and the sum of the components along the z-axis, Ez = P1 cos y1 + P2 cos y1⁄2 + P2 cos Y3 + P1 COS Y 4 The original system of forces may now be r by a system of three rectangular forces Σx, Σ Ez (Fig. 12). Finally, this system may be replaced by a resultant which T and its direction given by the angles α, B, and y. angles are given by the equations This gives three equations of condition from which th unknown quantities may be determined. In the prece ing case of Art. 16 there were only two equations condition Ex=0 and Zy=0; consequently, only two u known quantities could be determined. Problem 6. Three men (Fig. 14) are each pulling with a force P at the points a, b, and c, respectively. What weight Q can they raise with uniform motion if each man pulls 100 lb.? Each force makes an angle of 60° with the Problem 7. Three concurring forces act upon a rigid body. nitude and direction. as follows: The forces are defined 76° 14'; ẞs 147° 2′; 73 = ? = ? HINT. Yı, Y1⁄2, and y, may be found from either of the following relations: cos (a + B) cos (α — B) + cos2y = 0, cos2a + cos2ß + cos2y = 1. shears (Fig. 15) is 50 ft. long. The back stay is 75 ft. long. ? Problem 8. Each leg of a pair of They are spread 20 ft. at the foot. Find the forces acting on each member when lifting a load of 20 tons at a distance of 20 ft. from the foot of the shear legs, neglecting the weight of structure. 75' 50' 20 18. Moment of a Force. FIG. 15 The moment of a force respect to any point in its plane may be defined as the uct of the force and a perpendicular let fall from the point on the line of action of the force. Let P (Fig. 16) be the force and the point and a the perpendicular distance of the force from the point; then Pa is the moment P E F FIG. 16 of the force with respect to the point 0. This mome is measured in terms of the units of both force a length, viz. foot-pounds or inch-pounds, and is read foo pounds moment or inch-pounds moment to distinguis it from foot-pounds work or inch-pounds work. For convenience the algebraic sign of the moment с 20 T said to be positive when the moment tends to turn the body in a direction counter-clockwise, and negative when it tends to turn the body in the clockwise direction. The moment may be represented geometrically as follows: let EF represent the magnitude of P, drawn to the desired scale, and draw EO and FO. The area of the triangle OEF = {EFa, or EFa EFa, or EFa= 2 AOEF; that is, the moment of the force with respect to a point is geometrically represented by twice the area of the triangle, whose base is the line representing the magnitude of the force and whose vertex is the given point. 19. Varignon's Theorem of Moments. The moment of the resultant of two concurring forces with respect to any point in their plane is equal to the algebraic sum of the moments of the two forces with respect to the same point. The given forces P and P1 may be represented by CP with respect to O is twice the area of the triangle COP (Art. 18), 2 area of the triangle COB, since the tri = angles have the same base and the same altitude. That is, the moment of CP with respect to O is the same as the moment of CB with respect to 0 = CBa, where a is the perpendicular let fall from O on CR. In a similar way, it is seen that the moment of CP, with respect to O is equal to the moment of CA with respect to 0; that is, to CA a. Therefore, the sum of the moments of P and P1 with respect to equals (CB+ CA) · a. But (CB+ CA)α = (CA+AR) a = R. A, since CB= AR (equal triangles CPB and AP1R). When the point is taken between the P and P1, the moment of the resultant equals the difference of the moments of P and P1. Let the student show that this is true. COR. 1. If there are any number of concurring forces in a plane, it may be shown that Varignon's theorem holds by considering the resultant of two of them with the third, and so on. The more general theorem may then be stated as follows: The moment of the resultant of any number of concurring forces in a plane with respect to any point in that plane is equal to the algebraic sum of the moments of the forces with respect to the same point. COR. 2. If the point be taken in the line of action of R, then a 0, and therefore the sum of the positive moments equals the sum of the negative moments. The moment of a force with respect to a line at right angles to the line of action of the force is the product of the force and the shortest distance between the two lines. The moment of a force with respect to a line not at right angles to the line of action of the force is the same as the moment of the component of the force in a plane perpendicular to the line. |