19. Cases might be multiplied indefinitely by placing various amounts
of weight on the hooks, constructing the parallelogram on cardboard,
and comparing it with the cords as before. We shall, however, confine
ourselves to one more illustration, which is capable of very remarkable
applications. Attach 1 lb. to each of the hooks E and F; the cord
joining them remains straight until drawn down by placing a weight on
the centre hook. A very small weight will suffice to do this. Let us
put on half-a-pound; the position the cords then assume is indicated
in Fig. 7. As before, each force is equal and opposite to the resultant
of the other two. Hence a force of half-a-pound is the resultant of two
forces each of 1 lb. The apparent paradox is explained by noticing that
the forces of 1 lb. are very nearly opposite, and therefore to a large
extent counteract each other. Constructing the cardboard parallelogram
we may easily verify that the principle of the parallelogram of forces
holds in this case also.
[Illustration: FIG. 7.]
20. No matter how small be the weight we suspend from the middle of a
horizontal cord, you see that the cord is deflected: and no matter how
great a tension were applied, it would be impossible to straighten the
cord. The cord could break, but it could not again become horizontal.
Look at a telegraph wire; it is never in a straight line between two
consecutive poles, and its curved form is more evident the greater be
the distance between the poles. But in putting up a telegraph wire
great straining force is used, by means of special machines for the
purpose; yet the wires cannot be straightened: because the weight of
the heavy wire itself acts as a force pulling it downwards. Just as
the cord in our experiments cannot be straight when any force, however
small, is pulling it downwards at the centre, so it is impossible
by any exertion of force to straighten the long wire. Some further
illustrations of this principle will be given in our next lecture, and
with one application of it the present will be concluded.
21. One of the most important practical problems in mechanics is to
make a small force overcome a greater. There are a number of ways in
which this may be accomplished for different purposes, and to the
consideration of them several lectures of this course will be devoted.
Perhaps, however, there is no arrangement more simple than that which
is furnished by the principles we have been considering. We shall
employ it to raise a 28 lb. weight by means of a 2 lb. weight. I do not
say that this particular application is of much practical use. I show
it to you rather as a remarkable deduction from the parallelogram of
forces than as a useful machine.
[Illustration: FIG. 8.]
A rope is attached at one end of an upright, A (Fig. 8), and passes
over a pulley B at the same vertical height about 16' distant. A
weight of 28 lbs. is fastened to the free end of the rope, and the
supports must be heavily weighted or otherwise secured from moving.
The rope AB is apparently straight and horizontal, in consequence of
its weight being inappreciable in comparison with the strain (28 lbs.)
to which it is subjected; this position is indicated in the figure by
the dotted line AB. We now suspend from C at the middle of the rope a
weight of 2 lbs. Instantly the rope moves to the position represented
in the figure. But this it cannot do without at the same moment raising
slightly the 28 lbs., for, since two sides of a triangle, CB, CA, are
greater than the third side, AB, more of the rope must lie between the
supports when it is bent down by the 2 lb. weight than when it was
straight. But this can only have taken place by shortening the rope
between the pulley B and the 28 lb. weight, for the rope is firmly
secured at the other end. The effect on the heavy weight is so small
that it is hardly visible to you from a distance. We can, however,
easily show by an electrical arrangement that the big weight has been
raised by the little one.
22. When an electric current passes through this alarum you hear the
bell ring, and the moment I stop the current the bell stops. I have
fastened one piece of brass to the 28 lb. weight, and another to the
support close above it, but unless the weight be raised a little the
two will not be in contact; the electricity is intended to pass from
one of these pieces of brass to the other, but it cannot pass unless
they are touching. When the rope is straight the two pieces of brass
are separated, the current does not pass, and our alarum is dumb; but
the moment I hang on the 2 lb. weight to the middle of the rope it
raises the weight a little, brings the pieces of brass in contact, and
now you all hear the alarum. On removing the 2 lbs. the current is
interrupted and the noise ceases.
23. I am sure you must all have noticed that the 2 lb. weight descended
through a distance of many inches, easily visible to all the room; that
is to say, the small weight moved through a very considerable distance,
while in so doing it only raised the larger one a very small distance.
This is a point of the very greatest importance; I therefore take the
first opportunity of calling your attention to it.
LECTURE II.
_THE RESOLUTION OF FORCES._
Introduction.—One Force resolved into Two
Forces.—Experimental Illustrations.—Sailing.—One
Force resolved into Three Forces not in the same
Plane.—The Jib and Tie-rod.