Chapter VI - THE SOLUTION OF PROBLEMS

Summary

We have found that conditions sufficient to ensure the equilibrium of a rigid body under the action of coplanar forces are three in number, and may be summarized as—

two resolutions and one equation of moments,

or one resolution and two equations of moments,

or three equations of moments.

The student will do well to bear in mind that, in the solution of a problem, no additional information can be obtained by writing down more than three such equations of moments or resolutions for one and the same body.

It follows that many statical problems are indeterminate or insolvable without further hypotheses as to the elastic properties of the body acted upon. For example, if a beam or bar under the action of given forces has its ends fixed, it is not possible to determine the magnitude and direction of the reactions at the fixed ends, because this means four unknown quantities, two magnitudes and two directions, and the conditions of equilibrium give only three equations between the four unknown quantities.

Constraints and Degrees of Freedom. The position of a plane body in its plane is determined by three quantities, such as the rectangular co-ordinates of one specified point of the body and the angle which a line fixed in the body makes with a line fixed in the plane. Hence a body free to move in a plane has three degrees of freedom. We may also express this fact by saying that it is free to have a motion of translation with components in either or both of two perpendicular directions and also free to rotate.