Perpendicular axis theorem explained
The perpendicular axis theorem (or plane figure theorem) states that, "The moment of inertia (Iz) of a laminar body about an axis (z) perpendicular to its plane is the sum of its moments of inertia about two mutually perpendicular axes (x and y) in its plane, all the three axes being concurrent."
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. Then the perpendicular axis theorem states that
[1]
This rule can be applied with the parallel axis theorem and the stretch rule to find polar moments of inertia for a variety of shapes.
If a planar object has rotational symmetry such that
and
are equal,
[2] 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:
[3] Iz=\int(x2+y2)dm=\intx2dm+\inty2dm=Iy+Ix
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.
See also
Notes and References
- Book: Physics . Paul A. Tipler . Ch. 12: Rotation of a Rigid Body about a Fixed Axis . Worth Publishers Inc. . 0-87901-041-X . 1976 . registration .
- Book: Obregon
, Joaquin
. Mechanical Simmetry. 978-1-4772-3372-6. Author House. 2012.
- Book: Mathematical Methods for Physics and Engineering . registration . K. F. Riley, M. P. Hobson & S. J. Bence . Ch. 6: Multiple Integrals . Cambridge University Press . 978-0-521-67971-8 . 2006.