the properties just explained, the motion of rolling being as much
as possible substituted for that of sliding; and where the motion of
rolling cannot be applied, that of a wheel upon its axle is used. In
some cases both these motions are combined.
If a heavy load be drawn upon a plane in the manner of a sledge, the
motion will be that of sliding, the species which is attended with
the greatest quantity of friction; but if the load be placed upon
cylindrical rollers, the nature of the motion is changed, and becomes
that in which there is the least quantity of friction. Thus large
blocks of stone, or heavy beams of timber, which would require an
enormous power to move them on a level road, are easily advanced when
rollers are put under them.
When very heavy weights are to be moved through small spaces, this
method is used with advantage; but when loads ore to be transported to
considerable distances, the process is inconvenient and slow, owing to
the necessity of continually replacing the rollers in front of the load
as they are left behind by its progressive advancement.
The wheels of carriages may be regarded as rollers which are
continually carried forward with the load. In addition to the friction
of the rolling motion on the road, they have, it is true, the friction
of the axle in the nave; but, on the other hand, they are free from
the friction of the rollers with the under surface of the load, or
the carriage in which the load is transported. The advantages of
wheel carriages in diminishing the effects of friction is sometimes
attributed to the slowness with which that axle moves within the box,
compared with the rate at which the wheel moves over the road; but
this is erroneous. The quantity of friction does not in any case vary
considerably with the velocity of the motion, but least of all does it
in that particular kind of motion here considered.
In certain cases, where it is of great importance to remove the
effects of friction, a contrivance called _friction-wheels_, or
friction-rollers, is used. The axle of a friction-wheel, instead of
revolving within a hollow cylinder, which is fixed, rests upon the
edges of wheels which revolve with it; the species of motion thus
becomes that in which the friction is of least amount.
Let A B and D C, _fig. 179._, be two wheels revolving
on pivots P Q with as little friction as possible, and so placed
that the axle O of a third wheel E F may rest between their edges.
As the wheel E F revolves, the axle O, instead of grinding its
surface on the surface on which it presses, carries that surface with
it, causing the wheels A B, C D, to revolve.
In wheel carriages, the roughness of the road is more easily overcome
by large wheels than by small ones. The cause of this arises partly
from the large wheels not being so liable to sink into holes as small
ones, but more because, in surmounting obstacles, the load is elevated
less abruptly. This will be easily understood by observing the curves
in _fig. 180._, which represent the elevation of the axle in each
case.
(328.) If a carriage were capable of moving on a road without friction,
the most advantageous direction in which a force could be applied to
draw it would be parallel to the road. When the motion is impeded by
friction, it is better, however, that the line of draught should be
inclined to the road, so that the drawing force may be expended partly
in lessening the pressure on the road, and partly in advancing the load.
Let W, _fig. 181._, be a load which is to be moved upon the plane
surface A B. If the drawing force be applied in the direction
C D, parallel to the plane A B, it will have to overcome the
friction produced by the pressure of the whole weight of the load upon
the plane; but if it be inclined upwards in the direction C E,
it will be equivalent to two forces expressed (74.) by C G and
C F. The part C G has the effect of lightening the pressure
of the carriage upon the road, and therefore of diminishing the
friction in the same proportion. The part C F draws the load along
the plane. Since C F is less than C E or C D the whole
moving force, it is evident that a part of the force of draught is
lost by this obliquity; but, on the other hand, a part of the opposing
resistance is also removed. If the latter exceed the former, an
advantage will be gained by the obliquity; but if the former exceed the
latter, force will be lost.
By mathematical reasoning, founded on these considerations, it is
proved that the best angle of draught is exactly that obliquity which
should be given to the road in order to enable the carriage to move of
itself. This obliquity is sometimes called the _angle of repose_, and
is that angle which determines the proportion of the friction to the
pressure in the second method, explained in (323.). The more rough the
road is, the greater will this angle be; and therefore it follows, that
on bad roads the obliquity of the traces to the road should be greater
than on good ones. On a smooth Macadamised way, a very slight declivity
would cause a carriage to roll by its own weight: hence, in this case,
the traces should be nearly parallel to the road.
In rail roads, for like reasons, the line of draught should be parallel
to the road, or nearly so.
(329.) When ropes or cords form a part of machinery, the effects of
their imperfect flexibility are in a certain degree counteracted by
bending them over the grooves of wheels. But although this so far
diminishes these effects as to render ropes practically useful, yet
still, in calculating the powers of machinery, it is necessary to take
into account some consequences of the rigidity of cordage which even
by these means are not removed.
To explain the way in which the stiffness of a rope modifies the
operation of a machine, we shall suppose it bent over a wheel and
stretched by weights A B, _fig. 182._, at its extremities.
The weights A and B being equal, and acting at C and D in opposite
ways, balance the wheel. If the weight A receive an addition, it will
overcome the resistance of B, and turn the wheel in the direction
D E C. Now, for the present, let us suppose that the rope
is perfectly inflexible; the wheel and weights will be turned into
the position represented in _fig. 183._ The leverage by which
A acts will be diminished, and will become O F, having been
before O C; and the leverage by which B acts will be increased to
O G, having been before O D.
But the rope not being inflexible will yield partially to the effects
of the weights A and B, and the parts A C and B D will
be bent into the forms represented in _fig. 184._ The form of
the curvature which the rope on each side of the wheel receives is
still such that the descending weight A works with a diminished
leverage F O, while the ascending weight resists it with an
increased leverage G O. Thus so much of the moving power is lost,
by the stiffness of the rope, as is necessary to compensate this
disadvantageous change in the power of the machine.
CHAP. XX.
ON THE STRENGTH OF MATERIALS.
(330.) Experimental enquiries into the laws which regulate the strength
of solid bodies, or their power to resist forces variously applied
to tear or break them, are obstructed by practical difficulties, the
nature and extent of which are so discouraging that few have ventured
to encounter them at all, and still fewer have had the steadiness to
persevere until any result showing a general law has been obtained.
These difficulties arise, partly from the great forces which must be
applied, but more from the peculiar nature of the objects of those
experiments. The end to which such an enquiry must be directed is
the development of a _general law_; that is, such a rule as would be
rigidly observed if the materials, the strength of which is the object
of enquiry, were perfectly uniform in their texture, and subject to no
casual inequalities. In proportion as these inequalities are frequent,
experiments must be multiplied, that a long average may embrace cases
varying in both extremes, so as to eliminate each other’s effects in
the final result.
The materials of which structures and works of art are composed are
liable to so many and so considerable inequalities of texture, that
any rule which can be deduced, even by the most extensive series of
experiments, must be regarded as a mean result, from which individual
examples will be found to vary in so great a degree, that more than
usual caution must be observed in its practical application. The
details of this subject belong to engineering, more properly than
to the elements of mechanics. Nevertheless, a general view of the
most important principles which have been established respecting the
strength of materials will not be misplaced in this treatise.
A piece of solid matter may be submitted to the action of a force
tending to separate its parts in several ways; the principal of which
are,--
1. To a _direct pull_,--as when a rope or wire is stretched by a
weight. When a tie-beam resists the separation of the sides of a
structure, &c.
2. To a direct pressure or thrust,--as when a weight rests upon a
pillar.
3. To a transverse strain,--as when weights on the ends of a lever
press it on the fulcrum.
(331.) If a solid be submitted to a force which draws it in the
direction of its length, having a tendency to pull its ends in
opposite directions, its strength or power to resist such a force is
proportional to the magnitude of its transverse section. Thus, suppose
a square rod of metal A B, _fig. 185._, of the breadth and
thickness of one inch, be pulled by a force in the direction A B,
and that a certain force is found sufficient to tear it; a rod of the
same metal of twice the breadth and the same thickness will require
double the force to break it; one of treble the breadth and the same
thickness will require treble the force to break it, and so on.
The reason of this is evident. A rod of double or treble the thickness,
in this case, is equivalent to two or three equal and similar rods
which equally and separately resist the drawing force, and therefore
possess a degree of strength proportionate to their number.
It will easily be perceived, that whatever be the section, the same
reasoning will be applicable, and the power of resistance will, in
general, be proportional to its magnitude or area.
If the material were perfectly uniform throughout its dimensions, the
resistance to a direct pull would not be affected by the length of the
rod. In practice, however, the increase of length is found to lessen
the strength. This is to be attributed to the increased chance of
inequality.
(332.) No satisfactory results have been obtained either by theory or
experiment respecting the laws by which solids resist compression.
The power of a perpendicular pillar to support a weight placed upon
it evidently depends on its thickness, or the magnitude of its base,
and on its height. It is certain that when the height is the same,
the strength increases with every increase of the base, but it seems
doubtful whether the strength be exactly proportional to the base. That
is, if two columns of the same material have equal heights, and the
base of one be double the base of the other, the strength of one will
be greater, but it is not certain whether it will exactly double that
of the other. According to the theory of Euler, which is in a certain
degree verified by the experiments of Musschenbrock, the strength will
be increased in a greater proportion than the base, so that, if the
base be doubled, the strength will be more than doubled.
When the base is the same, the strength is diminished by increasing the
height, and this decrease of strength is proportionally greater than
the increase of height. According to Euler’s theory, the decrease of
strength is proportional to the square of the height; that is, when
the height is increased in a two-fold proportion, the strength is
diminished in a four-fold proportion.
(333.) The strain to which solids forming the parts of structures of
every kind are most commonly exposed is the lateral or transverse
strain, or that which acts at right angles to their lengths. If any
strain act obliquely to the direction of their length it may be
resolved into two forces (76.), one in the direction of the length, and
the other at right angles to the length. That part which acts in the
direction of the length will produce either compression or a direct
pull, and its effect must be investigated accordingly.
Although the results of theory, as well as those of experimental
investigations, present great discordances respecting the transverse
strength of solids, yet there are some particulars, in which they, for
the most part, agree; to this it is our object here to confine our
observations, declining all details relating to disputed points.
Let A B C D, _fig. 186._, be a beam, supported
at its ends A and B. Its strength to support a weight at E pressing
downwards at right angles to its length is evidently proportional to
its breadth, the other things being the same. For a beam of double or
treble breadth, and of the same thickness, is equivalent to two or
three equal and similar beams placed side by side. Since each of these
would possess the same strength, the whole taken together would possess
double or treble the strength of any one of them.
When the breadth and length are the same the strength obviously
increases with the depth, but not in the same proportion. The
increase of strength is found to be much greater in proportion than
the increase of depth. By the theory of Galileo, a double or treble
thickness ought to increase the strength in a four-fold or nine-fold
proportion, and experiments in most cases do not materially vary from
this rule.
If while the breadth and depth remain the same, the length of the
beam, or rather, the distance between the points of support, vary, the
strength will vary accordingly, decreasing in the same proportion as
the length increases.
From these observations it appears, that the transverse strength of
a beam depends more on its thickness than its breadth. Hence we find
that a broad thin board is much stronger when its edge is presented
upwards. On this principle the joists or rafters of floors and roofs
are constructed.
If two beams be in all respects similar, their strengths will be in the
proportion of the squares of their lengths. Let the length, breadth,
and depth of the one be respectively double the length, breadth,
and depth of the other. By the double breadth the beam doubles its
strength, but by doubling the length half this strength is lost. Thus
the increase of length and breadth counteract each other’s effects, and
as far as they are concerned the strength of the beam is not changed.
But by doubling the thickness the strength is increased in a four-fold
proportion, that is, as the square of the length. In the same manner it
may be shown, that when all the dimensions are trebled, the strength is
increased in a nine-fold proportion, and so on.
(334.) In all structures the materials have to support their own
weight, and therefore their available strength is to be estimated
by the excess of their absolute strength above that degree of
strength which is just sufficient to support their own weight. This
consideration leads to some conclusions, of which numerous and striking
illustrations are presented in the works of nature and art.
We have seen that the absolute strength with which a lateral strain is
resisted is in the proportion of the square of the linear dimensions of
similar parts of a structure, and therefore the amount of this strength
increases rapidly with every increase of the dimensions of a body. But
at the same time the weight of the body increases in a still more rapid
proportion. Thus, if the several dimensions be doubled, the strength
will be increased in a four-fold but the weight in an eight-fold
proportion. If the dimensions be trebled, the strength will be
multiplied nine times, but the weight twenty-seven times. Again, if the
dimensions be multiplied four times, the strength will be multiplied
sixteen times, and the weight sixty-four times, and so on.
Hence it is obvious, that although the strength of a body of small
dimensions may greatly exceed its weight, and, therefore, it may be
able to support a load many times its own weight; yet by a great
increase in the dimensions the weight increasing in a much greater
degree the available strength may be much diminished, and such a
magnitude may be assigned, that the weight of the body must exceed its
strength, and it not only would be unable to support any load, but
would actually fall to pieces by its own weight.
The strength of a structure of any kind is not, therefore, to be
determined by that of its model, which will always be much stronger in
proportion to its size. All works natural and artificial have limits
of magnitude which, while their materials remain the same, they cannot
surpass.
In conformity with what has just been explained, it has been observed,
that small animals are stronger in proportion than large ones; that the
young plant has more available strength in proportion than the large
forest tree; that children are less liable to injury from accident
than men, &c. But although to a certain extent these observations are
just, yet it ought not to be forgotten, that the mechanical conclusions
which they are brought to illustrate are founded on the supposition,
that the smaller and greater bodies which are compared are composed
of precisely similar materials. This is not the case in any of the
examples here adduced.
CHAP. XXI.
ON BALANCES AND PENDULUMS.
(335.) The preceding chapters have been confined almost wholly to
the consideration of the laws of mechanics, without entering into a
particular description of the machinery and instruments dependant upon
those laws. Such descriptions would have interfered too much with the
regular progress of the subject, and it therefore appeared preferable
to devote a chapter exclusively to this portion of the work.
Perhaps there are no ideas which man receives through the medium of
sense which may not be referred ultimately to matter and motion. In
proportion, therefore, as he becomes acquainted with the properties
of the one and the laws of the other, his knowledge is extended, his
comforts are multiplied; he is enabled to bend the powers of nature to
his will, and to construct machinery which effects with ease that which
the united labour of thousands would in vain be exerted to accomplish.
Of the properties of matter, one of the most important is its weight,
and the element which mingles inseparably with the laws of motion is
time.
In the present chapter it is our intention to describe such instruments
as are usually employed for determining the weight of bodies. To
attempt a description of the various machines which are used for the
measurement of time, would lead us into too wide a field for the
present occasion, and we shall, therefore, confine ourselves to an
account of the methods which have been practised to perfect, to perfect
that instrument which affords the most correct means of measuring time,
the pendulum.
The instrument by which we are enabled to determine, with greater
accuracy than by any other means, the relative weight of a body,
compared with the weight of another body assumed as a standard, is the
balance.
[Illustration: _H. Adlard, sc._
_London, Pubd. by Longman & Co._]
_Of the Balance._
The balance may be described as consisting of an inflexible rod or
lever, called the beam, furnished with three axes; one, the fulcrum or
centre of motion situated in the middle, upon which the beam turns,
and the other two near the extremities, and at equal distances from
the middle. These last are called the points of support, and serve to
sustain the pans or scales.
The points of support and the fulcrum are in the same right line, and
the centre of gravity of the whole should be a little below the fulcrum
when the position of the beam is horizontal.
The arms of the lever being equal, it follows that if equal weights be
put into the scales no effect will be produced on the position of the
balance, and the beam will remain horizontal.
If a small addition be made to the weight in one of the scales, the
horizontality of the beam will be disturbed; and after oscillating
for some time, it will, on attaining a state of rest, form an angle
with the horizon, the extent of which is a measure of the delicacy or
sensibility of the balance.
As the sensibility of a balance is of the utmost importance in
nice scientific enquiries, we shall enter somewhat at large into a
consideration of the circumstances by which this property is influenced.
In _fig. 187._ let A B represent the beam drawn from the
horizontal position by a very small weight placed in the scale
suspended from the point of support B; then the force tending to draw
the beam from the horizontal position may be expressed by P B,
multiplied by such very small weight acting upon the point B.
Let the centre of gravity of the whole be at G; then the force acting
against the former will be G P multiplied into the weight of the
beam and scales, and when these forces are equal, the beam will rest
in an inclined position. Hence we may perceive that as the centre of
gravity is nearer to or further from the fulcrum S, (every thing else
remaining the same) the sensibility of the balance will be increased or
diminished.
For, suppose the centre of gravity were removed to _g_, then to produce
an opposing force equal to that acting upon the extremity of the beam,
the distance _g p_ from the perpendicular line must be increased
until it becomes nearly equal to G P; but for this purpose
the end of the beam B must descend, which will increase the angle
H S B.
As all weights placed in the scales are referred to the line joining
the points of support, and as this line is above the centre of gravity
of the beam when not loaded, such weights will raise the centre of
gravity; but it will be seen that the sensibility of the balance, as
far as it depends upon this cause, will remain unaltered.
For, calling the distance S G unity, the distance of the centre of
gravity from the point S (to which the weight which has been added is
referred) will be expressed by the reciprocal of the weight of the beam
so increased; that is, if the weight of the beam be doubled by weights
placed in the scales, S _g_ will be one half of S G; and if the
weight of the beam be in like manner trebled, S _g_ will be one third
of S G, and so on. And as G P varies as S G, _g p_
will be inversely proportionate to the increased weight of the beam,
and consequently, the product obtained by multiplying _g p_ by the
weight of the beam and its load will be a constant quantity, and the
sensibility of the balance, as before stated, will suffer no alteration.
We will now suppose that the fulcrum S, _fig. 188._, is situated
below the line joining the points of support, and that the centre of
gravity of the beam when not loaded is at G. Also that when a very
small weight is placed in the scale suspended from the point B, the
beam is drawn from its horizontal position, the deviation being a
measure of the sensibility of the balance. Then, as before stated,
G P multiplied by the weight of the beam will be equal to
P′ B multiplied by the very small additional weight acting on the
point B.
Now if we place equal weights in both scales, such additional weights
will be referred to the point W, and the resulting distance of the
centre of gravity from the point W, calling W G unity, will be
expressed as before by the reciprocal of the increased weight of the
loaded beam. But G P will decrease in a greater proportion than
W G: thus, supposing the weight of the beam to be doubled, W _g_
would be one half of W G; but _g p_, as will be evident on
an inspection of the figure, will be less than half of G P; and
the same small weight which was before applied to the point B, if
now added, would depress the point B, until the distance _g p_
became such as that, when multiplied by the weight of the whole, the
product would be as before equal to P′ B, multiplied by the before
mentioned very small added weight. The sensibility of the balance,
therefore, in this case would be increased.