Rigid Bodies

Abstract

We start from the fundamental equation F = ma for the motion of a single particle. It follows immediately that

$$ \sum F = \sum {ma} ,\quad \sum {r \times } \,F = \sum {r \times \,} ma$$

((11.1.1))

, where the summation is over a system of particles. For economy of notation the suffix i labelling a particular particle is omitted in (11.1.1), so that F represents the resultant of the external and internal forces on a typical particle of mass m, position vector r and acceleration a. When the internal forces form a null system (that is, a system with zero resultant and zero moment), as is the case when they occur in equal and opposite pairs in the same line, they do not contribute to the left-hand sides of either of equations (11.1.1) and the summation is then confined to the external forces only. The same is true of a rigid body if it is regarded as a system of particles obeying the same rules. There are, however, logical difficulties associated with this assumption, one of which is that there are limiting processes involved in the mathematics since rigid bodies are modelled as continuous distributions of matter. Another difficulty is that a model in which the internal forces occur in equal and opposite pairs acting along the same line is not realistic, since there are electromagnetic forces involved which do not have this property. Indeed the internal forces in real solids are not adequately described by classical mechanics.