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. 1), are acting 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 =Q× BC, from which the distance of C from either A or B can be found.

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.

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. 2).

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 so on. 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.

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