Rigid body.

551. We commence with the case of a rigid body or system, that is, an ideal substance continuously occupying a given solid figure, admitting no change of shape, but free to move translationally and rotationally. It is sometimes convenient to regard a rigid body as a group of material particles maintained by mutual forces in definite positions relatively to each other, but free to move relatively to other bodies. The condition of perfect rigidity is approximately fulfilled in natural solid bodies, so long as the applied forces are not sufficiently powerful to break them or to distort them, or to condense or rarefy them to a sensible extent. To find the conditions of equilibrium of a rigid body under the influence of any number of forces, we follow the example of Language in using the principle of work (§ 289) and take advantage of our kinematic preliminary (§ 197).

Equilibrium of free rigid body.

552. First supposing the body to be perfectly free to take any motion possible to a rigid body:—Give it an infinitesimal translation in any direction, and an infinitesimal rotation round any line.

I. In respect to the translational displacement, the work done by the applied forces is equal to the product of the amount of the displacement (being the same for all the points of application) into the algebraic sum of the components of the forces in its direction.