But CM and CN are perpendiculars from C to the lines in which P and Q act when they are hung at A' and B'; therefore, by Prop. vi., P and Q will balance on A'CB'. Wherefore, "If two weights, &c." Q. E. D. CHAPTER III. THE COMPOSITION AND RESOLUTION OF FORCES. 27. DEFINITION of COMPONENT, and RESULTANT Forces. A body that is acted on by two forces applied at the same instant to the same point, instead of having a tendency to move in either of the lines the forces act in, has a tendency to move in a line that lies between them. Whence it appears that the two original pressures produce a third pressure by their mutual action, which is called, from the circumstance of its "resulting" out of the action of the original pressures, their "Resultant" with respect to them, while they are called, with respect to it, its "Components." The Resultant (R) which produces the same effect as the compound action of the original forces P and Q that are applied at the same point, is said to be "compounded" of P and Q. This Resultant (R) also, if conceived to be the sole original force, may be supposed to be "resolved" into the two forces P and Q; since those two forces, acting in the manner described, (namely, at the same point, and at the same instant), produce exactly the same effect on the body as the single force R does. [An instance of the " Composition of Forces" is a boat moored in a stream by ropes attached to the boat and to either bank; the Component forces here being the tensions of the ropes, and their Resultant being the force by which the pressure of the stream on the boat is counteracted.] 28. PROP. VIII. If the adjacent sides of a parallelogram represent the component forces, [that act at a point at the same instant] in direction [i. e. in their lines of action] and magnitude, the diagonal will represent the resultant force (1) in direction [line of action], and (2) in magnitude. M B Let AB and AC represent the two component forces that act at 4, in direction and in magnitude. Complete the parallelogram BC, and draw the diagonal AD; Then AD will represent the resultant of AB and AC in (1) direction, and also in E (2) magnitude. T From D draw DM and DN perpendicular to AB and AC, produced if necessary. (1) Then in the triangles DBM, dcn, = angle DNC, angle DMB = right angle And angle DBM = angle BAC (since BD, AC are parallel and MBA cuts them) = angle DCN; .. the third angle, BDM, of the one triangle = the third angle, CDN, of the other; and the triangles are equiangular. .. CD: DN :: BD: DM and alternately, CD BD :: DN: DM. 22 CHAPTER III. THE COMPOSITION AND RESOLUTION OF FORCES. 27. DEFINITION of COMPONENT, and RESULTANT Forces. A body that is acted on by two forces applied at the same instant to the same point, instead of having a tendency to move in either of the lines the forces act in, has a tendency to move in a line that lies between them. Whence it appears that the two original pressures produce a third pressure by their mutual action, which is called, from the circumstance of its "resulting" out of the action of the original pressures, their "Resultant" with respect to them, while they are called, with respect to it, its "Components." The Resultant (R) which produces the same effect as the compound action of the original forces P and Q that are applied at the same point, is said to be "compounded" of P and Q. This Resultant (R) also, if conceived to be the sole original force, may be supposed to be "resolved" into the two forces P and Q; since those two forces, acting in the manner described, (namely, at the same point, and at the same instant), produce exactly the same effect on the body as the single force R does. |