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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(x^{2} + y^{2}). The moment of inertia is related to the kinetic energy of rotation. A mass m moving at speed v has kinetic energy KE = ½ mv^{2}. Hence if m is rotating about the origin with angular velocity ω radians per second, its speed is
and Thus moment of inertia is the rotational analogue of mass. DEFINITION Given a plane object on the region D with continuous density ρ(x, y), the moment of inertia about the origin is JUSTIFICATION On an element of volume ΔD, the moment of inertia is ΔI ≈ (x^{2} + y^{2}) Δm ≈ ρ(x, y)(x^{2} + y^{2}) ΔA (compared to ΔA). The integral for I follows by the Infinite Sum Theorem.


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