Perpendicular axis theorem
Encyclopedia
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
, 
, 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 Ix, Iy and Iz be moments of inertia about axis x, y, z respectively, the perpendicular axis theorem states that
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
and 
are equal, then the perpendicular axes theorem provides the useful relationship:
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.
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, the perpendicular axis theorem (or plane figure theorem) can be used to determine the moment of inertia
Moment of inertia
In classical mechanics, moment of inertia, also called mass moment of inertia, rotational inertia, polar moment of inertia of mass, or the angular mass, is a measure of an object's resistance to changes to its rotation. It is the inertia of a rotating body with respect to its rotation...
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
Perpendicular
In geometry, two lines or planes are considered perpendicular to each other if they form congruent adjacent angles . The term may be used as a noun or adjective...
axes lying within the plane. The axes must all pass through a single point in the plane.
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 Ix, Iy and Iz be moments of inertia about axis x, y, z respectively, the perpendicular axis theorem states thatThis rule can be applied with the parallel axis theorem
Parallel axis theorem
In physics, the parallel axis theorem or Huygens-Steiner theorem can be used to determine the second moment of area or the mass moment of inertia of a rigid body about any axis, given the body's moment of inertia about a parallel axis through the object's centre of mass and the perpendicular...
and the stretch rule
Stretch rule
In physics, the stretch rule states that the moment of inertia of a rigid object is unchanged when the object is stretched parallel to the axis of rotation,...
to find moments of inertia for a variety of shapes.
If a planar object (or prism, by the stretch rule
Stretch rule
In physics, the stretch rule states that the moment of inertia of a rigid object is unchanged when the object is stretched parallel to the axis of rotation,...
) has rotational symmetry such that
and are equal, then the perpendicular axes theorem provides the useful relationship:Derivation
Working in Cartesian co-ordinates, 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.See also
- Parallel axis theoremParallel axis theoremIn physics, the parallel axis theorem or Huygens-Steiner theorem can be used to determine the second moment of area or the mass moment of inertia of a rigid body about any axis, given the body's moment of inertia about a parallel axis through the object's centre of mass and the perpendicular...
- Stretch ruleStretch ruleIn physics, the stretch rule states that the moment of inertia of a rigid object is unchanged when the object is stretched parallel to the axis of rotation,...