We can think of a rigid body as a collection of a large number of small mass elements
that 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
![$\displaystyle m_i\,\frac{d^{\,2}{\bf r}_i}{dt^{\,2}} = \sum_{j=1,N}^{j\neq i}
{\bf f}_{ij} + {\bf F}_i.$](img1485.png) |
(8.1) |
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 that develop within the body in order to ensure that its various
elements maintain a fixed spatial relationship with respect to one another.
Of course,
, by Newton's third law.
The external forces represent forces that originate outside the body.
Repeating the analysis of Section 2.6, we can
sum Equation (8.1) over all mass elements to obtain
![$\displaystyle M\,\frac{d^{\,2}{\bf r}_{cm}}{dt^{\,2}} = {\bf F}.$](img1487.png) |
(8.2) |
Here,
is the total mass,
the position vector of the center of mass [see Equation (2.27)],
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 2.6, we can sum
Equation (8.1) over all mass elements to obtain
Here,
is the
total angular momentum of the body (about the origin), and
![$\tau$](img279.png)
the
total external torque (about the origin). The preceding equation is
only valid if the internal forces are central in nature. However, this
is not a particularly onerous constraint. Equation (8.3) 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, because their translational motion is similar to that
of point particles [see Equation (8.2)] and, therefore, is fairly straightforward
in nature.