The perpendicular axis theorem (or plane figure theorem) states that the

moment of inertia The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular accele ...

of a

planar lamina In mathematics, a planar lamina (or plane lamina) is a figure representing a thin, usually uniform, flat layer of the solid. It serves also as an idealized model of a planar cross section of a solid body in integration. Planar laminas can be use ...

(i.e. 2-D body) about an axis perpendicular to the

plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * ''Planes'' ...

of the lamina is equal to the sum of the moments of inertia of the lamina about the two axes at right angles to each other, in its own plane intersecting each other at the point where the perpendicular axis passes through it. 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. Then the perpendicular axis theorem states that :

I_z = I_x + I_y

This rule can be applied with the

parallel axis theorem The parallel axis theorem, also known as Huygens–Steiner theorem, or just as Steiner's theorem, named after Christiaan Huygens and Jakob Steiner, can be used to determine the moment of inertia or the second moment of area of a rigid body ...

and the

stretch rule In classical mechanics, the stretch rule (sometimes referred to as Routh's rule) states that the moment of inertia of a rigid object is unchanged when the object is stretched parallel to an axis of rotation that is a principal axis, provided that ...

to find polar moments of inertia for a variety of shapes. If a planar object 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 coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured i ...

, the moment of inertia of the planar body about the

z

axis is given by: :

I_ = \int (x^2 + y^2) \,dm = \int x^2\,dm + \int y^2\,dm = I_ + I_

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_ \ne I_

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'' coordinate.

Derivation

References

See also