208. By increasing the number of sheaves in a pair of pulley-blocks the
power may be increased; but the length of rope (or chain) requisite
for several sheaves becomes a practical inconvenience. There are also
other reasons which make the differential pulley-block, which we shall
now consider, more convenient for many purposes than the common pulley
blocks when a considerable augmentation of power is required.
209. The principle of the differential pulley is very ancient, and in
modern times it has been embodied in a machine of practical utility.
The object is to secure, that while the power moves over a considerable
distance, the load shall only be raised a short distance. When this has
been attained, we then know by the principle of energy that we have
gained a mechanical advantage.
210. Let us consider the means by which this is effected in that
ingenious contrivance, Weston’s differential pulley-block. The
principle of this machine will be understood from Fig. 36 and Fig. 37.
[Illustration: FIG. 36.]
It consists of three parts,—an upper pulley-block, a moveable pulley,
and an endless chain. We shall briefly describe them. The upper block
P is furnished with a hook for attachment to a support. The sheave
it contains resembles two sheaves, one a little smaller than the
other, fastened together: they are in fact one piece. The grooves are
provided with ridges, adapted to prevent the chain from slipping. The
lower pulley Q consists of one sheave, which is also furnished with a
groove; it carries a hook, to which the load is attached. The endless
chain performs a part that will be understood from the sketch of the
principle in Fig. 36. The chain passes from the hand at A up to L over
the larger groove in the upper pulley, then downwards at B, under the
lower pulley, up again at C, over the smaller groove in the upper
pulley at A, and then back again by D to the hand at A. When the hand
pulls the chain downwards, the two grooves of the upper pulley begin
to turn together in the direction shown by the arrows on the chain.
The large groove is therefore winding up the chain, while the smaller
groove is lowering.
211. In the pulley which has been employed in the experiments to be
described, the effective circumference of the large groove is found to
be 11"·84, while that of the small groove is 10"·36. When the upper
pulley has made one revolution, the large groove must have drawn up
11"·84 of chain, since the chain cannot slip on account of the ridges;
but in the same time the small groove has lowered 10"·36 of chain:
hence when the upper pulley has revolved once, the chain between the
two must have been shortened by the difference between 11"·84 and
10"·36, that is by 1"·48; but this can only have taken place by raising
the moveable pulley through half 1"·48, that is, through a space 0"·74.
The power has then acted through 11"·84, and has raised the resistance
0"·74. The power has therefore moved through a space 16 times greater
than that through which the load moves. In fact, it is easy to verify
by actual trial that the power must be moved through 16 feet in order
that the load may be raised 1 foot. We express this by saying that the
_velocity ratio_ is 16.
212. By applying power to the chain at D proceeding from the smaller
groove, the chain is lowered by the large groove faster than it is
raised by the small one, and the lower pulley descends. The load is
thus raised or lowered by simply pulling one chain A or the other D.
[Illustration: FIG. 37.]
213. We shall next consider the mechanical efficiency of the
differential pulley-block. The block (Fig. 37) which we shall use
is intended to be worked by one man, and will raise any weight not
exceeding a quarter of a ton.
We have already learned that with this block the power must act through
sixteen feet for the load to be raised one foot. Hence, were it not
for friction, the power need only be the sixteenth part of the load.
A few trials will show us that the real efficiency is not so large,
and that in fact more than half the work exerted is merely expended
upon overcoming friction. This will lead afterwards to a result of
considerable practical importance.
214. Placing upon the load hook a weight of 200 lbs., I find that 38
lbs. attached to a hook fastened on the power chain is sufficient to
raise the load; that is to say, the power is about one-sixth of the
load. If I make the load 400 lbs. I find the requisite power to be
64 lbs., which is only about 3 lbs. less than one-sixth of 400 lbs.
We may safely adopt the practical rule, that with this differential
pulley-block a man would be able to raise a weight six times as great
as he could raise without such assistance.
215. A series of experiments carefully tried with different loads have
given the results shown in Table XI.