other, and the fulcrum F, _fig. 84._, is at the right angle. The
moment of the power, in this case, is P multiplied by A F, and
that of the weight W multiplied by B F. When the instrument is in
equilibrium these moments must be equal.
When the hammer is used for drawing a nail, it is a lever of this kind:
the claw of the hammer is the shorter arm; the resistance of the nail
is the weight; and the hand applied to the handle the power.
(245.) When a beam rests on two props A B, _fig. 85._, and
supports, at some intermediate place C, a weight W, this weight is
distributed between the props in a manner which may be determined by
the principles already explained. If the pressure on the prop B be
considered as a power sustaining the weight W, by means of the lever of
the second kind B A, then this power multiplied by B A must
be equal to the weight multiplied by C A. Hence the pressure on
B will be the same fraction of the weight as the part A C is of
A B. In the same manner it may be proved, that the pressure on A
is the same fraction of the weight as B C is of B A. Thus, if
A C be one third, and therefore B C two thirds of B A,
the pressure on B will be one third of the weight, and the pressure on
A two thirds of the weight.
It follows from this reasoning, that if the weight be in the middle,
equally distant from B and A, each prop will sustain half the weight.
The effect of the weight of the beam itself may be determined by
considering it to be collected at its centre of gravity. If this point,
therefore, be equally distant from the props, the weight of the beam
will be equally distributed between them.
According to these principles, the manner in which a load borne
on poles between two bearers is distributed between them may be
ascertained. As the efforts of the bearers and the direction of the
weight are always parallel; the position of the poles relatively to the
horizon makes no difference in the distribution of the weights between
the bearers. Whether they ascend or descend, or move on a level plane,
the weight will be similarly shared between them.
If the beam extend beyond the prop, as in _fig. 86._, and the
weight be suspended at a point not placed between them, the props must
be applied at different sides of the beam. The pressures which they
sustain may be calculated in the same manner as in the former case.
The pressure of the prop B may be considered as a power sustaining the
weight W by means of the lever B C. Hence, the pressure of B,
multiplied by B A, must be equal to the weight W multiplied by
A C. Therefore, the pressure on B bears the same proportion to the
weight as A C does to A B. In the same manner, considering B
as a fulcrum, and the pressure of the prop A as the power, it may be
proved that the pressure of A bears the same proportion to the weight
as the line B C does to A B. It therefore appears, that the
pressure on the prop A is greater than the weight.
(246.) When great power is required, and it is inconvenient to
construct a long lever, a combination of levers may be used. In
_fig. 87._ such a system of levers is represented, consisting of
three levers of the first kind. The manner in which the effect of the
power is transmitted to the weight may be investigated by considering
the effect of each lever successively. The power at P produces an
upward force at P′, which bears to P the same proportion as P′ F
to P F. Therefore, the effect at P′ is as many times the power
as the line P F is of P′ F. Thus, if P F be ten times
P′ F, the upward force at P′ is ten times the power. The arm
P′ F′ of the second lever is pressed upwards by a force equal
to ten times the power at P. In the same manner this may be shown to
produce an effect at P″ as many times greater than P′ as P′ F′
is greater than P″ F′. Thus, if P′ F′ be twelve times P″ F′, the
effect at P″ will be twelve times that of P′. But this last was ten
times the power, and therefore the P″ will be one hundred and twenty
times the power. In the same manner it may be shown that the weight is
as many times greater than the effect at P″ as P″ F″ is greater than
W F″. If P″ F″ be five times W F″, the weight will be five
times the effect at P″. But this effect is one hundred and twenty times
the power, and therefore the weight would be six hundred times the
power.
In the same manner the effect of any compound system of levers may be
ascertained by taking the proportion of the weight to the power in
each lever separately, and multiplying these numbers together. In the
example given, these proportions are 10, 12, and 5, which multiplied
together give 600. In _fig. 87._ the levers composing the system
are of the first kind; but the principles of the calculation will not
be altered if they be of the second or third kind, or some of one kind
and some of another.
(247.) That number which expresses the proportion of the weight to the
equilibrating power in any machine, we shall call the _power of the
machine_. Thus, if, in a lever, a power of one pound support a weight
of ten pounds, the power of the machine is _ten_. If a power of 2lbs.
support a weight of 11lbs., the power of the machine is 5-1/2, 2 being
contained in 11 5-1/2 times.
(248.) As the distances of the power and weight from the fulcrum of
a lever may be varied at pleasure, and any assigned proportion given
to them, a lever may always be conceived having a power equal to that
of any given machine. Such a lever may be called, in relation to that
machine, the _equivalent lever_.
As every complex machine consists of a number of simple machines acting
one upon another, and as each simple machine may be represented by an
equivalent lever, the complex machine will be represented by a compound
system of equivalent levers. From what has been proved in (246.), it
therefore follows that the power of a complex machine may be calculated
by multiplying together the powers of the several simple machines of
which it is composed.
CHAP. XIV.
OF WHEEL-WORK.
(249.) When a lever is applied to raise a weight, or overcome a
resistance, the space through which it acts at any one time is small,
and the work must be accomplished by a succession of short and
intermitting efforts. In _fig. 81._, after the weight has been
raised from W to W′, the lever must again return to its first position,
to repeat the action. During this return the motion of the weight is
suspended, and it will fall downwards unless some provision be made to
sustain it. The common lever is, therefore, only used in cases where
weights are required to be raised through small spaces, and under these
circumstances its great simplicity strongly recommends it. But where
a continuous motion is to be produced, as in raising ore from the
mine, or in weighing the anchor of a vessel, some contrivance must be
adopted to remove the intermitting action of the lever, and render
it continual. The various forms given to the lever, with a view to
accomplish this, are generally denominated the _wheel and axle_.
[Illustration: _H. Adlard, sc._
_London, Pubd. by Longman & Co._]
In _fig. 88._, A B is a horizontal axle, which rests in
pivots at its extremities, or is supported in gudgeons, and capable of
revolving. Round this axis a rope is coiled, which sustains the weight
W. On the same axis a wheel C is fixed, round which a rope is coiled
in a contrary direction, to which is appended the power P. The moment
of the power is found by multiplying it by the radius of a wheel, and
the moment of the weight, by multiplying it by the radius of its axle.
If these moments be equal (185.), the machine will be in equilibrium.
Whence it appears that the power of the machine (247.) is expressed by
the proportion which the radius of the wheel bears to the radius of
the axle; or, what is the same, of the diameter of the wheel to the
diameter of the axle. This principle is applicable to the wheel and
axle in every variety of form under which it can be presented.
(250.) It is evident that as the power descends continually, and the
rope is uncoiled from the wheel, the weight will be raised continually,
the rope by which it is suspended being at the same time coiled upon
the axle.
When the machine is in equilibrium, the forces of both the weight and
power are sustained by the axle, and distributed between its props, in
the manner explained in (245.)
When the machine is applied to raise a weight, the velocity with which
the power moves is as many times greater than that with which the
weight rises, as the weight itself is greater than the power. This is
a principle which has already been noticed, and which is common to all
machines whatsoever. It may hence be proved, that in the elevation of
the weight a quantity of power is expended equal to that which would be
necessary to elevate the weight if the power were immediately applied
to it, without the intervention of any machine. This has been explained
in the case of the lever in (241.), and may be explained in the
present instance in nearly the same words.
In one revolution of the machine the length of rope uncoiled from
the wheel is equal to the circumference of the wheel, and through
this space the power must therefore move. At the same time the length
of rope coiled upon the axle is equal to the circumference of the
axle, and through this space the weight must be raised. The spaces,
therefore, through which the power and weight move in the same time,
are in the proportion of the circumferences of the wheel and axle; but
these circumferences are in the same proportion as their diameters.
Therefore the velocity of the power will bear to the velocity of the
weight the same proportion as the diameter of the wheel bears to the
diameter of the axle, or, what is the same, as the weight bears to the
power (249).
(251.) We have here omitted the consideration of the thickness of the
rope. When this is considered, the force must be conceived as acting in
the direction of the centre of the rope, and therefore the thickness
of the rope which supports the power ought to be added to the diameter
of the wheel, and the thickness of the rope which supports the weight
to the diameter of the axle. It is the more necessary to attend to
this circumstance, as the strength of the rope necessary to support
the weight causes its thickness to bear a considerable proportion to
the diameter of the axle; while the rope which sustains the power not
requiring the same strength, and being applied to a larger circle,
bears a very inconsiderable proportion to its diameter.
(252.) In numerous forms of the wheel and axle, the weight or
resistance is applied by a rope coiled upon the axle; but the manner in
which the power is applied is very various, and not often by means of a
rope. The circumference of a wheel sometimes carries projecting pins,
as represented in _fig. 88._, to which the hand is applied to
turn the machine. An instance of this occurs in the wheel used in the
steerage of a vessel.
In the common _windlass_, the power is applied by means of a _winch_,
which is a rectangular lever, as represented in _fig. 89._ The arm
B C of the winch represents the radius of the wheel, and the power
is applied to C D at right angles to B C.
In some cases no wheel is attached to the axle; but it is pierced with
holes directed towards its centre, in which long levers are incessantly
inserted, and a continuous action produced by several men working at
the same time; so that while some are transferring the levers from hole
to hole, others are working the windlass.
The axle is sometimes placed in a vertical position, the wheel or
levers being moved horizontally. The _capstan_ is an example of this:
a vertical axis is fixed in the deck of the ship; the circumference is
pierced with holes presented towards its centre. These holes receive
long levers, as represented in _fig. 90._ The men who work the
capstan walk continually round the axle, pressing forward the levers
near their extremities.
In some cases the wheel is turned by the weight of animals placed at
its circumference, who move forward as fast as the wheel descends,
so as to maintain their position continually at the extremity of the
horizontal diameter. The _treadmill_, _fig. 91._, and certain
_cranes_, such as _fig. 92._, are examples of this.
In water-wheels, the power is the weight of water contained in
buckets at the circumference, as in _fig. 93._, which is called
an over-shot wheel: and sometimes by the impulse of water against
float-boards at the circumference, as in the under-shot wheel,
_fig. 94._ Both these principles act in the breast-wheel,
_fig. 95._
In the paddle-wheel of a steam-boat, the power is the resistance which
the water offers to the motion of the paddle-boards.
In windmills, the power is the force of the wind acting on various
parts of the arms, and may be considered as different powers
simultaneously acting on different wheels having the same axle.
(253.) In most cases in which the wheel and axle is used, the action of
the power is liable to occasional suspension or intermission, in which
case some contrivance is necessary to prevent the recoil of the weight.
A ratchet wheel R, _fig. 88._, is provided for this purpose, which
is a contrivance which permits the wheel to turn in one direction;
but a catch which falls between the teeth of a fixed wheel prevents
its motion in the other direction. The effect of the power or weight
is sometimes transmitted to the wheel or axle by means of a straight
bar, on the edge of which teeth are raised, which engage themselves in
corresponding teeth on the wheel or axle. Such a bar is called a rack;
and an instance of its use may be observed in the manner of working the
pistons of an air-pump.
(254.) The power of the wheel and axle being expressed by the number
of times the diameter of the axle is contained in that of the wheel,
there are obviously only two ways by which this power may be increased;
viz. either by increasing the diameter of the wheel, or diminishing
that of the axle. In cases where great power is required, each of these
methods is attended with practical inconvenience and difficulty. If the
diameter of the wheel be considerably enlarged, the machine will become
unwieldy, and the power will work through an unmanageable space. If,
on the other hand, the power of the machine be increased by reducing
the thickness of the axle, the strength of the axle will become
insufficient for the support of that weight, the magnitude of which had
rendered the increase of the power of the machine necessary. To combine
the requisite strength with moderate dimensions and great mechanical
power is, therefore, impracticable in the ordinary form of the wheel
and axle. This has, however, been accomplished by giving different
thicknesses to different parts of the axle, and carrying a rope, which
is coiled on the thinner part, through a wheel attached to the weight,
and coiling it in the opposite direction on the thicker part, as in
_fig. 96._ To investigate the proportion of the power to the
weight in this case, let _fig. 97._ represent a section of the
apparatus at right angles to the axis. The weight is equally suspended
by the two parts of the rope, S and S′, and therefore each part is
stretched by a force equal to half the weight. The moment of the force,
which stretches the rope S, is half the weight multiplied by the radius
of the thinner part of the axle. This force being at the same side of
the centre with the power, co-operates with it in supporting the force
which stretches S′, and which acts at the other side of the centre. By
the principle established in (185.), the moments of P and S must be
equal to that of S′; and therefore if P be multiplied by the radius of
the wheel, and added to half the weight multiplied by the radius of the
thinner part of the axle, we must obtain a sum equal to half the weight
multiplied by the radius of the thicker part of the axle. Hence it is
easy to perceive, that the power multiplied by the radius of the wheel
is equal to half the weight multiplied by the difference of the radii
of the thicker and thinner parts of the axle; or, what is the same, the
power multiplied by the diameter of the wheel, is equal to the weight
multiplied by half the difference of the diameters of the thinner and
thicker parts of the axle.
A wheel and axle constructed in this manner is equivalent to an
ordinary one, in which the wheel has the same diameter, and whose axle
has a diameter equal to half the difference of the diameters of the
thicker and thinner parts. The power of the machine is expressed by the
proportion which the diameter of the wheel bears to half the difference
of these diameters; and therefore this power, when the diameter of the
wheel is given, does not, as in the ordinary wheel and axle, depend
on the smallness of the axle, but on the smallness of the difference
of the thinner and thicker parts of it. The axle may, therefore,
be constructed of such a thickness as to give it all the requisite
strength, and yet the difference of the diameters of its different
parts may be so small as to give it all the requisite power.
(255.) It often happens that a varying weight is to be raised, or
resistance overcome by a uniform power. If, in such a case, the weight
be raised by a rope coiled upon a uniform axle, the action of the
power would not be uniform, but would vary with the weight. It is,
however, in most cases desirable or necessary that the weight or
resistance, even though it vary, shall be moved uniformly. This will
be accomplished if by any means the leverage of the weight is made
to increase in the same proportion as the weight diminishes, and to
diminish in the same proportion as the weight increases: for in that
case the moment of the weight will never vary, whatever it gains by the
increase of weight being lost by the diminished leverage, and whatever
it loses by the diminished weight being gained by the increased
leverage. An axle, the surface of which is curved in such a manner,
that the thickness on which the rope is coiled continually increased
or diminishes in the same proportion as the weight or resistance
diminishes or increases, will produce this effect.
It is obvious that all that has been said respecting a variable
weight or resistance, is also applicable to a variable power, which,
therefore, may, by the same means, be made to produce a uniform effect.
An instance of this occurs in a watch, which is moved by a spiral
spring. When the watch has been wound up, this spring acts with its
greatest intensity, and as the watch goes down, the elastic force of
the spring gradually loses its energy. This spring is connected by a
chain with an axle of varying thickness, called a _fusee_. When the
spring is at its greatest intensity, the chain acts upon the thinnest
part of the fusee, and as it is uncoiled it acts upon a part of the
fusee which is continually increasing in thickness, the spring at the
same time losing its elastic power in exactly the same proportion. A
representation of the fusee, and the cylindrical box which contains
the spring, is given in _fig. 98._, and of the spring itself in
_fig. 99._
(256.) When great power is required, wheels and axles may be combined
in a manner analogous to a compound system of levers, explained
in (246.) In this case the power acts on the circumference of the
first wheel, and its effect is transmitted to the circumference
of the first axle. That circumference is placed in connection with
the circumference of the second wheel, and the effect is thereby
transmitted to the circumference of the second axle, and so on. It
is obvious from what was proved in (248.), that the power of such a
combination of wheels and axles will be found by multiplying together
the powers of the several wheels of which it is composed. It is
sometimes convenient to compute this power by numbers expressing the
proportions of the circumferences or diameters of the several wheels,
to the circumferences or diameters of the several axles respectively.
This computation is made by first multiplying the numbers together
which express the circumferences or diameters of the wheels, and then
multiplying together the numbers which express the circumferences or
diameters of the several axles. The proportion of the two products
will express the power of the machine. Thus, if the circumferences or
diameters be as the numbers 10, 14, and 15, their product will be 2100;
and if the circumferences or diameters of the axles be expressed by the
numbers 3, 4, and 5, their product will be 60, and the power of the
machine will be expressed by the proportion of 2100 and 60, or 35 to 1.
[Illustration: _H. Adlard, sc._
_London, Pubd. by Longman & Co._]
(257.) The manner in which the circumferences of the axles act upon
the circumferences of the wheels in compound wheel-work is various.
Sometimes a strap or cord is applied to a groove in the circumference
of the axle, and carried round a similar groove in the circumference
of the succeeding wheel. The friction of this cord or strap with the
groove is sufficient to prevent its sliding and to communicate the
force from the axle to the wheel, or _vice versa_. This method of
connecting wheel-work is represented in _fig. 100._
Numerous examples of wheels and axles driven by straps or cords occur
in machinery applied to almost every department of the arts and
manufactures. In the turning lathe, the wheel worked by the treddle is
connected with the mandrel by a catgut cord passing through grooves
in the wheel and axle. In all great factories, revolving shafts are
carried along the apartments, on which, at certain intervals, straps
are attached passing round their circumferences and carried round
the wheels which give motion to the several machines. If the wheels,
connected by straps or cords, are required to revolve in the same
direction, these cords are arranged as in _fig. 100._; but if they
are required to revolve in contrary directions, they are applied as in
_fig. 101._
One of the chief advantages of the method of transmitting motion
between wheels and axles by straps or cords, is that the wheel and
axle may be placed at any distance from each other which may be found
convenient, and may be made to turn either in the same or contrary
directions.
(258.) When the circumference of the wheel acts immediately on the
circumference of the succeeding axle, some means must necessarily be
adopted to prevent the wheel from moving in contact with the axle
without compelling the latter to turn. If the surfaces of both were
perfectly smooth, so that all friction were removed, it is obvious that
either would slide over the surface of the other, without communicating
motion to it. But, on the other hand, if there were any asperities,
however small, upon these surfaces, they would become mutually inserted
among each other, and neither the wheel nor axle could move without
causing the asperities with which its edge is studded to encounter
those asperities which project from the surface of the other; and
thus, until these projections should be broken off, both wheel and
axle must be moved at the same time. It is on this account that if the
surfaces of the wheels and axles are by any means rendered rough, and
pressed together with sufficient force, the motion of either will turn
the other, provided the load or resistance be not greater than the
force necessary to break off these small projections which produce the
friction.
In cases where great power is not required, motion is communicated in
this way through a train of wheel-work, by rendering the surface of the
wheel and axle rough, either by facing them with buff leather, or with
wood cut across the grain. This method is sometimes used in spinning
machinery, where one large buffed wheel, placed in a horizontal
position, revolves in contact with several small buffed rollers, each
roller communicating motion to a spindle. The position of the wheel W,
and the rollers R R, &c., are represented in _fig. 102._ Each
roller can be thrown out of contact with the wheel, and restored to it
at pleasure.
The communication of motion between wheels and axles by friction has
the advantage of great smoothness and evenness, and of proceeding with
little noise; but this method can only be used in cases where the
resistance is not very considerable, and therefore is seldom adopted in
works on a large scale. Dr. Gregory mentions an instance of a saw mill
at Southampton, where the wheels act upon each other by the contact of
the end grain of wood. The machinery makes very little noise, and wears
very well, having been used not less than 20 years.