Moment of Inertia for a cylinder

This page discusses the moment of inertia specifically in the case of a cylinder or rod.

Written by Jack Corelli

The Main Idea

See The Moments of Inertia for a more general explanation of moments of inertia.

The moment of inertia of an object relates the mass of the object, the distance between the center of mass and the shape of rotation. In a ring, or uniform loop, the center of mass is exactly at the center of the ring, and the moment of inertia can be found by approximating the exterior ring as a single line, akin to a circle.

A Mathematical Model

In its simplest form, the moment of inertia can be found for a simple, constant density and shape rod by the equation:

[math]\displaystyle{ I = \frac{1}{2}*M*R^2 }[/math] Where I is the moment of inertia, M is the mass of the object being rotated and R is the radius of the rod. This describes the moment of inertia calculated when a rod is spun about its center axis.

As can be seen here:

If the axis of rotation is no longer the center of the cylinder or rod, the equations for moment of inertia change distinctly. At the center of a rod, with the axis of rotation perpendicular to the length:

[math]\displaystyle{ I = \frac{1}{4}*M*R^2 + \frac{1}{12}*M*L^2 }[/math]

As can be seen here:

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