S R tively when it falls upon the prolongation of the force, as Op'. The product obtained by multiplying any force by its virtual velocity is called the virtual moment of the force Assume the figure and notation of Article 36. Op, Oq, and Or are the virtual velocities of the forces P, Q, and R. Let us denote the virtual velocity of o any force by the symbol of variation d, followed by a small letter of the same name as that which designates the force. a Fig. 18. We have from the figure, as in Article 36, the relations, Multiplying both members of the first by cos p, and of the second by sin, and adding the resultant equations, we have, R cos = P(cos a cos + sin a sin o̟) + Or, by reduction, R cos o P cos (p − a) + Q cos (9 + B). But, from the right-angled triangles C'Op, COq, and C'Or, Substituting these in the preceding equation, and reducing, we have, R&r= Pop + Q&q. Hence, the virtual moment of the resultant of two forces, is equal to the algebraic sum of the virtual moments of the two forces taken separately. If we regard the force as the resultant of two other forces, and one of these as the resultant of two others, and so on, the principle may be extended to any number of forces, applied at the same point. This principle may be expressed by the following equation: Hence, the virtual moment of the resultant of any number of forces applied at the same point, is equal to the algebraio sum of the virtual moments of the forces taken separately. This is called the principle of virtual moments. If the resultant is equal to 0, the system is in equilibrium, and the algebraic sum of the virtual moments is equal to 0; conversely, if the algebraic sum of the virtual moments of the forces is equal to 0, the resultant is also equal to 0, and the forces are in equilibrium. This principle, and the preceding one, are much used in discussing the subject of machines. Resultant of parallel Forces. N 39. Let P and Q be two forces lying in the same plane, and applied at points invariably connected, for example, at the points M and N of a solid body. Their lines of direction being prolonged, will meet at some point O; and if we suppose the points of application to be transferred to O, their resultant may be deter M mined by the parallelogram of forces. The direction of the resultant will pass through O. (Art. 27.) Whether the forces be transferred to O or not, the direction of the resultant will always pass through O, and this whatever may be the value of the included angle. Now, supposing the points of application to be at M and N, let the force Q be turned about N as an axis. As it approaches parallelism with P, the point will recede from M and N, and the resultant will also approach parallelism with P. Finally, when becomes parallel to P, the point O will be at an infinite distance from M and N, and the resultant will also be parallel to P and Q. In any position of P and Q, the value of the resultant, denoted by R, will be given by the equation (Art. 36), R = Pcosa +Qcosß. M Fig. 19. When the forces are parallel, and lying in the same direction, we shall have a = 0, and B cos 1. Hence, B = = 0; or, cos a = 1, and R = P+Q. If the forces lie in opposite directions, we shall have If we regard as the resultant of two parallel forces, and one of these as the resultant of two others, and so on, the principle may be extended to any number of parallel forces. Denoting the resultant of a group of parallel forces, 'P, P', P'', &c., by R, we have, That is, the resultant of a group of parallel forces is equal in intensity to the algebraic sum of the forces. Its line of direction is also parallel to that of the given forces. Point of Application of the Resultant. 40. Let P and Q be two parallel forces, and R their resultant. Let M and N be the points of application of the two forces, and S the point in which the direction of R cuts the line MN. Through N draw NL per pendicular to the general di -B P I Fig. 21. rection of the forces, and assume the point C, in which it intersects the line of direction of R, as a centre of moments. Since the centre of moments is on the line of direction of the resultant, the lever arm of the resultant will be 0, and we shall have, N M from the principle of moments (Art. 36), or, PX CL Q× CN; = P: Q: CN; CL. Fig. 22. But, from the similar triangles CNS and LNM, we have, CN: CL:: SN: SM. Combining the two proportions, we have, P: Q: SN : SM. That is, the line of direction of the resultant divides the line joining the points of application of the components, inversely as the components. From the last proportion, we have, by composition, P: Q: P+Q:: SN: SM: SN + SM; and, by division, P: Q: P- Q:: SN: SM: SN-SM. When the forces act in the same direction, P + Q will be their resultant, and SN + SM will equal MN. Since P+Q is greater than either P or Q, MN will be greater than either SN or SM, which shows that the resultant lies between the components. When the forces act in contrary directions, P - Q will be their resultant, and SN-SM will equal MN. Since P-Q is less than P (supposed the greater of the components), MN will be less than SN, which shows that the resultant lies without both components, and on the side of the greater. Substituting in the preceding proportions, for P + Q, P-Q, SN+ SM, and SN-SM, their values, we have, P: QR SN: SM: MN.... (8)'. That is, of two parallel forces and their resultant, each is proportional to the distance between the other two. Geometrical Composition and Resolution of Parallel Forces. 41. The preceding principles give rise to the following geometrical constructions: 1. To find the resultant of two parallel forces lying in the same direction: Let P and Q be the forces, M and IN their points of application. Make MQ' Q, and NP' P; draw P'Q', = = cutting MN in S; through S draw SR parallel to MP, and make it equal to P+Q: it will be the resultant. For, from the similar triangles P'SN and Q'SM, we have, Mo R Fig. 23. P'N: QM:: SN: SM; or, P: Q:: SN: SM. P After the construction is made, the distances MS and NS may be measured by a seale of equal parts. |