72. In one case, however, two parallel forces have no resultant; this
occurs when the two forces are equal, and in opposite directions. A
pair of forces of this kind is called _a couple_; there is no single
force which could balance a couple,—it can only be counterbalanced by
another couple acting in an opposite manner. This remarkable case, may
be studied by the arrangement of Fig. 23.
[Illustration: FIG. 23.]
A wooden rod, A B 48" × 0"·5 × 0"·5, has strings attached to it at
points A and D, one foot distant. The string at D passes over a fixed
pulley E, and at the end P a hook is attached for the purpose of
receiving weights, while a similar hook descends from A; the weight of
the rod itself, which only amounts to three ounces, may be neglected,
as it is very small compared with the weights which will be used.
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.