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.
[Illustration: 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.
80. Suppose, for example, that the two arms be 10 inches and 11 inches
long, then, if 1 lb. weight be placed in the pan of the 10-inch end,
its moment is 10; and if ¹⁰/₁₁ of 1 lb. be placed in the pan belonging
to the 11-inch end, its moment is also 10: hence 1 lb. at the short
end balances ¹⁰/₁₁ of 1 lb. at the long end; and therefore, if the
shopkeeper placed his weight in the short arm, his customers would
lose ¹/₁₁ part of each pound for which they paid; on the other hand,
if the shopkeeper placed his 1 lb. weight on the long arm, then not
less than ¹¹/₁₀ lb. would be required in the pan belonging to the short
arm. Hence in this case the customer would get ¹/₁₀ lb. too much. It
follows, that if a shopman placed his weights and his goods alternately
in the one scale and in the other he would be a loser on the whole;
for, though every second customer gets ¹/₁₁ lb. less than he ought, yet
the others get ¹/₁₀ lb. more than they have paid for.
LECTURE IV.
_THE FORCE OF GRAVITY._
Introduction.—Specific Gravity.—The Plummet and
Spirit-Level.—The Centre of Gravity.—Stable and
Unstable Equilibrium.—Property of the Centre of
Gravity in a Revolving Wheel.