Parallel axis theorem: Difference between revisions

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Revision as of 09:54, 23 February 2020

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[[File:Steiner.png|thumb|right|The mass moment of inertia of a body around an axis can be determined from the mass moment of inertia around a parallel axis through the centre of mass.]]

Suppose a body of mass {{math|''m''}} is rotated about an axis {{math|''z''}} passing through the body's [[centre of gravity]]. The body has a moment of inertia {{math|''I''_cm}} with respect to this axis.

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The parallel axis theorem states that if the body is made to rotate instead about a new axis {{math|''z′''}} which is parallel to the first axis and displaced from it by a distance {{math|''d''}}, then the moment of inertia {{math|''I''}} with respect to the new axis is related to {{math|''I''_cm}} by

:<math> I = I_\mathrm{cm} + md^2.</math>

Explicitly, {{math|''d''}} is the perpendicular distance between the axes {{math|''z''}} and {{math|''z&prime;''}l}o.

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===Derllloivation===

The parallel axis theorem can be applied with the [[stretch rule]] and [[perpendicular axis theorem]] to find moments of inertia for a variety of shapes.

[[Image:Parallelaxes-1.png|thumb|right|Parallel axes rule for area moment of inertia]]

===Derivation===

We may assume, without loss of generality, that in a [[Cartesian coordinate system]] the perpendicular distance between the axes lies along the ''x''-axis and that the center of mass lies at the origin. The moment of inertia relative to the ''z''-axis is