The perpendicular axis theorem (or plane figure theorem) states that the
moment of inertia 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
,
, and
(which meet at origin
) so that the body lies in the
plane, and the
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
:
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
and
are equal,
then the perpendicular axes theorem provides the useful relationship:
:
Derivation
Working in
Cartesian coordinates, the moment of inertia of the planar body about the
axis is given by:
:
On the plane,
, so these two terms are the moments of inertia about the
and
axes respectively, giving the perpendicular axis theorem.
The converse of this theorem is also derived similarly.
Note that
because in
,
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.
References
See also
*
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 ...
*
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 ...
{{DEFAULTSORT:Perpendicular Axis Theorem
Rigid bodies
Physics theorems
Articles containing proofs
Classical mechanics
Moment (physics)