The perpendicular axis theorem (or plane figure theorem) states that for a planar lamina the moment of inertia about an axis perpendicular to the plane of the lamina is equal to the sum of the moments of inertia about two mutually perpendicular axes in the plane of the lamina, which intersect at the point where the perpendicular axis passes through. This theorem applies only to planar bodies and is valid when the body lies entirely in a single 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. 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 abo ...

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 th ...

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 In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...

, 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