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# Moment of inertia

Moment of inertia is rotational inertia, i.e., moment of inertia is to rotational motion as mass is to linear motion. Rotational versions of Newton's Second Law, momentum, and the formula for kinetic energy use this value (with torque, angular velocity and angular acceleration replacing force, velocity and acceleration, respectively). The moment of inertia for an object depends on its shape and distribution of mass within that shape: the more the mass is on the outside with respect to the axis of rotation, the larger the moment of inertia. For a given mass M and radius r, in order of increasing moment of inertia we have a solid sphere, a solid cylinder, a hollow sphere and a hollow cylinder, namely cMr2, with c=2/5, 1/2, 2/3 and 1, respectively. The general form of the moment of inertia involves an integral.

The moment of inertia is often represented by the letter I.

For an bunch of infinitely small particles with mass $m_n$, and distance $r_n$ from one particular axis, the moment of inertia for that bunch of particles around that axis will be:

$I = \sum_n r_n^2 m_n$

Continuous mass distributions require an infinite sum over all the point mass moments which make up the whole. This is accomplished by an integration over all the mass:

$I = \int_0^M r^2\,dm$

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