the power sustains the weight; but this, in fact, is not the case, and
hence arises that appearance of paradox which has already been alluded
to. If, for example, it is said that a power of one ounce sustains the
weight of one ton, astonishment is not unnaturally excited, because
the fact, as thus stated, if the terms be literally interpreted, is
physically impossible. No power less than a ton can, in the ordinary
acceptation of the word, support the weight of a ton. It will, however,
be asked how it happens that a machine _appears_ to do this? how it
happens that by holding a silken thread, which an ounce weight would
snap, many hundred weight may be sustained? To explain this it will
only be necessary to consider the effect of a machine, when the power
and weight are in equilibrium.
(225.) In every machine there are some fixed points or props; and the
arrangement of the parts is always such, that the pressure, excited by
the power or weight, or both, is distributed among these props. If the
weight amount to twenty hundred, it is possible so to distribute it,
that any proportion, however great, of it may be thrown on the fixed
points or props of the machine; the remaining part only can properly be
said to be supported by the power, and this part can never be greater
than the power. Considering the effect in this way, it appears that
the power supports just so much of the weight and no more as is equal
to its own force, and that all the remaining part of the weight is
sustained by the machine. The force of these observations will be more
apparent when the nature and properties of the mechanic powers and
other machines have been explained.
(226.) When a machine is considered dynamically, its effects are
explained on different principles. It is true that, in this case, a
very small power may elevate a very great weight; but nevertheless,
in so doing, whatever be the machine used, the total expenditure of
power, in raising the weight through any height, is never less than
that which would be expended if the power were immediately applied to
the weight without the intervention of any machine. This circumstance
arises from an universal property of machines by which the velocity of
the weight is always less than that of the power, in exactly the same
proportion as the power itself is less than the weight; so that when
a certain power is applied to elevate a weight, the rate at which the
elevation is effected is always slow in the same proportion as the
weight is great. From a due consideration of this remarkable law, it
will easily be understood, that a machine can never diminish the total
expenditure of power necessary to raise any weight or to overcome any
resistance. In such cases, all that a machine ever does or ever can
do, is to enable the power to be expended at a slow rate, and in a
more advantageous direction than if it were immediately applied to the
weight or the resistance.
Let us suppose that P is a power amounting to an ounce, and that W is
a weight amounting to 50 ounces, and that P elevates W by means of a
machine. In virtue of the property already stated, it follows, that
while P moves through 50 feet, W will be moved through 1 foot; but
in moving P through 50 feet, 50 distinct efforts are made, by each
of which 1 ounce is moved through 1 foot, and by which collectively
50 distinct ounces might be successively raised through 1 foot. But
the weight W is 50 ounces, and has been raised through 1 foot; from
whence it appears, that the expenditure of power is equal to that which
would be necessary to raise the weight without the intervention of any
machine.
This important principle may be presented under another aspect, which
will perhaps render it more apparent. Suppose the weight W were
actually divided into 50 equal parts, or suppose it were a vessel of
liquid weighing 50 ounces, and containing 50 equal measures; if these
50 measures were successively lifted through a height of 1 foot; the
efforts necessary to accomplish this would be the same as those used
to move the power P through 50 feet, and it is obvious, that the total
expenditure of force would be the same as that which would be necessary
to lift the entire contents of the vessel through 1 foot.
When the nature and properties of the mechanic powers and other
machines have been explained, the force of these observations will be
more distinctly perceived. The effects of props and fixed points in
sustaining a part of the weight, and sometimes the whole, both of the
weight and power, will then be manifest, and every machine will furnish
a verification of the remarkable proportion between the velocities
of the weight and power, which has enabled us to explain what might
otherwise be paradoxical and difficult of comprehension.
(227.) The most simple species of machines are those which are commonly
denominated the MECHANIC POWERS. These have been differently enumerated
by different writers. If, however, the object be to arrange in distinct
classes, and in the smallest possible number of them, those machines
which are alike in principle, the mechanic powers may be reduced to
three.
1. The lever.
2. The cord.
3. The inclined plane.
To one or other of these classes all simple machines whatever may be
reduced, and all complex machines may be resolved into simple elements
which come under them.
(228.) The first class includes every machine which is composed of
a solid body revolving on a fixed axis, although the name lever has
been commonly confined to cases where the machine affects certain
particular forms. This is by far the most useful class of machines, and
will require in subsequent chapters very detailed development. The
general principle, upon which equilibrium is established between the
power and weight in machines of this class has been already explained
in (183.) The power and weight are always supposed to be applied in
directions at right angles to the axis. If lines be drawn from the axis
perpendicular to the directions of power and weight, equilibrium will
subsist, provided the power multiplied by the perpendicular distance
of its direction from the axis, be equal to the weight multiplied by
the perpendicular distance of its direction from the axis. This is a
principle to which we shall have occasion to refer in explaining the
various machines of this class.
(229.) If the moment of the power (184.) be greater than that of the
weight, the effect of the power will prevail over that of the weight,
and elevate it; but if, on the other hand, the moment of the power be
less than that of the weight, the power will be insufficient to support
the weight, and will allow it to fall.
(230.) The second class of simple machines includes all those cases
in which force is transmitted by means of flexible threads, ropes,
or chains. The principle, by which the effects of these machines are
estimated, is, that the tension throughout the whole length of the same
cord, provided it be perfectly flexible, and free from the effects
of friction, must be the same. Thus, if a force acting at one end be
balanced by a force acting at the other end, however the cord may be
bent, or whatever course it may be compelled to take, by any causes
which may affect it between its ends, these forces must be equal,
provided the cord be free to move over any obstacles which may deflect
it.
Within this class of machines are included all the various forms of
_pulleys_.
(231.) The third class of simple machines includes all those cases in
which the weight or resistance is supported or moved on a hard surface
inclined to the vertical direction.
The effects of such machines are estimated by resolving the whole
weight of the body into two elements by the parallelogram of forces.
One of these elements is perpendicular to the surface, and supported
by its resistance; the other is parallel to the surface, and supported
by the power. The proportion, therefore, of the power to the weight
will always depend on the obliquity of the surface to the direction of
the weight. This will be easily understood by referring to what has
been already explained in Chapter VIII.
Under this class of machines come the inclined plane, commonly so
called, the wedge, the screw, and various others.
(232.) In order to simplify the development of the elementary theory
of machines, it is expedient to omit the consideration of many
circumstances, of which, however, a strict account must be taken before
any practically useful application of that theory can be attempted.
A machine, as we must for the present contemplate it, is a thing
which can have no real or practical existence. Its various parts are
considered to be free from friction: all surfaces which move in contact
are supposed to be infinitely smooth and polished. The solid parts are
conceived to be absolutely inflexible. The weight and inertia of the
machine itself are wholly neglected, and we reason upon it as if it
were divested of these qualities. Cords and ropes are supposed to have
no stiffness, to be infinitely flexible. The machine, when it moves, is
supposed to suffer no resistance from the atmosphere, and to be in all
respects circumstanced as if it were _in vacuo_.
It is scarcely necessary to state, that, all these suppositions
being false, none of the consequences deduced from them can be true.
Nevertheless, as it is the business of art to bring machines as near
to this state of ideal perfection as possible, the conclusions which
are thus obtained, though false in a strict sense, yet deviate from
the truth in but a small degree. Like the first outline of a picture,
they resemble in their general features that truth to which, after many
subsequent corrections, they must finally approximate.
After a first approximation has been made on the several false
suppositions which have been mentioned, various effects, which have
been previously neglected, are successively taken into account.
Roughness, rigidity, imperfect flexibility, the resistance of air and
other fluids, the effects of the weight and inertia of the machine,
are severally examined, and their laws and properties detected. The
modifications and corrections, thus suggested as necessary to be
introduced into our former conclusions, are applied, and a second
approximation, but still _only_ an approximation, to truth is made.
For, in investigating the laws which regulate the several effects
just mentioned, we are compelled to proceed upon a new group of false
suppositions. To determine the laws which regulate the friction of
surfaces, it is necessary to assume that every part of the surfaces of
contact are uniformly rough; that the solid parts which are imperfectly
rigid, and the cords which are imperfectly flexible, are constituted
throughout their entire dimensions of a uniform material; so that the
imperfection does not prevail more in one part than another. Thus,
all irregularity is left out of account, and a general average of the
effects taken. It is obvious, therefore, that by these means we have
still failed in obtaining a result exactly conformable to the real
state of things; but it is equally obvious, that we have obtained
one much more conformable to that state than had been previously
accomplished, and sufficiently near it for most practical purposes.
This apparent imperfection in our instruments and powers of
investigation is not peculiar to mechanics: it pervades all departments
of natural science. In astronomy, the motions of the celestial bodies,
and their various changes and appearances as developed by theory,
assisted by observation and experience, are only approximations to the
real motions and appearances which take place in nature. It is true
that these approximations are susceptible of almost unlimited accuracy;
but still they are, and ever will continue to be, only approximations.
Optics and all other branches of natural science are liable to the same
observations.
CHAP. XIII.
OF THE LEVER.
(233.) An inflexible, straight bar, turning on an axis, is commonly
called a _lever_. The _arms_ of the lever are those parts of the bar
which extend on each side of the axis.
The axis is called the _fulcrum_ or _prop_.
(234.) Levers are commonly divided into three kinds, according to the
relative positions of the power, the weight, and the fulcrum.
In a lever of the first kind, as in _fig. 78._, the fulcrum is
between the power and weight.
In a lever of the second kind, as in _fig. 79._, the weight is
between the fulcrum and power.
In a lever of the third kind, as in _fig. 80._, the power is
between the fulcrum and weight.
(235.) In all these cases, the power will sustain the weight in
equilibrium, provided its moment be equal to that of the weight. (184.)
But the moment of the power is, in this case, equal to the product
obtained by multiplying the power by its distance from the fulcrum; and
the moment of the weight by multiplying the weight by its distance from
the fulcrum. Thus, if the number of ounces in P, being multiplied by
the number of inches in P F, be equal to the number of ounces in
W, multiplied by the number of inches in W F, equilibrium will be
established. It is evident from this, that as the distance of the power
from the fulcrum increases in comparison to the distance of the weight
from the fulcrum, in the same degree exactly will the proportion of the
power to the weight diminish. In other words, the proportion of the
power to the weight will be always the same as that of their distances
from the fulcrum taken in a reverse order.
In cases where a small power is required to sustain or elevate a great
weight, it will therefore be necessary either to remove the power to a
great distance from the fulcrum, or to bring the weight very near it.
(236.) Numerous examples of levers of the first kind may be given. A
crow-bar, applied to elevate a stone or other weight, is an instance.
The fulcrum is another stone placed near that which is to be raised,
and the power is the hand placed at the other end of the bar.
A handspike is a similar example.
A poker applied to raise fuel is a lever of the first kind, the fulcrum
being the bar of the grate.
Scissors, shears, nippers, pincers, and other similar instruments are
composed of two levers of the first kind; the fulcrum being the joint
or pivot, and the weight the resistance of the substance to be cut or
seized; the power being the fingers applied at the other end of the
levers.
The brake of a pump is a lever of the first kind; the pump-rods and
piston being the weight to be raised.
(237.) Examples of levers of the second kind, though not so frequent as
those just mentioned, are not uncommon.
An oar is a lever of the second kind. The reaction of the water against
the blade is the fulcrum. The boat is the weight, and the hand of the
boatman the power.
The rudder of a ship or boat is an example of this kind of lever, and
explained in a similar way.
The chipping knife is a lever of the second kind. The end attached
to the bench is the fulcrum, and the weight the resistance of the
substance to be cut, placed beneath it.
A door moved upon its hinges is another example.
Nut-crackers are two levers of the second kind; the hinge which unites
them being the fulcrum, the resistance of the shell placed between
them being the weight, and the hand applied to the extremity being the
power.
A wheelbarrow is a lever of the second kind; the fulcrum being the
point at which the wheel presses on the ground, and the weight being
that of the barrow and its load, collected at their centre of gravity.
The same observation may be applied to all two-wheeled carriages, which
are partly sustained by the animal which draws them.
(238.) In a lever of the third kind, the weight, being more distant
from the fulcrum than the power, must be proportionably less than
it. In this instrument, therefore, the power acts upon the weight to
a mechanical disadvantage, inasmuch as a greater power is necessary
to support or move the weight than would be required if the power
were immediately applied to the weight, without the intervention of a
machine. We shall, however, hereafter show that the advantage which
is lost in force is gained in despatch, and that in proportion as the
weight is less than the power which moves it, so will the speed of its
motion be greater than that of the power.
Hence a lever of the third kind is only used in cases where the
exertion of great power is a consideration subordinate to those of
rapidity and despatch.
The most striking example of levers of the third kind is found in the
animal economy. The limbs of animals are generally levers of this
description. The socket of the bone is the fulcrum; a strong muscle
attached to the bone near the socket is the power; and the weight
of the limb, together with whatever resistance is opposed to its
motion, is the weight. A slight contraction of the muscle in this case
gives a considerable motion to the limb: this effect is particularly
conspicuous in the motion of the arms and legs in the human body; a
very inconsiderable contraction of the muscles at the shoulders and
hips giving the sweep to the limbs from which the body derives so much
activity.
The treddle of the turning lathe is a lever of the third kind. The
hinge which attaches it to the floor is the fulcrum, the foot applied
to it near the hinge is the power, and the crank upon the axis of the
fly-wheel, with which its extremity is connected, is the weight.
Tongs are levers of this kind, as also the shears used in shearing
sheep. In these cases the power is the hand placed immediately below
the fulcrum or point where the two levers are connected.
(239.) When the power is said to support the weight by means of a lever
or any other machine, it is only meant that the power keeps the machine
in equilibrium, and thereby enables it to sustain the weight. It is
necessary to attend to this distinction, to remove the difficulty which
may arise from the paradox of a small power sustaining a great weight.
In a lever of the first kind, the fulcrum F, _fig. 78._, or axis,
sustains the united forces of the power and weight.
In a lever of the second kind, if the power be supposed to act over
a wheel R, _fig. 79._, the fulcrum F sustains a pressure equal
to the difference between the power and weight, and the axis of the
wheel R sustains a pressure equal to twice the power; so that the total
pressures on F and R are equivalent to the united forces of the power
and weight.
In a lever of the third kind similar observations are applicable. The
wheel R, _fig. 80._, sustains a pressure equal to twice the power,
and the fulcrum F sustains a pressure equal to the difference between
the power and weight.
These facts may be experimentally established by attaching a string
to the lever immediately over the fulcrum, and suspending the lever
by that string from the arm of a balance. The counterpoising weight,
when the fulcrum is removed, will, in the first case, be equal to the
sum of the weight and power, and in the last two cases equal to their
difference.
(240.) We have hitherto omitted the consideration of the effect of the
weight of the lever itself. If the centre of gravity of the lever be
in the vertical line through the axis, the weight of the instrument
will have no other effect than to increase the pressure on the axis by
its own amount. But if the centre of gravity be on the same side of
the axis with the weight, as at G, it will oppose the effect of the
power, a certain part of which must therefore be allowed to support
it. To ascertain what part of the power is thus expended, it is to
be considered that the moment of the weight of the lever collected
at G, is found by multiplying that weight by the distance G F.
The moment of that part of the power which supports this must be
equal to it; therefore, it is only necessary to find how much of the
power multiplied by P F will be equal to the weight of the lever
multiplied by G F. This is a question in common arithmetic.
If the centre of gravity of the lever be at a different side of the
axis from the weight, as at G′, the weight of the instrument will
co-operate with the power in sustaining the weight W. To determine what
portion of the weight W is thus sustained by the weight of the lever,
it is only necessary to find how much of W, multiplied by the distance
W F, is equal to the weight of the lever multiplied by G′ F.
In these cases the pressure on the fulcrum, as already estimated, will
always be increased by the weight of the lever.
(241.) The sense in which a small power is said to sustain a great
weight, and the manner of accomplishing this, being explained, we
shall now consider how the power is applied in moving the weight. Let
P W, _fig. 81._, be the places of the power and weight, and
F that of the fulcrum, and let the power be depressed to P′ while the
weight is raised to W′. The space P P′ evidently bears the same
proportion to W W′, as the arm P F to W F. Thus if
P F be ten times W F, P P′ will be ten times W W′.
A power of one pound at P being moved from P to P′, will carry a weight
of ten pounds from W to W′. But in this case it ought not to be said,
that a lesser weight moves a greater, for it is not difficult to show,
that the total expenditure of force in the motion of one pound from P
to P′ is exactly the same as in the motion of ten pounds from W to W′.
If the space P P′ be ten inches, the space W W′ will be one
inch. A weight of one pound is therefore moved through ten successive
inches, and in each inch the force expended is that which would be
sufficient to move one pound through one inch. The total expenditure
of force from P to P′ is ten times the force necessary to move one
pound through one inch, or what is the same, it is that which would be
necessary to move ten pounds through one inch. But this is exactly what
is accomplished by the opposite end W of the lever; for the weight W is
ten pounds, and the space W W′ is one inch.
If the weight W of ten pounds could be conveniently divided into ten
equal parts of one pound each, each part might be separately raised
through one inch, without the intervention of the lever or any other
machine. In this case, the same quantity of power would be expended,
and expended in the same manner as in the case just mentioned.
It is evident, therefore, that when a machine is applied to raise a
weight or to overcome resistance, as much force must be really used as
if the power were immediately applied to the weight or resistance. All
that is accomplished by the machine is to enable the power to do that
by a succession of distinct efforts which should be otherwise performed
by a single effort. These observations will be found to be applicable
to all machines whatever.
(242.) Weighing machines of almost every kind, whether used for
commercial or philosophical purposes, are varieties of the lever. The
common balance, which, of all weighing machines, is the most perfect
and best adapted for ordinary use, whether in commerce or experimental
philosophy, is a lever with equal arms. In the steel-yard one weight
serves as a counterpoise and measure of others of different amount, by
receiving a leverage variable according to the varying amount of the
weight against which it acts. A detailed account of such instruments
will be found in Chapter XXI.
(243.) We have hitherto considered the power and weight as acting on
the lever, in directions perpendicular to its length and parallel to
each other. This does not always happen. Let A B, _fig. 83._,
be a lever whose fulcrum is F, and let A R be the direction of the
power, and B S the direction of the weight. If the lines R A
and S B be continued, and perpendiculars F C and F D
drawn from the fulcrum to those lines, the moment of the power will be
found by multiplying the power by the line F C, and the moment of
the weight by multiplying the weight by F D. If these moments be
equal, the power will sustain the weight in equilibrium. (185).
It is evident, that the same reasoning will be applicable when the
arms of the lever are not in the same direction. These arms may be of
any figure or shape, and may be placed relatively to each other in any
position.