to our attention, is that which we have called the _cord_. If a rope
were perfectly flexible, and were capable of being bent over a sharp
edge, and of moving upon it without friction, we should be enabled by
its means to make a force in any one direction overcome resistance, or
communicate motion in any other direction. Thus if P, _fig. 112._,
be such an edge, a perfectly flexible rope passing over it would be
capable of transmitting a force S F to a resistance Q R,
so as to support or overcome R, or by a motion in the direction of
S F to produce another motion in the direction R Q. But as
no materials of which ropes can be constructed can give them perfect
flexibility, and as in proportion to the strength by which they are
enabled to transmit force their rigidity increases, it is necessary,
in practice, to adopt means to remove or mitigate those effects which
attend imperfect flexibility, and which would otherwise render cords
practically inapplicable as machines.
When a cord is used to transmit a force from one direction to another,
its stiffness renders some force necessary in bending it over the
angle P, which the two directions form; and if the angle be sharp,
the exertion of such a force may be attended with the rupture of the
cord. If, instead of bending the rope at one point over a single angle,
the change of direction were produced by successively deflecting it
over several angles, each of which would be less sharp than a single
one could be, the force requisite for the deflection, as well as the
liability of rupturing the cord, would be considerably diminished. But
this end will be still more perfectly attained if the deflection of the
cord be produced by bending it over the surface of a curve.
If a rope were applied only to sustain, and not to move a weight,
this would be sufficient to remove the inconveniences arising from
its rigidity. But when motion is to be produced, the rope, in passing
over the curved surface, would be subject to excessive friction, and
consequently to rapid wear. This inconvenience is removed by causing
the surface on which the rope runs to move with it, so that no more
friction is produced than would arise from the curved surface rolling
upon the rope.
(268.) All these ends are attained by the common pulley, which consists
of a wheel called a _sheave_, fixed in a block and turning on a pivot.
A groove is formed in the edge of the wheel in which the rope runs,
the wheel revolving with it. Such an apparatus is represented in
_fig. 113._
We shall, for the present, omit the consideration of that part of the
effects of the stiffness and friction of the machine which is not
removed by the contrivance just explained, and shall consider the rope
as perfectly flexible and moving without friction.
From the definition of a flexible cord, it follows, that its tension,
or the force by which it is stretched throughout its entire length,
must be uniform. From this principle, and this alone, all the
mechanical properties of pulleys may be derived.
Although, as already explained, the whole mechanical efficacy of this
machine depends on the qualities of the cord, and not on those of the
block and sheave, which are only introduced to remove the accidental
effects of stiffness and friction; yet it has been usual to give the
name pulley to the block and sheave, and a combination of blocks,
sheaves, and ropes is called a _tackle_.
(269.) When the rope passes over a single wheel, which is fixed in
its position, as in _fig. 113._, the machine is called a _fixed
pulley_. Since the tension of the cord is uniform throughout its
length, it follows, that in this machine the power and weight are
equal. For the weight stretches that part of the cord which is between
the weight and pulley, and the power stretches that part between the
power and the pulley. And since the tension throughout the whole length
is the same, the weight must be equal to the power.
Hence it appears that no mechanical advantage is gained by this
machine. Nevertheless, there is scarcely any engine, simple or complex,
attended with more convenience. In the application of power, whether
of men or animals, or arising from natural forces, there are always
some directions in which it may be exerted to much greater convenience
and advantage than others, and in many cases the exertion of these
powers is limited to a single direction. A machine, therefore, which
enables us to give the most advantageous direction to the moving power,
whatever be the direction of the resistance opposed to it, contributes
as much practical convenience as one which enables a small power to
balance or overcome a great weight. In directing the power against the
resistance, it is often necessary to use two fixed pulleys. Thus, in
elevating a weight A, _fig. 114._, to the summit of a building,
by the strength of a horse moving below, two fixed pulleys B and C may
be used. The rope is carried from A over the pulley B; and, passing
downwards, is brought under C, and finally drawn by the animal on
the horizontal plane. In the same manner sails are spread, and flags
hoisted on the yards and masts of a ship, by sailors pulling a rope on
the deck.
By means of the fixed pulley a man may raise himself to a considerable
height, or descend to any proposed depth. If he be placed in a chair
or bucket attached to one end of a rope which is carried over a fixed
pulley, by laying hold of this rope on the other side, as represented
in _fig. 115._, he may, at will, descend to a depth equal to half
of the entire length of the rope, by continually yielding rope on the
one side, and depressing the bucket or chair by his weight on the
other. Fire-escapes have been constructed on this principle, the fixed
pulley being attached to some part of the building.
(270.) A _single moveable pulley_ is represented in _fig. 116._
A cord is carried from a fixed point F, and passing through a block
B, attached to a weight W, passes over a fixed pulley C, the power
being applied at P. We shall first suppose the parts of the cord on
each side the wheel B to be parallel; in this case, the whole weight W
being sustained by the parts of the cords B C and B F, and
these parts being equally stretched (268.), each must sustain half the
weight, which is therefore the tension of the cord. This tension is
resisted by the power at P, which must, therefore, be equal to half the
weight. In this machine, therefore, the weight is twice the power.
(271.) If the parts of the cord B C and B F be not parallel,
as in _fig. 117._, a greater power than half the weight is
therefore necessary to sustain it. To determine the power necessary
to support a given weight, in this case take the line B A in the
vertical direction, consisting of as many inches as the weight consists
of ounces; from A draw A D parallel to B C, and A E
parallel to B F; the force of the weight represented by A B
will be equivalent to two forces represented by B D and B E.
(74.) The number of inches in these lines respectively will represent
the number of ounces which are equivalent to the tensions of the parts
B F and B C of the cord. But as these tensions are equal,
B D and B E must be equal, and each will express the amount
of the power P, which stretches the cord at P C.
It is evident that the four lines, A E, E B, B D, and
D A, are equal. And as each of them represents the power, the
weight which is represented by A B must be less than twice the
power which is represented by A E and E B taken together. It
follows, therefore, that as parts of the ropes which support the weight
depart from parallelism the machine becomes less and less efficacious;
and there are certain obliquities at which the equilibrating power
would be much greater than the weight.
(272.) The mechanical power of pulleys admits of being almost
indefinitely increased by combination. Systems of pulleys may be
divided into two classes; those in which a single rope is used, and
those which consist of several distinct ropes. _Fig. 118._ and
_119._ represent two systems of pulleys, each having a single rope.
The weight is in each case attached to a moveable block, B, in which
are fixed two or more wheels; A is a fixed block, and the rope is
successively passed over the wheels above and below, and, after passing
over the last wheel above, is attached to the power. The tension of
that part of the cord to which the power is attached is produced by
the power, and therefore equivalent to it, and the same tension must
extend throughout its whole length. The weight is sustained by all
those parts of the cord which pass from the lower block, and as the
force which stretches them all is the same, viz. that of the power,
the effect of the weight must be equally distributed among them, their
directions being supposed to be parallel. It will be evident, from
this reasoning, that the weight will be as many times greater than the
power as the number of cords which support the lower block. Thus, if
there be six cords, each cord will support a sixth part of the weight,
that is, the weight will be six times the tension of the cord, or six
times the power. In _fig. 118._ the cord is represented as being
finally attached to a hook on the upper block. But it may be carried
over an additional wheel fixed in that block, and finally attached
to a hook in the lower block, as in _fig. 119._, by which one
will be added to the power of the machine, the number of cords at
the lower block being increased by one. In the system represented in
_fig. 118._ the wheels are placed in the blocks one above the
other; in _fig. 119._ they are placed side by side. In all systems
of pulleys of this class, the weight of the lower block is to be
considered as a part of the weight to be raised, and in estimating the
power of the machine, this should always be attended to.
(273.) When the power of the machine, and therefore the number of
wheels, is considerable, some difficulty arises in the arrangement of
the wheels and cords. The celebrated Smeaton contrived a tackle, which
takes its name from him, in which there are ten wheels in each block:
five large wheels placed side by side, and five smaller ones similarly
placed above them in the lower block, and below them in the upper.
_Fig. 120._ represents Smeaton’s blocks without the rope. The
wheels are marked with the numbers 1, 2, 3, &c., in the order in which
the rope is to be passed over them. As in this pulley 20 distinct parts
of the rope support the lower block, the weight, including the lower
block, will be 20 times the equilibrating power.
(274.) In all these systems of pulleys, every wheel has a separate
axle, and there is a distinct wheel for every turn of the rope at each
block. Each wheel is attended with friction on its axle, and also with
friction between the sheave and block. The machine is by this means
robbed of a great part of its efficacy, since, to overcome the friction
alone, a considerable power is in most cases necessary.
An ingenious contrivance has been suggested, by which all the advantage
of a large number of wheels may be obtained without the multiplied
friction of distinct sheaves and axles. To comprehend the excellence
of this contrivance, it will be necessary to consider the rate at
which the rope passes over the several wheels of such a system, as
_fig. 118._ If one foot of the rope G F pass over the
pulley F, two feet must pass over the pulley E, because the distance
between F and E being shortened one foot, the total length of the rope
G F E must be shortened two feet. These two feet of rope
must pass in the direction E D, and the wheel D, rising one foot,
three feet of rope must consequently pass over it. These three feet of
rope passing in the direction D C, and the rope D C being
also shortened one foot by the ascent of the lower block, four feet of
rope must pass over the wheel C. In the same way it may be shown that
five feet must pass over B, and six feet over A. Thus, whatever be
the number of wheels in the upper and lower blocks, the parts of the
rope which pass in the same time over the wheels in the lower block
are in the proportion of the odd numbers 1, 3, 5, &c.; and those which
pass over the wheels in the upper block in the same time, are as the
even numbers 2, 4, 6, &c. If the wheels were all of equal size, as in
_fig. 119._, they would revolve with velocities proportional to
the rate at which the rope passes over them. So that, while the first
wheel below revolves once, the first wheel above will revolve twice;
the second wheel below three times; the second wheel above, four times,
and so on. If, however, the wheels differed in size in proportion to
the quantity of rope which must pass over them, they would evidently
revolve in the same time. Thus, if the first wheel above were twice the
size of the first wheel below, one revolution would throw off twice the
quantity of rope. Again, if the second wheel below were thrice the size
of the first wheel below, it would throw off in one revolution thrice
the quantity of rope, and so on. Wheels thus proportioned, revolving
in exactly the same time, might be all placed on one axle, and would
partake of one common motion, or, what is to the same effect, several
grooves might be cut upon the face of one solid wheel, with diameters
in the proportion of the odd numbers 1, 3, and 5, &c., for the lower
pulley, and corresponding grooves on the face of another solid wheel
represented by the even numbers 2, 4, 6, &c., for the upper pulley. The
rope being passed successively over the grooves of such wheels, would
be thrown off exactly in the same manner as if every groove were upon a
separate wheel, and every wheel revolved independently of the others.
Such is White’s pulley, represented in _fig. 121._
The advantage of this machine, when accurately constructed, is very
considerable. The friction, even when great resistances are to be
opposed, is very trifling; but, on the other hand, it has corresponding
disadvantages which greatly circumscribe its practical utility. In the
workmanship of the grooves great difficulty is found in giving them
the exact proportions. In doing which, the thickness of the rope must
be accurately allowed for; and consequently it follows, that the same
pulley can never act except with a rope of a particular diameter. A
very slight deviation from the true proportion of the grooves will
cause the rope to be unequally stretched, and will throw on some parts
of it an undue proportion of the weight, while other parts become
nearly, and sometimes altogether slack. Besides these defects, the rope
is so liable to derangement by being thrown out of the grooves, that
the pulley can scarcely be considered portable.
For these and other reasons, this machine, ingenious as it
unquestionably is, has never been extensively used.
(275.) In the several systems of pulleys just explained, the hook to
which the fixed block is attached supports the entire of both the power
and weight. When the machine is in equilibrium, the power only supports
so much of the weight as is equal to the tension of the cord, all the
remainder of the weight being thrown on the fixed point, according to
what was observed in (225.)
If the power be moved so as to raise the weight, it will move with a
velocity as many times greater than that of the weight as the weight
itself is greater than the power. Thus in _fig. 118._ if the
weight attached to the lower block ascend one foot, six feet of line
will pass over the pulley A, according to what has been already proved.
Thus, the power will descend through six feet, while the weight rises
one foot. But, in this case, the weight is six times the power. All the
observations in (226.) will therefore be applicable to the cases of
great weights raised by small powers by means of the system of pulleys
just described.
(276.) When two or more ropes are used, pulleys may be combined in
various ways so as to produce any degree of mechanical effect. If
to any of the systems already described a single moveable pulley
be added, the power of the machine would be doubled. In this case,
the second rope is attached to the hook of the lower block, as in
_fig. 122._, and being carried through a moveable pulley
attached to the weight, it is finally brought up to a fixed point. The
tension of the second cord is equal to half the weight (270.); and
therefore the power P, by means of the first cord, will have only half
the tension which it would have if the weight were attached to the
lower block. A moveable pulley thus applied is called a _runner_.
[Illustration: _C. Varley, del._ _H. Adlard, sc._
_London, Pubd. by Longman & Co._]
(277.) Two systems of pulleys, called _Spanish bartons_, having
each two ropes, are represented in _fig. 123._ The tension of
the rope P A B C in the first system is equal to
the power; and therefore the parts B A and B C support a
portion of the weight equal to twice the power. The rope E A
supports the tensions of A P and A B; and therefore the
tension of A E D is twice the power. Thus, the united
tensions of the ropes which support the pulley B is four times the
power, which is therefore the amount of the weight. In the second
system, the rope P A D is stretched by the power. The rope
A E B C acts against the united tensions A P and
A D; and therefore the tension of A E or E B is twice
the power. Thus, the weight acts against three tensions; two of which
are equal to twice the power, and the remaining one is equal to the
power. The weight is therefore equal to five times the power.
A single rope may be so arranged with one moveable pulley as to support
a weight equal to three times the power. In _fig. 124._ this
arrangement is represented, where the numbers sufficiently indicate the
tension of the rope, and the proportion of the weight and power. In
_fig. 125._ another method of producing the same effect with two
ropes is represented.
(278.) If several single moveable pulleys be made successively to act
upon each other, the effect is doubled by every additional pulley:
such a system as this is represented in _fig. 126._ The tension
of the first rope is equal to the power; the second rope acts against
twice the tension of the first, and therefore it is stretched with
a force equal to twice the power: the third rope acts against twice
this tension, and therefore it is stretched with a force equal
to four times the power, and so on. In the system represented in
_fig. 126._ there are three ropes, and the weight is eight times
the power. Another rope would render it sixteen times the power, and so
on.
In this system, it is obvious that the ropes will require to have
different degrees of strength, since the tension to which they are
subject increases in a double proportion from the power to the weight.
(279.) If each of the ropes, instead of being attached to fixed points
at the top, are carried over fixed pulleys, and attached to the several
moveable pulleys respectively, as in _fig. 127._, the power of
the machine will be greatly increased; for in that case the forces
which stretch the successive ropes increase in a treble instead of a
double proportion, as will be evident by attending to the numbers which
express the tensions in the figure. One rope would render the weight
three times the power, two ropes nine times, three ropes twenty-seven
times, and so on. An arrangement of pulleys is represented in _fig.
128._, by which each rope, instead of being finally attached to a fixed
point, as in _fig. 126._, is attached to the weight. The weight
is in this case supported by three ropes; one stretched with a force
equal to the power; another with a force equal to twice the power;
and a third with a force equal to four times the power. The weight is
therefore, in this case, seven times the power.
(280.) If the ropes, instead of being attached to the weight, pass
through wheels, as in _fig. 129._, and are finally attached to the
pulleys above, the power of the machine will be considerably increased.
In the system here represented the weight is twenty-six times the power.
(281.) In considering these several combinations of pulleys, we have
omitted to estimate the effects produced by the weights of the sheaves
and blocks. Without entering into the details of this computation,
it may be observed generally, that in the systems represented in
_figs. 126._, _127._ the weight of the wheel and blocks acts
against the power; but that in _figs. 128._ and _129._ they
assist the powers in supporting the weight. In the systems represented
in _fig. 123._ the weight of the pulleys, to a certain extent,
neutralise each other.
(282.) It will in all cases be found, that that quantity by which the
weight exceeds the power is supported by fixed points; and therefore,
although it be commonly stated that a small power supports a great
weight, yet in the pulley, as in all other machines, the power supports
no more of the weight than is exactly equal to its own amount. It
will not be necessary to establish this in each of the examples which
have been given: having explained it in one instance, the student
will find no difficulty in applying the same reasoning to others. In
_fig. 126._, the fixed pulley sustains a force equal to twice the
power, and by it the power giving tension to the first rope sustains a
part of the weight equal to itself. The first hook sustains a portion
of the weight equal to the tension of the first string, or to the
power. The second hook sustains a force equal to twice the power; and
the third hook sustains a force equal to four times the power. The
three hooks therefore sustain a portion of the weight equal to seven
times the power; and the weight itself being eight times the power, it
is evident that the part of the weight which remains to be supported by
the power is equal to the power itself.
(283.) When a weight is raised by any of the systems of pulleys which
have been last described, the proportion between the velocity of
the weight and the velocity of the power, so frequently noticed in
other machines, will always be observed. In the system of pulleys
represented in _fig. 126._ the weight being eight times the power,
the velocity of the power will be eight times that of the weight. If
the power be moved through eight feet, that part of the rope between
the fixed pulley and the first moveable pulley will be shortened by
eight feet. And since the two parts which lie above the first moveable
pulley must be equally shortened, each will be diminished by four feet;
therefore the first pulley will rise through four feet while the power
moves through eight feet. In the same way it may be shown, that while
the first pulley moves through four feet, the second moves through two;
and while the second moves through two, the third, to which the weight
is attached, is raised through one foot. While the power, therefore, is
carried through eight feet, the weight is moved through one foot.
By reasoning similar to this, it may be shown that the space through
which the power is moved in every case is as many times greater than
the height through which the weight is raised, as the weight is greater
than the power.
(284.) From its portable form, cheapness of construction, and the
facility with which it may be applied in almost every situation,
the pulley is one of the most useful of the simple machines. The
mechanical advantage, however, which it appears in theory to possess
is considerably diminished in practice, owing to the stiffness of the
cordage, and the friction of the wheels and blocks. By this means it
is computed that in most cases so great a proportion as two thirds of
the power is lost. The pulley is much used in building, where weights
are to be elevated to great heights. But its most extensive application
is found in the rigging of ships, where almost every motion is
accomplished by its means.
(285.) In all the examples of pulleys, we have supposed the parts of
the rope sustaining the weight and each of the moveable pulleys to be
parallel to each other. If they be subject to considerable obliquity,
the relative tensions of the different ropes must be estimated
according to the principle applied in (271.)
CHAP. XVI.
ON THE INCLINED PLANE, WEDGE, AND SCREW.
(286.) The inclined plane is the most simple of all machines. It is
a hard plane surface forming some angle with a horizontal plane,
that angle not being a right angle. When a weight is placed on such
a plane, a two-fold effect is produced. A part of the effect of the
weight is resisted by the plane, and produces a pressure upon it; and
the remainder urges the weight down the plane, and would produce a
pressure against any surface resisting its motion placed in a direction
perpendicular to the plane (131.)