Fig. 67.—Law of parallel forces illustrated.
85. Parallel Forces.—Objects are frequently supported by two or more upward forces acting at different points and forming in this way a system of parallel forces; as when two boys carry a string of fish on a rod between them or when a bridge is supported at its ends. The principle of moments just described aids in determining the magnitude of such forces and of their resultant. To illustrate this take a wooden board 4 in. wide and 4 ft. long of uniform dimensions. (See Fig. 67.) Place several screw hooks on one edge with one set at O where the board will hang horizontally when the board is suspended there. Weigh the board by a spring balance hung at O. This will be the resultant in the following tests. Now hang the board from two spring balances at M and N and read both balances. Call readings f and f´. To test the forces consider M as a fixed point (see Fig. 67) and the weight of the board to act at O. Then the moment of the weight of the board should be equal the moment of the force at N since the board does not move, or w times OM equals f´ times NM. If N is considered the fixed point then the moment of the weight of the board and of f with reference to the point N should be equal, or w times ON = f times NM. Keeping this illustration in mind, the law of parallel forces may be stated at follows: 1. The resultant of two parallel forces acting in the same direction at different points in a body is equal to their sum and has the same direction as the components.
The moment of one of the components about the point of application of the other is equal and opposite to the moment of the supported weight about the other.
Problem.—If two boys carry a string of fish weighing 40 lbs. on a rod 8 ft. long between them, what force must each boy exert if the string is 5 ft. from the rear boy?
Solution.—The moment of the force F exerted about the opposite end by the rear boy is F × 8. The moment of the weight about the same point is 40 × (8 - 5) = 120. Therefore F × 8 = 120, or F = 15, the force exerted by the rear boy. The front boy exerts a force of F whose moment about the other end of the rod is F × 8. The moment of the weight about the same point is 40 × 5 = 200. Since the moment of F equals this, 200 = F × 8, or F = 25. Hence the front boy exerts 25 lbs. and the rear boy 15 lbs.
Fig. 68.—A couple.
86. The Couple.—If two equal parallel forces act upon a body along different lines in opposite directions, as in Fig. 68, they have no single resultant or there is no one force that will have the same effect as the two components acting together. A combination of forces of this kind is called a couple. Its tendency is to produce change of rotation in a body. An example is the action upon a compass needle which is rotated by a force which urges one end toward the north and by an equal force which urges the other end toward the south.