horizontal base, A C its height, and A B C its angle
of elevation. Let W be a weight placed upon it. This weight acts in
the vertical direction W D, and is equivalent to two forces,
W F perpendicular to the plane, and W E directed down the
plane (74.) If a plane be placed at right angles to the inclined
plane below W, it will resist the descent of the weight, and sustain
a pressure expressed by W E. Thus, the weight W resting in the
corner, instead of producing one pressure in the direction W D,
will produce two pressures, one expressed by W F upon the inclined
plane, and the other expressed by W E upon the resisting plane.
These pressures respectively have the same proportion to the entire
weight as W F and W E have to W D, or as D E and
W E have to W D, because D E is equal to W F. Now
the triangle W E D is in all respects similar to the triangle
A B C, the one differing from the other only in the scale on
which it is constructed. Therefore, the three lines A C, C B,
and B A, are in the same proportion to each other as the lines
W E, E D, and W D. Hence, A B has to A C the
same proportion as the whole weight has to the pressure directed toward
B, and A B has to B C the same proportion as the whole
weight has to the pressure on the inclined plane.
We have here supposed the weight to be sustained upon the inclined
plane by a hard plane fixed at right angles to it. But the power
necessary to sustain the weight will be the same in whatever way it is
applied, provided it act in the direction of the plane. Thus, a cord
may be attached to the weight, and stretched towards A, or the hands of
men may be applied to the weight below it, so as to resist its descent
towards B. But in whatever way it be applied, the amount of the power
will be determined in the same manner. Suppose the weight to consist
of as many pounds as there are inches in A B, then the power
requisite to sustain it upon the plane will consist of as many pounds
as there are inches in A C, and the pressure on the plane will
amount to as many pounds as there are inches in B C.
From what has been stated it may easily be inferred that the less the
elevation of the plane is, the less will be the power requisite to
sustain a given weight upon it, and the greater will be the pressure
upon it. Suppose the inclined plane A B to turn upon a hinge
at B, and to be depressed so that its angle of elevation shall be
diminished, it is evident that as this angle decreases the height of
the plane decreases, and its base increases. Thus, when it takes the
position B A′, the height A′ C′ is less than the former
height A C, while the base B C′ is greater than the former
base B C. The power requisite to support the weight upon the plane
in the position B A′ is represented by A′ C′, and is as much
less than the power requisite to sustain it upon the plane A B,
as the height A′ C′ is less than the height A C. On the
other hand, the pressure upon the plane in the position B A′ is
as much greater than the pressure upon the plane B A, as the base
B C′ is greater than the base B C.
(287.) The power of an inclined plane, considered as a machine, is
therefore estimated by the proportion which its length bears to its
height. This power is always increased by diminishing the elevation of
the plane.
Roads which are not level may be regarded as inclined planes, and
loads drawn upon them in carriages, considered in reference to the
powers which impel them, are subject to all the conditions which have
been established for inclined planes. The inclination of the road is
estimated by the height corresponding to some proposed length. Thus it
is said to rise one foot in fifteen, one foot in twenty, &c., meaning
that if fifteen or twenty feet of the road be taken as the length of
an inclined plane, such as A B, the corresponding height will be
one foot. Or the same may be expressed thus: that if fifteen or twenty
feet be measured upon the road, the difference of the levels of the two
extremities of the distance measured is one foot. According to this
method of estimating the inclination of roads, the power requisite to
sustain a load upon them (setting aside the effect of friction), is
always proportional to that elevation. Thus, if a road rise one foot in
twenty, a power of one ton will be sufficient to sustain twenty tons,
and so on.
On a horizontal plane the only resistance which the power has
to overcome is the friction of the load with the plane, and the
consideration of this being for the present omitted, a weight once put
in motion would continue moving for ever, without any further action of
the power. But if the plane be inclined, the power will be expended in
raising the weight through the perpendicular height of the plane. Thus,
in a road which rises one foot in ten, the power is expended in raising
the weight through one perpendicular foot for every ten feet of the
road over which it is moved. As the expenditure of power depends upon
the rate at which the weight is raised perpendicularly, it is evident
that the greater the inclination of the road is, the slower the motion
must be with the same force. If the energy of the power be such as to
raise the weight at the rate of one foot per minute, the weight may be
moved in each minute through that length of the road which corresponds
to a rise of one foot. Thus, if two roads rise one at the rate of a
foot in fifteen feet, and the other at the rate of one foot in twenty
feet, the same expenditure of power will move the weight through
fifteen feet of the one, and twenty feet of the other at the same rate.
From such considerations as these, it will readily appear that it may
often be more expedient to carry a road through a circuitous route
than to continue it in the most direct course; for though the measured
length of road may be considerably greater than in the former case, yet
more may be gained in speed with the same expenditure of power than is
lost by the increase of distance. By attending to these circumstances,
modern road-makers have greatly facilitated and expedited the
intercourse between distant places.
(288.) If the power act obliquely to the plane, it will have a twofold
effect; a part being expended in supporting or drawing the weight,
and a part in diminishing or increasing the pressure upon the plane.
Let W P, _fig. 130._, be the power. This will be equivalent
to two forces, W F′, perpendicular to the plane, and W E′
in the direction of the plane. (74.) In order that the power should
sustain the weight, it is necessary that that part W E′ of the
power which acts in the direction of the plane should be equal to that
part W E, _fig. 130._, of the weight which acts down the
plane. The other part W F′ of the power acting perpendicular to
the plane is immediately opposed to that part W F of the weight
which produces pressure. The pressure upon the plane will therefore
be diminished by the amount of W F′. The amount of the power
which will equilibrate with the weight may, in this case, be found
as follows. Take W E′ equal to W E, and draw E′ P
perpendicular to the plane, and meeting the direction of the power.
The proportion of the power to the weight will be that of W P to
W D. And the proportion of the pressure to the weight will be that
of the difference between W F and W F′ to W D. If the
amount of the power have a less proportion to the weight than W P
has to W D, it will not support the body on the plane, but will
allow it to descend. And if it have a greater proportion, it will draw
the weight up the plane towards A.
(289.) It sometimes happens that a weight upon one inclined plane is
raised or supported by another weight upon another inclined plane.
Thus, if A B and A B′, _fig. 131._, be two inclined
planes forming an angle at A, and W W′ be two weights placed
upon these planes, and connected by a cord passing over a pulley at
A, the one weight will either sustain the other, or one will descend,
drawing the other up. To determine the circumstances under which these
effects will ensue, draw the lines W D and W′ D′ in the
vertical direction, and take upon them as many inches as there are
ounces in the weights respectively. W D and W′ D′ being the
lengths thus taken, and therefore representing the weights, the lines
W E and W′ E′ will represent the effects of these weights
respectively down the planes. If W E and W′ E′ be equal, the
weights will sustain each other without motion. But if W E be
greater than W′ E′, the weight W will descend, drawing the weight
W′ up. And if W′ E′ be greater than W E, the weight W′ will
descend, drawing the weight W up. In every case the lines W F and
W′ F′ will represent the pressures upon the planes respectively.
It is not necessary, for the effect just described, that the inclined
planes should, as represented in the figure, form an angle with each
other. They may be parallel, or in any other position, the rope being
carried over a sufficient number of wheels placed so as to give it the
necessary deflection. This method of moving loads is frequently applied
in great public works where rail-roads are used. Loaded waggons descend
one inclined plane, while other waggons, either empty or so loaded as
to permit the descent of those with which they are connected, are drawn
up the other.
(290.) In the application of the inclined plane which we have hitherto
noticed, the machine itself is supposed to be fixed in its position,
while the weight or load is moved upon it. But it frequently happens
that resistances are to be overcome which do not admit of being thus
moved. In such cases, instead of moving the load upon the planes,
the plane is to be moved under or against the load. Let D E,
_fig. 132._, be a heavy beam secured in a vertical position
between guides F G and H I, so that it is free to move
upwards and downwards, but not laterally. Let A B C be an
inclined plane, the extremity of which is placed beneath the end of
the beam. A force applied to the back of this plane A C, in the
direction C B, will urge the plane under the beam so as to raise
the beam to the position represented in _fig. 133._ Thus, while
the inclined plane is moved through the distance C B, the beam is
raised through the height C A.
(291.) When the inclined plane is applied in this manner, it is called
a _wedge_. And if the power applied to the back were a continued
pressure, its proportion to the weight would be that of A C to
C B. It follows, therefore, that the more acute the angle B is,
the more powerful will be the wedge.
In some cases, the wedge is formed of two inclined planes, placed base
to base, as represented in _fig. 134._ The theoretical estimation
of the power of this machine is not applicable in practice with any
degree of accuracy. This is in part owing to the enormous proportion
which the friction in most cases bears to the theoretical value of
the power, but still more to the nature of the power generally used.
The force of a blow is of a nature so wholly different from continued
forces, such as the pressure of weights, or the resistance offered by
the cohesion of bodies, that it admits of no numerical comparison with
them. Hence we cannot properly state the proportion which the force
of a blow bears to the amount of a weight or resistance. The wedge is
almost invariably urged by percussion; while the resistances which it
has to overcome are as constantly forces of the other kind. Although,
however, no exact numerical comparison can be made, yet it may be
stated in a general way that the wedge is more and more powerful as its
angle is more acute.
[Illustration: _C. Varley, del._ _H. Adlard, sc._
_London, Pubd. by Longman & Co._]
In the arts and manufactures, wedges are used where enormous force
is to be exerted through a very small space. Thus it is resorted to
for splitting masses of timber or stone. Ships are raised in docks
by wedges driven under their keels. The wedge is the principal agent
in the oil-mill. The seeds from which the oil is to be extracted are
introduced into hair bags, and placed between planes of hard wood.
Wedges inserted between the bags are driven by allowing heavy beams to
fall on them. The pressure thus excited is so intense, that the seeds
in the bags are formed into a mass nearly as solid as wood. Instances
have occurred in which the wedge has been used to restore a tottering
edifice to its perpendicular position.
All cutting and piercing instruments, such as knives, razors, scissors,
chisels, &c., nails, pins, needles, awls, &c. are wedges. The angle
of the wedge, in these cases, is more or less acute, according to
the purpose to which it is to be applied. In determining this, two
things are to be considered--the mechanical power, which is increased
by diminishing the angle of the wedge; and the strength of the tool,
which is always diminished by the same cause. There is, therefore,
a practical limit to the increase of the power, and that degree of
sharpness only is to be given to the tool which is consistent with the
strength requisite for the purpose to which it is to be applied. In
tools intended for cutting wood, the angle is generally about 30°. For
iron it is from 50° to 60°; and for brass, from 80° to 90°. Tools which
act by pressure may be made more acute than those which are driven by a
blow; and in general the softer and more yielding the substance to be
divided is, and the less the power required to act upon it, the more
acute the wedge may be constructed.
In many cases the utility of the wedge depends on that which is
entirely omitted in its theory, viz. the friction which arises between
its surface and the substance which it divides. This is the case when
pins, bolts, or nails are used for binding the parts of structures
together; in which case, were it not for the friction, they would
recoil from their places, and fail to produce the desired effect. Even
when the wedge is used as a mechanical engine, the presence of friction
is absolutely indispensable to its practical utility. The power, as
has already been stated, generally acts by successive blows, and is
therefore subject to constant intermission, and but for the friction
the wedge would recoil between the intervals of the blows with as much
force as it had been driven forward. Thus the object of the labour
would be continually frustrated. The friction in this case is of the
same use as a ratchet wheel, but is much more necessary, as the power
applied to the wedge is more liable to intermission than in the cases
where ratchet wheels are generally used.
(292.) When a road directly ascends the side of a hill, it is to be
considered as an inclined plane; but it will not lose its mechanical
character, if, instead of directly ascending towards the top of the
hill, it winds successively round it, and gradually ascends so as after
several revolutions to reach the top. In the same manner a path may be
conceived to surround a pillar by which the ascent may be facilitated
upon the principle of the inclined plane. Winding stairs constructed in
the interior of great columns partake of this character; for although
the ascent be produced by successive steps, yet if a floor could
be made sufficiently rough to prevent the feet from slipping, the
ascent would be accomplished with equal facility. In such a case the
winding path would be equivalent to an inclined plane, bent into such
a form as to accommodate it to the peculiar circumstances in which it
would be required to be used. It will not be difficult to trace the
resemblance between such an adaptation of the inclined plane and the
appearances presented by the thread of a _screw_: and it may hence be
easily understood that a screw is nothing more than an inclined plane
constructed upon the surface of a cylinder.
This will, perhaps, be more apparent by the following contrivance:
Let A B, _fig. 135._, be a common round ruler, and let
C D E be a piece of white paper cut in the form of an
inclined plane, whose height C D is equal to the length of the
ruler A B, and let the edge C E of the paper be marked with
a broad black line: let the edge C D be applied to the ruler
A B, and being attached thereto, let the paper be rolled round
the ruler; the ruler will then present the appearance of a screw,
_fig. 136._ the thread of the screw being marked by the black
line C E, winding continually round the ruler. Let D F,
_fig. 135._, be equal to the circumference of the ruler, and draw
F G parallel to D C, and G H parallel to D E, the
part C G F D of the paper will exactly surround the
ruler once: the part C G will form one convolution of the thread,
and may be considered as the length of one inclined plane surrounding
the cylinder, C H being the corresponding height, and G H
the base. The power of the screw does not, as in the ordinary cases of
the inclined plane, act parallel to the plane or thread, but at right
angles to the length of the cylinder A B, or, what is to the same
effect, parallel to the base H G; therefore the proportion of the
power to the weight will be, according to principles already explained,
the same as that of C H to the space through which the power
moves parallel to H G in one revolution of the screw. H C is
evidently the distance between the successive positions of the thread
as it winds round the cylinder; and it appears from what has been just
stated, that the less this distance is, or, in other words, the finer
the thread is, the more powerful the machine will be.
(293.) In the application of the screw the weight or resistance is
not, as in the inclined plane and wedge, placed upon the surface of
the plane or thread. The power is usually transmitted by causing the
screw to move in a concave cylinder, on the interior surface of which
a spiral cavity is cut, corresponding exactly to the thread of the
screw, and in which the thread will move by turning round the screw
continually in the same direction. This hollow cylinder is usually
called the _nut_ or _concave screw_. The screw surrounded by its
spiral thread is represented in _fig. 137._; and a section of the
same playing in the nut is represented in _fig. 138._
There are several ways in which the effect of the power may be conveyed
to the resistance by this apparatus.
First, let us suppose that the nut A B is fixed. If the screw be
continually turned on its axis, by a lever E F inserted in one
end of it, it will be moved in the direction C D, advancing every
revolution through a space equal to the distance between two contiguous
threads. By turning the lever in an opposite direction, the screw will
be moved in the direction D C.
If the screw be fixed, so as to be incapable either of moving
longitudinally or revolving on its axis, the nut A B may be turned
upon the screw by a lever, and will move on the screw towards C or
towards D, according to the direction in which the lever is turned.
In the former case we have supposed the nut to be absolutely
immoveable, and in the latter case the screw to be absolutely
immoveable. It may happen, however, that the nut, though capable of
revolving, is incapable of moving longitudinally; and that the screw,
though incapable of revolving, is capable of moving longitudinally. In
that case, by turning the nut A B upon the screw by the lever, the
screw will be urged in the direction C D or D C, according to
the way in which the nut is turned.
The apparatus may, on the contrary, be so arranged, that the nut,
though incapable of revolving, is capable of moving longitudinally;
and the screw, though capable of revolving, is incapable of moving
longitudinally. In this case, by turning the screw in the one direction
or in the other, the nut A B will be urged in the direction
C D or D C.
All these various arrangements may be observed in different
applications to the machine.
(294.) A screw may be cut upon a cylinder by placing the cylinder in
a turning lathe, and giving it a rotatory motion upon its axis. The
cutting point is then presented to the cylinder, and moved in the
direction of its length, at such a rate as to be carried through the
distance between the intended thread, while the cylinder revolves
once. The relative motions of the cutting point and the cylinder being
preserved with perfect uniformity, the thread will be cut from one end
to the other. The shape of the threads may be either square, as in
_fig. 137._, or triangular, as in _fig. 139._
(295.) The screw is generally used in cases where severe pressure
is to be excited through small spaces; it is therefore the agent in
most presses. In _fig. 140._, the nut is fixed, and by turning
the lever, which passes through the head of the screw, a pressure is
excited upon any substance placed upon the plate immediately under
the end of the screw. In _fig. 141._, the screw is incapable of
revolving, but is capable of advancing in the direction of its length.
On the other hand, the nut is capable of revolving, but does not
advance in the direction of the screw. When the nut is turned by means
of the screw inserted in it, the screw advances in the direction of its
length, and urges the board which is attached to it upwards, so as to
press any substance placed between it and the fixed board above.
In cases where liquids or juices are to be expressed from solid bodies,
the screw is the agent generally employed. It is also used in coining,
where the impression of a die is to be made upon a piece of metal, and
in the same way in producing the impression of a seal upon wax or other
substance adapted to receive it. When soft and light materials, such
as cotton, are to be reduced to a convenient bulk for transportation,
the screw is used to compress them, and they are thus reduced into hard
dense masses. In printing, the paper is urged by a severe and sudden
pressure upon the types, by means of a screw.
(296.) As the mechanical power of the screw depends upon the relative
magnitude of the circumference through which the power revolves, and
the distance between the threads, it is evident, that, to increase
the efficacy of the machine, we must either increase the length
of the lever by which the power acts, or diminish the magnitude of
the thread. Although there is no limit in theory to the increase of
the mechanical efficacy by these means, yet practical inconvenience
arises which effectually prevents that increase being carried beyond
a certain extent. If the lever by which the power acts be increased,
the same difficulty arises as was already explained in the wheel and
axle (254.); the space through which the power should act would be so
unwieldy, that its application would become impracticable. If, on the
other hand, the power of the machine be increased by diminishing the
size of the thread, the strength of the thread will be so diminished,
that a slight resistance will tear it from the cylinder. The cases
in which it is necessary to increase the power of the machine, being
those in which the greatest resistances are to be overcome, the object
will evidently be defeated, if the means chosen to increase that power
deprive the machine of the strength which is necessary to sustain the
force to which it is to be submitted.
(297.) These inconveniences are removed by a contrivance of Mr. Hunter,
which, while it gives to the machine all the requisite strength and
compactness, allows it to have an almost unlimited degree of mechanical
efficacy.
This contrivance consists in the use of two screws, the threads of
which may have any strength and magnitude, but which have a very small
difference of breadth. While the working point is urged forward by
that which has the greater thread, it is drawn back by that which has
the less; so that during each revolution of the screw, instead of
being advanced through a space equal to the magnitude of either of
the threads, it moves through a space equal to their difference. The
mechanical power of such a machine will be the same as that of a single
screw having a thread, whose magnitude is equal to the difference of
the magnitudes of the two threads just mentioned.
Thus, without inconveniently increasing the sweep of the power, on the
one hand, or, on the other, diminishing the thread until the necessary
strength is lost, the machine will acquire an efficacy limited by
nothing but the smallness of the difference between the two threads.
This principle was first applied in the manner represented in
_fig. 142._ A is the greater thread, playing in the fixed nut; B
is the lesser thread, cut upon a smaller cylinder, and playing in a
concave screw, cut within the greater cylinder. During every revolution
of the screw, the cylinder A descends through a space equal to the
distance between its threads. At the same time the smaller cylinder
B ascends through a space equal to the distance between the threads
cut upon it: the effect is, that the board D descends through a space
equal to the difference between the threads upon A and the threads upon
B, and the machine has a power proportionate to the smallness of this
difference.