172. In connection with the subject of friction, and also as
introductory to the mechanical powers, the notion of “work,” or as
it is more properly called “energy,” is of great importance. The
meaning of this word as employed in mechanics will require a little
consideration.
173. In ordinary language, whatever a man does that can cause fatigue,
whether of body or mind, is called work. In mechanics, we mean by
energy that particular kind of work which is directly or indirectly
equivalent to raising weights.
174. Suppose a weight is lying on the floor and a stool is standing
beside it: if a man raise the weight and place it upon the stool, the
exertion that he expends is energy in the sense in which the word is
used in mechanics. The amount of exertion necessary to place the weight
upon the stool depends upon two things, the magnitude of the weight and
the height of the stool. It is clear that both these things must be
taken into account, for although we know the weight which is raised, we
cannot tell the amount of exertion that will be required until we know
the height through which it is to be raised; and if we know the height,
we cannot appreciate the quantity of exertion until we know the weight.
175. The following plan has been adopted for expressing quantities
of energy. The small amount of exertion necessary to raise 1 lb.
avoirdupois through one British foot is taken as a standard, compared
with which all other quantities of energy are estimated. This quantity
of exertion is called in mechanics the unit of energy, and sometimes
also the “foot-pound.”
176. If a weight of 1 lb. has to be raised through a height of 2 feet,
or a weight of 2 lbs. through a height of 1 foot, it will be necessary
to expend twice as much energy as would have raised a weight of 1 lb.
through 1 foot, that is, 2 foot-pounds.
If a weight of 5 lbs. had to be raised from the floor up to a stool 3
feet high, how many units of energy would be required? To raise 5 lbs.
through 1 foot requires 5 foot-pounds, and the process must be again
repeated twice before the weight arrive at the top of the stool. For
the whole operation 15 foot-pounds will therefore be necessary.
If 100 lbs. be raised through 20 feet, 100 foot-pounds of energy is
required for the first foot, the same for the second, third, &c., up to
the twentieth, making a total of 2,000 foot-pounds.
Here is a practical question for the sake of illustration. Which would
it be preferable to hoist, by a rope passing over a single fixed
pulley, a trunk weighing 40 lbs. to a height of 20 feet, or a trunk
weighing 50 lbs. to a height of 15 feet? We shall find how much energy
would be necessary in each case: 40 times 20 is 800; therefore in the
first case the energy would be 800 foot-pounds. But 50 times 15 is 750;
therefore the amount of work, in the second case, is only 750 lbs.
Hence it is less exertion to carry 50 lbs. up 15 feet than 40 lbs. up
20 feet.
177. The rate of working of every source of energy, whether it lie in
the muscles of men or other animals, in water-wheels, steam-engines,
or other prime movers, is to be measured by the number of foot-pounds
produced in the unit of time.
The power of a steam-engine is defined by its equivalent in
horse-power. For example, it is meant that a steam-engine of 3
horse-power, could, when working for an hour, do as much work as 3
horses could do when working for the same time. The power of a horse
is, however, an uncertain quantity, differing in different animals and
not quite uniform in the same individual; accordingly the selection of
this measure for the efficiency of the steam-engine is inconvenient.
We replace it by a convenient standard horse-power, which is, however,
a good deal larger than that continuously obtainable from any ordinary
horse. A one horse-power steam-engine is capable of accomplishing
33,000 foot-pounds per minute.
178. We shall illustrate the numerical calculation of horse-power by
an example: if a mine be 1,000 feet deep, how much water per minute
would a 50 horse-power engine be capable of raising to the surface? The
engine would yield 50 × 33,000 units of work per minute, but the weight
has to be raised 1,000 feet, consequently the number of pounds of water
raised per minute is
50 × 33,000
———————————— = 1,650.
1,000
179. We shall apply the principle of work to the consideration of the
pulley already described (p. 90). In order to raise the weight of 14
lbs., it is necessary that the rope to which the power is applied
should be pulled downwards by a force of 15 lbs., the extra pound
being on account of the friction. To fix our ideas, we shall suppose
the 14 lbs. to be raised 1 foot; to lift this load directly, without
the intervention of the pulley, 14 foot-pounds would be necessary,
but when it is raised by means of the pulley, 15, foot-pounds are
necessary. Hence there is an absolute loss of ¹/₁₅th of the energy when
the pulley is used. If a steam-engine of 1 horse-power were employed
in raising weights by a rope passing over a pulley similar to that on
which we have experimented, only ¹⁴/₁₅ths of the work would be usefully
employed; but we find
33,000 × 14 = 30,800.
————
15
The engine would therefore perform 30,800 foot-pounds of useful work
per minute.
180. The effect of friction on a pulley, or on any other machine, is
always to waste energy. To perform a piece of work directly requires
a certain number of foot-pounds, while to do it by a machine requires
more, on account of the loss by friction. This may at first sight
appear somewhat paradoxical, as it is well known that, by levers,
pulleys, &c., an enormous mechanical advantage may be gained. This
subject will be fully explained in the next and following lectures,
which relate to the mechanical powers.
181. We shall conclude with a few observations on a point of the
greatest importance. We have seen a case where 15 foot-pounds of
energy only accomplished 14 foot-pounds of work, and thus 1 foot-pound
appeared to be lost. We say that this was expended upon the friction;
but what is the friction? The axle is gradually worn away by rubbing in
its bearings, and, if it be not properly oiled, it becomes heated. The
amount of energy that seems to disappear is partly expended in grinding
down the axle, and is partly transformed into heat; it is thus not
really lost, but unfortunately assumes a form which we do not require
and in which it is rather injurious than otherwise. Indeed we know
that energy cannot be destroyed, however it may be transformed; if it
disappear in one shape, it is only to reappear in another. A so-called
loss of energy by friction only means a diversion of a part of the
work to some purpose other than that which we wish to accomplish. It
has long been known that matter is indestructible: it is now equally
certain that the same may be asserted of energy.
LECTURE VII.
_THE PULLEY-BLOCK._
Introduction.—The Single Moveable Pulley.—The
Three-sheave Pulley-block.—The Differential
Pulley-block.—The Epicycloidal Pulley-block.