COMPOSITION Ais-n RESOLUTION OF FORCES AND MOTIONS. 1. The fundamental problem in statics is to find the magnitude and direction of the resultant of two forces; in other words, to compound them into a single force, which shall be in every respect their equivalent. Intensity and direction only elements necessary to entirely describe a force, forces in statics are represented by lines, which are obviously capable of being made to represent them both in magnitude and direction. When two forces act along the same straight line on a particle, it is sufficiently obvious that if they act in the same direction, the resultant will be their algebraical sum; if in opposite direc tions, their algebraical difference. This being premised, the relation between two forces acting at the same point, but not in the same line, and their resultant, is set forth in the following theorem, which is known as the parallelogram of forces: If two forces, P, Q. acting on a particle A, be represented in direction and magnitude by the lines Ap, Aq, then the resultant will be represented in direc tion and magnitude by the diagonal Ar of the parallelo gram described upon Ap, Aq. The proof of this depends upon the simple principles, that a force may be supposed to act at any point of its direction, that point being con ceived to be rigidly attached to the particle on which the force acts; and what may be accepted as an axiom of universal experience, that when any number of forces are impressed on a particle or body, each exerts itself, as if the others were not acting, to produce its full effect. See any elementary treatise on mechanics. The doctrine of the parallelogram of forces has given rise to much controversy, not as to its truth, but as to its derivation, some appear ing to contend that it is directly deducible from the axiom above stated, without the necessity of further reasoning.—Knowing how to compound two forces acting at a point, we are able to compound or determine the resultant of any number. If the forces, though in the same plane, do not act at the same point of a body, those of them whose directions meet may be compounded by the preceding rule; if they arc parallel, their resultant is a force parallel to them and equal to their algebraical sum, counting those acting in one direction as positive, and in the opposite direction as negative. For the
position of the resultant in this case, sec PARALLEL FORCES. The singular case is that of equal parallel forces acting in opposite directions. These constitute a couple, and cannot be represented by any single force. See COUPLE.
2. The resolution of forces is the converse problem. To resolve a given force R, whose direction and magnitude is Ar, into two forces acting in any directions that may be chosen, as AP, AQ, we have only to draw parallels through r, which determine the lines Ap, Aq, representing the magnitude of the forces required. It is evident that there is an indefinite number of pairs of forces into which Ar might be resolved, accord ing to the direction in which the new forces arc to act. It is usual, however, to resolve a force into forces that are at right angles to each other.
3. The composition of motions is analogous in every *-ay to that of forces; motions are the results of forces, and the analogy might be expected. If a body be actuated simultaneously by two velocities having different directions, it will evidently move in a direction intermediate to the two, and with a velocity which will in some way depend on each of them, and which is called their resultant. The proposition which gets forth how to find the resultant, is called the parallelogram of 'velocities. It is: If two veloci ties, with which a particle is simultaneously impressed, be represented in direction and magnitude by two straight lines drawn from the particle, the resultant velocity of the particle will be represented in direction and magnitude by the diagonal of the parallelo gram described on those two straight lines. The proof is very simple. There is no reason why the full effect of both velocities should not be produced, as if the body moved first with one of them, and then with the other, in their respective directions. If in one second the body moving with the one velocity would reach p, and if we sup pose it then to move on pr for another second, parallel with the other velocity, it would at the end of the second second be at r. Hence, under their joint influence, it will be at r at the end of one second.
4. The resolution of motions is altogether analogous to that of forces.