[Illustration: FIG. 15.]
33. Up to the present we have only been considering forces which lie
in the same plane, but in nature we meet with forces acting in all
directions, and therefore we must not be satisfied with confining our
inquiries to the simpler case. We proceed to show, in two different
ways, how a force can be decomposed into three forces not in the same
plane, though passing through the same point. The first mode of doing
so is as follows. To three points A, B, C (Fig. 15) three spring
balances are attached; A, B, C are not in the same straight line,
though they are at the same vertical height: to the spring balances
cords are attached, which unite in a point O, from which a weight W is
suspended. This weight is supported by the three cords, and the strains
along these cords are indicated by the spring balances. The greatest
strain is on the shortest cord and the least strain on the longest.
Here the force W lbs. produces three forces which, taken together,
exceed its own amount. If I add an equal weight W, I find, as we
might have anticipated, that the strains indicated by the scales are
precisely double what they were before. Thus we see that the proportion
of the force to each of the components into which it is decomposed does
not depend on the actual magnitude of the force, but on the relative
direction of the force and its components.
[Illustration: FIG. 16.]
34. Another mode of showing the decomposition of one force into three
forces not in the same plane is represented in Fig. 16. The tripod is
formed of three strips of pine, 4' × 0"·5 × 0"·5, secured by a piece of
wire running through each at the top; one end of this wire hangs down,
and carries a hook to which is attached a weight of 28 lbs. This weight
is supported by the wire, but the strain on the wire must be borne by
the three wooden rods: hence there is a force acting downwards through
the wooden rods. We cannot render this manifest by a contrivance like
the spring scales, because it is a push instead of a pull. However, by
raising one of the legs I at once become aware that there is a force
acting downwards through it. The weight is, then, decomposed into three
forces, which act downwards through the legs; these three forces are
not in a plane, and the three forces taken together are larger than the
weight.
35. The tripod is often used for supporting weights; it is convenient
on account of its portability, and it is very steady. You may judge of
its strength by the model represented in the figure, for though the
legs are very slight, yet they support very securely a considerable
weight. The pulleys by means of which gigantic weights are raised
are often supported by colossal tripods. They possess stability and
steadiness in addition to great strength.
36. An important point may be brought out by contrasting the
arrangements of Figs. 15 and 16. In the one case three cords are used,
and in the other three rods. Three rods would have answered for both,
but three cords would not have done for the tripod. In one the cords
are strained, and the tendency of the strain is to break the cords,
but in the other the nature of the force down the rods is entirely
different; it does not tend to pull the rod asunder, it is trying to
crush the rod, and had the weight been large enough the rods would bend
and break. I hold one end of a pencil in each hand and then try to pull
the pencil asunder; the pencil is in the condition of the cords of Fig.
15; but if instead of pulling I push my hands together, the pencil is
like the rods in Fig. 16.
37. This distinction is of great importance in mechanics. A rod or cord
in a state of tension is called a “tie”; while a rod in a state of
compression is called a “strut.” Since a rod can resist both tension
and compression it can serve either as a tie or as a strut, but a
cord or chain can only act as a tie. A pillar is always a strut, as
the superincumbent load makes it to be in a state of compression.
These distinctions will be very frequently used during this course of
lectures, and it is necessary that they be thoroughly understood.