Perpendicular axis theorem

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 axes lying within the plane. The axes must all pass through a single point in the plane.

Define perpendicular axes $x\,$, $y\,$, and $z\,$ (which meet at origin $O\,$) so that the body lies in the $xy\,$ plane, and the $z\,$ axis is perpendicular to the plane of the body. Let I_x, I_y and I_z be moments of inertia about axis x, y, z respectively, the perpendicular axis theorem states that^[1]

$I_z = I_x + I_y\,$

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

If a planar object (or prism, by the stretch rule) has rotational symmetry such that $I_x\,$ and $I_y\,$ are equal, then the perpendicular axes theorem provides the useful relationship:

$I_z = 2I_x = 2I_y\,$

Derivation

Working in Cartesian co-ordinates, the moment of inertia of the planar body about the $z\,$ axis is given by:^[2]

$I_{z} = \int \left(x^2 + y^2\right)\, dm = \int x^2\,dm + \int y^2\,dm = I_{y} + I_{x}$

On the plane, $z=0\,$, so these two terms are the moments of inertia about the $x\,$ and $y\,$ axes respectively, giving the perpendicular axis theorem. The converse of this theorem is also derived similarly.

Note that $\int x^2\,dm = I_{y} \ne I_{x}$ because in $\int r^2\,dm$, 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.

References

1. Paul A. Tipler (1976). "Ch. 12: Rotation of a Rigid Body about a Fixed Axis". Physics. Worth Publishers Inc. ISBN 0-87901-041-X.
2. K. F. Riley, M. P. Hobson & S. J. Bence (2006). "Ch. 6: Multiple Integrals". Mathematical Methods for Physics and Engineering. Cambridge University Press. ISBN 978-0-521-67971-8.

See also

• Parallel axis theorem
• Stretch rule