Parallel axis theorem: Difference between revisions

Revision as of 20:42, 7 August 2005

Revision as of 21:30, 22 December 2005

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In [[physics]], the '''parallel axes rule''' can be used to determine the [[moment of inertia]] of a [[rigid object]] about any axis, given the moment of inertia of the object about the [[Parallel (geometry)|parallel]] axis through the object's [[center of mass]] and the [[perpendicular]] [[distance]] between the axes.

Let ''I''_''GCM'' denote the moment of inertia of the object about the center of mass, ''M'' the object's mass and ''d'' the perpendicular distance between the two axes. Then the moment of inertia about the new axis ''z'' is given by:

:<math>I_z = I_GI_{CM} + M d^2.\,</math>

This rule can be applied with the [[stretch rule]] and [[perpendicular axes rule]] to find moments of inertia for a variety of shapes.

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[[Image:Parallelaxes.jpg|thumb|right|250px|Parallel axes rule for area moment of inertia.]]

The parallel axes rule also applies to the [[second moment of area]] (area moment of inertia);

:<math>I_z = I_xI_{CM} + Ad^2.\,</math>

where ''I_z'' is the parallel axis, ''I_xCM'' is the area moment of inertia through the [[center of gravity]] of the [[area]], ''A'' is the surface of the area, and ''d'' is the distance from the new axis ''z'' to the center of gravity of the area.