VELOCITY

The general problem of statics may be thus enunciated: Tujittd tlie equations of equilibrium of a body urged by any number hall of briefly forces indicate having any the steps directions by which whatever the equationst'n space. which expre the conditions of equilibrium are obtained, principle of the composition Of forces. 1. A force which acts on any point of a solid body may be regarded as applied at any point in the line of its direction, provided the different points of the line are invariably connected, as is the case with any two points of a rigid body. This, as already staled, may be a umed as an axiom. •J. If a material point is urged by two forces acting in the at line, and in the same direction, the etlect will be the same as if the point were urged in the same direction by a eingle force equal to their sum; and if the two forces net in opposite directions, the effect will be the same as if the two forces were replaced by a single force equal to their difference acting in the direction of the greater. The single Into which the others are compounded is called the resultant; and as a third force may lie compounded with the resultant of two, another with the new resultant, and so on, it is obvious that any number of parallel forces acta point have a single resultant. 1). It has been shown h 'Vne article Force (p. 4C2) that a force acting in any direction can always be decomposed into three others respectively parallel to three straight lines u space, provided no two of these lines be parallel. ■ it follows that whatever be the number or the dim of the forces which act upon a solid body, they may be all replaced by three sets of parallel forces acting in directions. We have, therefore, only to consider the theory of parallel forces. 1 When two parallel forces P and Q, acting in the same „, direction, are applied to the extremities of a / rigid straight line A B, the resultant It will / bo equal to their sum, parallel to their com- / mon direction, and will divide the line A B ■*■ c * into two parts in C, so that the distance of / / / the points of application from C are reciproL / «' cally as the forces, or so that A C: C B:: Q: P. Thus, if A P and B a be taken to '* represent the forces, and CR the resultant, we have CR=AP + BQ; and the two forces will be in equilibrium with the force CR', equal to C It, ami acting in the opposite direction. 5. When two parallel forces P and Q., acting in opposite directions, are applied to the extremities q of a rigid straight line A B, there are two K- I cases for consideration. If P and Q are un- /. / equal, they have a resultant which is par -/* rallel to their direction, equal to their difference,and applied at a point C in A B produced, which is such that A (J: C B:: Q,: P, 'P or A C: A B:: a: P — (i, and the two forces represented by AP and B CI will be in equilibrium with a force C R' equal and opposite to C R. But if the two forces P and Q. are equal, the point C, determined as above, is at an infinite distance, and there is consequently no resultant. In this case, therefore, the two forces force. I' and (i cannot make equilibrium with any single The system of two equal and parallel forces applied in opposite directions at the extremities of a rigid straight line is very important in statics. For the sake of facilitating calculation,the two forces may be transferred to other points in the lines of their direction, so that the line joining the new points of application shall be perpendicular to the direction of the forces. Under this form, the system has been aVuignntrd by the French mathematicians a couple; and its consideration has introduced great simplicity into the theory •>f equilibrium. We shall here state the principal properties of couples, referring for the details of demonstration to the Elemens de Statique of Poinsot. 6. Def.: A couple is a system of two equal parallel forces acting in opposite directions, and applied perpendicularly to the extremities of an inflexible straight line of a given length [s. 1169]