act upon a body in opposite directions, their resultant will be zero, the forces will be in equilibrium, and the body will be at rest. A P C (c) When the forces act at an angle to each other. THE PARALLELOGRAM OF FORCES.(1) Suppose the force P (Fig. 23), of 3 dynes, to act toward the east, at a right angle to the force Q, of 2 dynes, acting toward the south. Represent P by AC, and Q by AB. Complete the parallelogram by drawing the dotted lines BD and CD (parallel to AC and AB, respectively), Q BY FIG. 23 and their intersection will locate the point D and determine both the magnitude and the direction of the resultant, AD. For D is the only point that is as far east as C and as far south as B. In each figure all lines representing forces must be measured by the same scale. (2) Suppose the force Q to act at an angle CAB to the force P (Fig. 24). Complete the parallelogram to determine the point D. Then, for reasons similar to the above, AD or R will be the resultant required. The resultant of any two forces acting at an angle to each other may be found by completing the parallelogram upon the forces as sides and drawing the diagonal from the common point of application. Q A BY P R FIG. 24 (d) When there are more than two forces. The resultant of any number of forces can be found by a repetition of the parallelogram of forces. Suppose three forces, P, Q, and S (Fig. 25), to be acting on a body at A. Complete the paral A P lelogram ACDB; then AD or R' will be the resultant of P and Q. Find the resultant of S and R' by completing the parallelogram ADHE; then AH or R will be the resultant of P, Q, and S. 56. Equilibrant.-The equilibrant of any number of forces is a force equal in magnitude, and opposite in direction, to S R H R' their resultant. If the forces and their equilibrant act upon on the blackboard and the direction of the lines leading to C and D. Take Hc to represent the reading of the balance C, and on the same scale lay off Hd to represent the reading of the balance D. Complete the parallelogram, and Hk, the resultant, will be found to represent an amount equal to W, the equilibrant, and to be vertical. 58. Forces not Lying in the Same Plane. When three forces having the same point of application do not lie in the ABEC, while the diagonal AH is the resultant of R' and S in the plane AEHD, and is the required resultant. 59. Resolution of Forces. In the composition of forces we have given the component forces to find the resultant, while in the resolution of forces the resultant is given and the components are to be found. When two components are to be found, the problem is to construct a parallelogram on the resultant as the diagonal, such that the desired components will be sides of the parallelogram. There are several cases of this problem, of which the following are the most important: (a) Given, the resultant and one of two components, to find the other component (Fig. 29). Suppose the force P and one of its components Q are given, and it is required to find the E B Q FIG. 29 other component. Complete the parallelogram on P as the diagonal and Q as one side (by connecting B and E, and drawing ED parallel to BA, and AD parallel to BE). Then C will be the required component. A E C B (b) Given, the resultant and the direction of each of two components, to find the components. Let AB (Fig. 30) be the given resultant and AC and AD the directions of the required components. From B draw two lines, one parallel to AD and the other parallel to AC. Their intersections with AC and AD will determine the points E and F; and AE and AF will be the required components of AB. F D FIG. 30 So far we have 60. General Condition of Equilibrium. considered only motion of translation, or motion in which all the parts of the moving body move in the same direction and with the same speed. But there may also be a motion of rotation, as when a body turns on an axis. In rotation, parts of the body on opposite sides of the axis move at any instant in opposite directions; and each part moves continuously in a curved path, with a speed that varies with its distance from the axis. In order to have complete equilibrium, not only must the resultant of all the forces tending to produce translation of the body be zero, but the resultant of all the forces tending to produce rotation must also be zero. 61. The Moment of a Force. Suppose there are two forces, F and F', acting at right angles to the bar AB, each tending to rotate it about the pivot C (Fig. 31). It is evident that the tendency of each force to produce rotation depends A C о d' B Q not only upon the magnitude of the force, but also upon its distance from the point C, about which it tends to turn the bar. If the distance CA is d, and CB is d', the tendency to produce rotation exerted by F is proportional to the product Fd, which is called the moment of the force F; so, too, the moment of the force F' is F'd'. The point C, about which the rotation takes place, FIG. 31 is called the center of moments. The force F' tends to produce a clockwise and the force F a counterclockwise rota pendiculars drawn from C to the directions of the forces. The moment of a force is the product of the force by the perpendicular distance from the center of moments to the direction of the force. 62. Parallel Forces are two or more forces that act upon a body in parallel directions but at different points of application. Suppose two parallel forces, P and Q (Fig. 33), are acting |