Fundamental Equations

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 $N$ elements, and let the $i$th element be of mass $m_i$, and instantaneous position vector ${\bf r}_i$. The equation of motion of the $i$th element is written

$$m_i\,\frac{d^2{\bf r}_i}{dt^2} = \sum_{j=1,N}^{j\neq i} {\bf f}_{ij} + {\bf F}_i.$$ (526)

Here, ${\bf f}_{ij}$ is the internal force exerted on the $i$th element by the $j$th element, and ${\bf F}_i$ the external force acting on the $i$th element. The internal forces ${\bf f}_{ij}$ 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, ${\bf f}_{ij} = - {\bf f}_{ji}$, by Newton's third law. The external forces represent forces which originate outside the body.

Generalizing the analysis contained in Sect. 3.6, we can sum Eq. (526) over all mass elements to obtain

$$M\,\frac{d^2{\bf r}_{cm}}{dt^2} = {\bf F}.$$ (527)

Here, $M=\sum_{i=1,N} m_i$ is the total mass, ${\bf r}_{cm}$ the position vector of the center of mass [see Eq. (125)], and ${\bf F} = \sum_{i=1,N} {\bf F}_i$ 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 as a point particle whose mass is identical with that of the body.

Again generalizing the analysis of Sect. 3.6, we can sum ${\bf r}_i\times$ Eq. (526) over all mass elements to obtain

$$\frac{d{\bf L}}{dt} = {\bf T}.$$ (528)

Here, ${\bf L}=\sum_{i=1,N} m_i\,{\bf r}_i\times d{\bf r}_i/dt$ is the total angular momentum of the body (about the origin), and ${\bf T} = \sum_{i=1, N} {\bf r}_i\times{\bf F}_i$ 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 (528) 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 Eq. (527)], and, therefore, fairly straightforward in nature.