the origin is the mass times the square of the distance to the origin,

I = m(x2 + y2).

The moment of inertia is related to the kinetic energy of rotation. A mass m moving at speed v has kinetic energy

KE = ½ mv2.

Hence if m is rotating about the origin with angular velocity ω radians per second, its speed is




Thus moment of inertia is the rotational analogue of mass.


Given a plane object on the region D with continuous density ρ(x, y), the moment of inertia about the origin is



On an element of volume ΔD, the moment of inertia is

ΔI ≈ (x2 + y2) Δm ≈ ρ(x, y)(x2 + y2) ΔA (compared to ΔA).

The integral for I follows by the Infinite Sum Theorem.

Example 4: Moment of Inertia

Last Update: 2010-11-25