73. Supposing 2 lbs. to be placed at p, and 1 lb. at q, we have two parallel forces acting in opposite directions; and since their difference is 1 lb., it follows from our rule that the point f, where d f is equal to a d, is the point where the resultant is applied. You see this is easily verified, for by placing my finger over the rod at f it remains horizontal and in equilibrium; whereas, when I move my finger to one side or the other, equilibrium is impossible. If I move it nearer to b, the end a ascends. If I move it towards a, the end b ascends.
74. To study the case when the two forces are equal, a load of 2 lbs. may be placed on each of the hooks p and q. It will then be found that the finger cannot be placed in any position along the rod so as to keep it in equilibrium; that is to say, no single force can counteract the two forces which form the couple. Let o be the point midway between a and d. The forces evidently tend to raise ob and turn the part o a downwards; but if I try to restrain o b by holding my finger above, as at the point x, instantly the rod begins to turn round x and the part from a to x descends. I find similarly that any attempt to prevent the motion by holding my finger underneath is equally unsuccessful. But if at the same time I press the rod downwards at one point, and upwards at another with suitable force, I can succeed in producing equilibrium; in this case the two pressures form a couple; and it is this couple which neutralizes the couple produced by the weights. We learn, then, the important result that a couple can be balanced by a couple, and by a couple only.
75. The moment of a couple is the product of one of the two equal forces into their perpendicular distance. Two couples tending to turn the body to which they are applied in the same direction will be equivalent if their moments are equal. Two couples which tend to turn the body in opposite directions will be in equilibrium if their moments are equal. We can also compound two couples in the same or in opposite directions into a single couple of which the moment is respectively either the sum or the difference of the original moments.
THE WEIGHING SCALES.
76. Another apparatus by which the nature of parallel forces may be investigated is shown in [Fig. 24]; it consists of a slight frame of wood a b c, 4' long. At e, a pair of steel knife-edges is clamped to the frame. The knife-edges rest on two pieces of steel, one of which is shown at o f. When the knife-edges are suitably placed the frame is balanced, so that a small piece of paper laid at a will cause that side to descend.
Fig. 24.
77. We suspend two small hooks from the points a and b: these are made of fine wire, so that their weight may be left out of consideration. With this apparatus we can in the first place verify the principle of equality of moments: for example, if I place the hook a at a distance of 9" from the centre o and load it with 1 lb., I find that when b is laden with 0·5 lb. it must be at a distance of 18" from o in order to counterbalance a; the moment in the one case is 9 × 1, in the other 18 × 0·5, and these are obviously equal.
78. Let a weight of 1 lb. be placed on each of the hooks, the frame will only be in equilibrium when the hooks are at precisely the same distance from the centre. A familiar application of this principle is found in the ordinary weighing scales; the frame, which in this case is called a beam, is sustained by two knife-edges, smaller, however, than those represented in the figure. The pans p, p are suspended from the extremities of the beam, and should be at equal distances from its centre. These scale-pans must be of equal weight, and then, when equal weights are placed in them, the beam will remain horizontal. If the weight in one slightly exceed that in the other, the pan containing the heavier weight will of course descend.
79. That a pair of scales should weigh accurately, it is necessary that the weights be correct; but even with correct weights, a balance of defective construction will give an inaccurate result. The error frequently arises from some inequality in the lengths of the arms of the beam. When this is the case, the two weights which really balance are not equal. Supposing, for instance, that with an imperfect balance I endeavour to weigh a pound of shot. If I put the weight on the short side, then the quantity of shot balanced is less than 1 lb.; while if the 1 lb. weight be placed at the long side, it will require more than 1 lb. of shot to produce equilibrium. The mode of testing a pair of scales is then evident. Let weights be placed in the pans which balance each other; if the weights be interchanged and the balance still remains horizontal, it is correct.