187. We commence with the most simple case, that of the single moveable
pulley (Fig. 35). The rope is firmly secured at one end A; it then
passes down under the moveable pulley B, and upwards over a fixed
pulley. To the free end, which depends from the fixed pulley, the
power is applied while the load to be raised is suspended from the
moveable pulley. We shall first study the relation between the power
and the load in a simple way, and then we shall describe a few exact
experiments.
188. When the load is raised the moveable pulley itself must of course
be also raised, and a part of the power is expended for this purpose.
But we can eliminate the weight of the moveable pulley, so far as our
calculations are concerned, by first attaching to the power end of
the rope a sufficient weight to lift up the moveable pulley when not
carrying a load. The weight necessary for doing this is found by trial
to be a little over 1·5 lbs. So that when a load is being raised we
must reduce the apparent power by 1·5 lbs. to obtain the power really
effective.
189. Let us suspend 14 lbs. from the load hook at B, and ascertain
what power will raise the load. We leave the weight of the moveable
pulley and 1·5 lbs. of the power at C out of consideration. I then
find by experiment that 7 lbs. of effective power is not sufficient to
raise the load, but if one pound more be added, the power descends, and
the load is raised. Here, then, is a remarkable result; a weight of 8
lbs. has overcome 14 lbs. In this we have the first application of the
mechanical powers to increase our available forces.
[Illustration: FIG. 35.]
190. Let us examine the reason of this mechanical advantage. If the
load be raised one foot, it is plain that the power must descend
two feet: for in order to raise the load the two parts of the rope
descending to the moveable pulley must each be shortened one foot, and
this can only be done by the power descending two feet. Hence when
the load of 14 lbs. is lifted by the machine, for every foot it is
raised the power must descend two feet: this simple point leads to a
conception of the greatest importance, on which depends the efficiency
of the pulley. In the study of the mechanical powers it is essential to
examine the number of feet through which the power must act in order to
raise the load one foot: this number we shall always call the _velocity
ratio_.
191. To raise 14 lbs. one foot requires 14 foot-pounds of energy.
Hence, were there no such thing as friction, 7 lbs. on the power hook
would be sufficient to raise the load; because 7 lbs. descending
through two feet yields 14 foot-pounds. But there is a loss of energy
on account of friction, and a power of 7 lbs. is not sufficient: 8
lbs. are necessary. Eight lbs. in descending two feet performs 16
foot-pounds; of these only 14 are utilised on the load, the remainder
being the quantity of energy that has been diverted by friction. We
learn, then, that in the moveable pulley the quantity of _energy_
employed is really greater than that which would lift the weight
directly, but that the actual _force_ which has to be exerted is less.
192. Suppose that 28 lbs. be placed on the load hook, a few trials
assure us that a power of 16 lbs. (but not less) will be sufficient for
motion; that is to say, when the load is doubled, we find, as we might
have expected, that the power must be doubled also. It is easily seen
that the loss of energy by friction then amounts to 4 foot-pounds. We
thus verify, in the case of the moveable pulley, the approximate law
that the _friction is proportional to the load_.
193. By means of a moveable pulley a man is able to raise a weight
nearly double as great as he could lift directly. From a series of
careful experiments it has been found that when a man is employed in
the particular exertion necessary for raising weights over a pulley, he
is able to work most efficiently when the pull he is required to make
is about 40 lbs. A man could, of course, exert greater force than this,
but in an ordinary day’s work he is able to perform more foot-pounds
when the pull is 40 lbs. than when it is larger or smaller. If
therefore the weights to be lifted amount to about 80 lbs., energy may
really be economized by the use of the single moveable pulley, although
by so doing a greater quantity of energy would be actually expended
than would have been necessary to raise the weights directly.
194. Some experiments on larger loads have been tried with the moveable
pulley we have just described; the results are recorded in Table IX.