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| In physics, the '''perpendicular axis theorem''' (or '''plane figure theorem''') can be used to determine the [[moment of inertia]] of a [[rigid object]] that lies entirely within a plane, about an axis perpendicular to the plane, given the moments of inertia of the object about two [[perpendicular]] [[Coordinate axis|axes]] lying within the plane. The axes must all pass through a single point in the plane.
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| Define perpendicular axes <math>x\,</math>, <math>y\,</math>, and <math>z\,</math> (which meet at origin <math>O\,</math>) so that the body lies in the <math>xy\,</math> plane, and the <math>z\,</math> axis is perpendicular to the plane of the body. Let ''I''<sub>''x''</sub>, ''I''<sub>''y''</sub> and ''I''<sub>''z''</sub> be moments of inertia about axis x, y, z respectively, the perpendicular axis theorem states that<ref>{{cite book |title=Physics |author=Paul A. Tipler |chapter=Ch. 12: Rotation of a Rigid Body about a Fixed Axis |publisher=Worth Publishers Inc. |isbn=0-87901-041-X |year=1976}}</ref>
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| :<math>I_z = I_x + I_y\,</math>
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| This rule can be applied with the [[parallel axis theorem]] and the [[stretch rule]] to find moments of inertia for a variety of shapes.
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| If a planar object (or prism, by the [[stretch rule]]) has rotational symmetry such that <math>I_x\,</math> and <math>I_y\,</math> are equal, then the perpendicular axes theorem provides the useful relationship:
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| :<math>I_z = 2I_x = 2I_y\,</math> | |
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| == Derivation ==
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| Working in Cartesian co-ordinates, the moment of inertia of the planar body about the <math>z\,</math> axis is given by:<ref>{{cite book |title=Mathematical Methods for Physics and Engineering |author=K. F. Riley, M. P. Hobson & S. J. Bence |chapter=Ch. 6: Multiple Integrals |publisher=Cambridge University Press |isbn=978-0-521-67971-8 |year=2006}}</ref>
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| :<math>I_{z} = \int \left(x^2 + y^2\right)\, dm = \int x^2\,dm + \int y^2\,dm = I_{y} + I_{x} </math>
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| On the plane, <math>z=0\,</math>, so these two terms are the moments of inertia about the <math>x\,</math> and <math>y\,</math> axes respectively, giving the perpendicular axis theorem.
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| The converse of this theorem is also derived similarly.
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| Note that <math>\int x^2\,dm = I_{y} \ne I_{x} </math> because in <math>\int r^2\,dm </math>, r measures the distance from the ''axis of rotation'', so for a y-axis rotation, deviation distance from the axis of rotation of a point is equal to its x co-ordinate.
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| == References ==
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| {{reflist}}
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| ==See also==
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| * [[Parallel axis theorem]]
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| * [[Stretch rule]]
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| {{DEFAULTSORT:Perpendicular Axis Theorem}}
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| [[Category:Rigid bodies]]
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| [[Category:Physics theorems]]
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| [[Category:Articles containing proofs]]
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| [[Category:Classical mechanics]]
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