Calculating angular momentum for
a rigid body
Alternate forms
for calculating angular momentum: A useful expression for calculating the angular
momentum is based on introducing a point A
fixed to the body and using the kinematic relations
_{}

where
_{} is the angular
velocity of the body as seen by an observer on the reference frame. Introducing
these relations into the expression for the angular velocity yields the
expression
_{}
Angular momentum
of a body rotating around a fixed point on the reference frame: If the body is pinned or
ball and socket connected to the reference frame at O, then one can simplify the above expressions by taking A equal to O to get
_{}
Using an
arbitrary intermediate point A on the
rigid body: If
A is an arbitrary point on the rigid
body, one can use the relation
_{}
to
get
_{}
Using the
intermediate point A as the center of
mass: If
one sets A as the center of mass,
then the expression for the angular momentum about O becomes
_{}
Since
the mass of the body is “evenly” distributed around the center of mass, one can
conclude that
_{}
Therefore,
one obtains the expression for angular momentum about an arbitrary point O as
_{}
Note: in all the above
expressions for angular momentum an integral needs to be calculated of the form
_{}
A
general expression for this can be found by selecting
_{}
and
directly calculating the integral to get
_{}
where
the moments of inertia are given by
_{}
and
the products of inertia are given by
_{}
ã Mehrdad Negahban and the University of Nebraska, 19962002.
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Copy and distribute freely for personal use only
Department of Engineering Mechanics, University of Nebraska, Lincoln, NE 685880347