But R is the resultant of P and Q; therefore Ac is the diagonal of the parallelogram on Aa, Ab (see Art. 17). Hence if two straight lines, having a common extremity, represent the axes of two couples, that diagonal of the parallelogram described on these lines which passes through their common extremity is equal in magnitude and direction to the axis of the resultant couple. 50. PROP. To find the magnitude and position of the couple which is the resultant of three couples which act in planes at right angles to each other. Let AB, AC, AD be the axes of the given couples (see fig. to Art. 24). Complete the parallelogram CB, and draw AE the diagonal. Then AE is the axis of the couple which is the resultant of the two couples of which the axes are AB, AC. Complete the parallelogram DE, and draw AF the diagonal. Then AF is the axis of the couple which is the resultant of the couples of which the axes are AE, AD, or of those of which the axes are AB, AC, AD. Now AF AE + AD* = AB+ AC*+ AD". Let G be the moment of the resultant couple; L, M, N those of the given couples; therefore G2 = L2 + M2 + N2; and if λ, μ, v be the angles the axis of the resultant makes with those of the components, 51. Hence conversely any couple may be replaced by three couples acting in planes at right angles to each other; their moments being G cosλ, G cosμ, & cosv; where G is the moment of the given couple, and λ, μ, v the angles its axis makes with the axis of the three couples. Thus couples follow, as to their composition and resolution, laws similar to those which apply to forces, the axis of the couple corresponding to the direction of the force and the moment of the couple to the intensity of the force. Hence for example, by Art. (29), the resolved part of a resultant couple in any direction is equal to the sum of the resolved parts of the component couples in the same direction. CHAPTER IV. RESULTANT OF FORCES IN ONE PLANE. EQUILIBRIUM. CONDITIONS OF MOMENTS. 52. PROP. To find the resultant of any number of parallel forces acting on a rigid body in one plane. Let P1, P2, P....... denote the forces. Take any point in 27 the plane of the forces as origin and draw rectangular axes Ox, Oy, the latter parallel to the forces. Let A, be the point where Ox meets the direction of P1, and let OA, = x. Apply at O two forces each equal and parallel to P1, in Thus the force P is replaced by P opposite directions. at O along Oy, and a couple of which the moment is P1.OA1, that is P.,. Transform the other forces in a similar manner, using a similar notation, and the whole system will be reduced to a force P1 + P2 + P2+ ......... or EP along Oy, and a couple P11 + P12+ Px +......... or ΣPx in the plane of the forces and tending to turn the body from the axis of x to the axis of y. 53. PROP. To find the conditions of equilibrium of a system of parallel forces acting on a rigid body in one plane. A system of parallel forces can be reduced to a single force and a couple. If neither of these vanish equilibrium is impossible, because a single force cannot neutralise a couple (Art. 40). If the single force alone vanish equilibrium is impossible, because there remains an unbalanced couple. If the couple alone vanish equilibrium is impossible, because there remains an unbalanced force. Hence, for equilibrium it is necessary that both the force and the couple should vanish; that is ΣP: = 0 and ΣPx = 0. 54. DEF. The product of a force into the perpendicular drawn upon it from any point, is called the moment of the force with respect to that point. Hence the conditions of equilibrium which have just been obtained for a system of parallel forces acting in one plane may be thus enunciated. (1) The sum of the forces must vanish. (2) The sum of the moments of the forces with respect to any point in the plane of the forces must vanish. The word sum must be understood algebraically; forces which act in one direction being considered positive, those in the opposite direction must be considered negative. Also a moment being considered positive when the force tends to urge the point at which it is applied from right to left when the eye is placed at the origin and looks along the perpendicular on the force, a force tending to urge the point of application from left to right will have a negative moment. 55. When the sum of the forces vanishes in Art. (52), the forces reduce to a couple. ΣΡ When EP is not zero, the from A is ΣP' y' A ΣΡ Let EP acting at A and EP acting The latter force is destroyed Hence the single resultant is a point the distance of which 56. PROP. To find the resultant of any number of forces which act upon a rigid body in one plane. Let the system be referred to any rectangular axes Ox, Oy in the plane of the forces. |