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We can think of a rigid body as a collection of a large number of small mass elements
which all maintain a fixed spatial relationship with respect to one another.
Let there be elements, and let the th element be of mass , and instantaneous position
vector . The equation of motion of the th element
is written



(454) 
Here, is the internal force exerted on the th element by the
th element, and the external force acting on the th
element. The internal forces represent the
stresses which develop within the body in order to ensure that its various
elements maintain a constant spatial relationship with respect to one another.
Of course,
, by Newton's third law.
The external forces represent forces which originate outside the body.
Repeating the analysis of Section 2.7, we can
sum Equation (454) over all mass elements to obtain

(455) 
Here,
is the total mass,
the position vector of the center of mass [see Equation (27)],
and
the total external force.
It can be seen that the center of mass of a rigid body moves under the action of the external forces like
a point particle whose mass is identical with that of the body.
Again repeating the analysis of Section 2.7, we can sum
Equation (454) over all mass elements to obtain

(456) 
Here,
is the
total angular momentum of the body (about the origin), and
the
total external torque (about the origin). Note that the above equation is
only valid if the internal forces are central in nature. However, this
is not a particularly onerous constraint. Equation (456) describes
how the angular momentum of a rigid body evolves in time under the action
of
the external torques.
In the following, we shall only consider the rotational motion of rigid bodies, since their translational motion is similar to that
of point particles [see Equation (455)], and, therefore, fairly straightforward
in nature.
Next: Moment of Inertia Tensor
Up: Rigid Body Rotation
Previous: Introduction
Richard Fitzpatrick
20110331