= upon a rigid bar at the points A and B. Then the resultant, not only as regards translation but also with respect to rotation about any point in the bar, will be equal to the sum of the forces in magnitude, and parallel to them in direction, and will be applied at a point C, between A and B, such that BC: AC P: Q. The equilibrant will also be applied at C, and is equal to R in magnitude and opposite to it in direction. This means that whenever three parallel forces are in equilibrium, one of them is between the other two, is equal to their sum, and is opposite them in direction. Since the moment of P equals the moment of Q, P× AC = QX BC, from which the distance of C from either A or B can be found. B Demonstration. The truth of the above equation for determining the point of application may be verified by suspending from a meter stick two weights, P and Q, and supporting the stick and its load by a spring balance or scale, as in Fig. 34. The weights can be supported from the meter stick by cords with loops passing over the stick, and the position of the scale can be found by slipping the loop to which it is attached along the stick until the stick balances in a horizontal direction. Before the proportion P: Q = BC: AC is tested, a small weight should be suspended from the short end near A so that the stick will balance when the weights P and Q are removed. The scale will read not only the sum of P and Q, but the weight of the stick and small weight also. FIG. 34 The application of this principle is useful in determining the pressure upon the abutments of a bridge when a load is passing over it. If a train passes over the bridge, the pressures upon the abutments (in addition to the weight of the bridge) are constantly varying from the whole weight to zero, and vice versa, while the sum of the two pressures is equal to the weight of the train (Fig. 35). The resultant of any number of parallel forces can be found by finding first the resultant of two of them, then combining this resultant with a third force to find their resultant, and Any number of parallel forces are in equilibrium when the resultant of all the forces in one direction is equal to and has the same point of application as the resultant of all the forces in the opposite direction. so on. 63. Couples. The parallel forces P and Q shown in Fig. 33 can be replaced by a single force, R, or counterbalanced by a single force, E. Parallel forces, however, can be applied to a body in such a way that neither of these things can be done. If the parallel forces are equal and in opposite A directions, their resultant is zero so far as motion of translation is concerned, but they tend to turn the bar AB (Fig. 36) into the position shown by the dotted lines. This combination of forces is called a couple and produces rotation only. B FIG. 36 64. The Graphical Representation and Composition of Velocities. Any velocity of which the starting point, the rate, and the direction are known can be represented graphically by a vector line (§ 54). If the directions of two component velocities are in the same straight line, the resultant velocity is either the sum or the difference of the two velocities. A person walking through a railway train from the rear to the front has a greater velocity with respect to the earth than a person sitting in a seat, while a person walking from the front to the rear has a smaller velocity. A E B If the directions of two component velocities are not in one straight line, the resultant velocity can be found by combining the velocities as forces are combined in the parallelogram of forces. For example, if a man rows a boat across a stream with a uniform velocity of 2 miles per hour while the stream flows with a uniform velocity of 1.5 miles per hour, the direction taken by the boat will be determined by these velocities independently of the width of the stream. If the boat starts from A, Fig. 37, the direction of the path will be found by laying off AB to represent the velocity of 2 miles per hour and FIG. 37 AC at a right angle to it to represent a velocity of 1.5 miles per hour; then AD will be the direction the boat will take. If the width, AE, of the stream is known, the length of the path, AF, is easily determined. A velocity, as well as a force, can also be resolved into components, as in § 59. 65. Reflected Motion. If an elastic ball strikes against a fixed body, it will rebound. This is called reflected motion, and is caused by the reaction of the body against which it strikes. If a moving body is not elastic, the reaction of the body against which it strikes will flatten it, as, for example, when a ball of putty is dropped upon the floor. If both the bodies are highly elastic, the direction of the rebound will be such that the angle of reflection will equal the angle of incidence. This is the Law of Reflection, and may be verified as follows: of board on its edge upon a table resting against the wall. Roll an elastic ball across the table along the line AB (Fig. 38) against the board. From B, where the ball strikes, draw BD perpendicular to the board. Then the angle CBD, the angle of reflection, will be found equal to the angle ABD, the angle of incidence. (The path. of the ball can be readily traced by dusting the table with crayon dust.) FIG. 38 66. Curvilinear Motion. The path of a body whose motion is that imparted by a single impulsive force is rectilinear. If its motion is due to two impulsive forces that have acted upon it, its path will still be rectilinear; but if B FIG. 39 E the motion due to an impulsive force is combined with that due to a constant force, not acting in the same straight line, its path will be curvilinear. If a stone is tied to a cord and swung around, a curved path is the result. If the cord should break, the stone would go off in a straight line the tangent line AB (Fig. 39); but the cord prevents the stone from taking such a course and compels it to go in the curved path ADE. 67. Centripetal and Centrifugal Forces. - The pull of the string that compels the body in Fig. 39 to move in a circular path is directed toward the center, C, and is called the centripetal force. Since this force acts constantly in a direction at a right angle to the direction of the motion of the body, it does not affect its velocity, but does give it an acceleration towards the center. The reaction that the moving body offers to the centripetal pull of the cord is called the centrifugal force and is equal, in amount, to the centripetal force. If part of the cord is replaced by a spring scale, its reading will be a measure of this force, which depends upon the mass of the body, its velocity, and the radius of the circle. The acceleration given by centripetal force is equal to Since force equals Ma or Wa, the expression for either ༧2 g |