Fundamental Equations

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 $N$ elements, and let the $i$th element have mass $m_i$, instantaneous displacement ${\bf r}_i$, and instantaneous velocity ${\bf v}_i$. The equation of motion of the $i$th element is written

$$m_i\,\frac{d{\bf v}_i}{dt} = \sum_{j=1,N}^{j\neq i} {\bf f}_{ij} + {\bf F}_i.$$ (1.178)

(See Section 1.4.1.) 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 that 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 that originate outside the body.

Repeating the analysis of Section 1.4.2, we can sum Equation (1.178) over all mass elements to obtain

$$M\,\frac{d^{2}{\bf R}}{dt^{2}} = {\bf F}.$$ (1.179)

Here, $M=\sum_{i=1,N} m_i$ is the total mass, ${\bf R}$ the displacement of the center of mass [see Equation (1.68)], 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 like a point particle whose mass is identical to that of the body.

Repeating the analysis of Section 1.4.5, we can sum ${\bf r}_i\,\times$ Equation (1.178) over all mass elements to obtain

$$\frac{d {\bf L}}{dt} = \mbox{\boldmath$\tau$} .$$ (1.180)

Here, ${\bf L}=\sum_{i=1,N} m_i\,{\bf r}_i\times {\bf v}_i$ is the total angular momentum of the body (about the origin), and $\tau$$= \sum_{i=1, N} {\bf r}_i\times{\bf F}_i$ the total external torque (about the origin). Note that the previous equation is only valid if the internal forces are central in nature. However, this is not a particularly onerous constraint. Equation (1.180) 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 (1.179)], and, therefore, fairly straightforward in nature.