the suspending thread. The rudest experiments will demonstrate the
fact, that every increase in the length of this thread will produce a
corresponding increase in the time of vibration; but according to what
law does this increase proceed? If the length of the thread be doubled
or trebled, will the time of vibration also be increased in a double
or treble proportion? This problem is capable of exact mathematical
solution, and the result shows that the time of vibration increases not
in the proportion of the increased length of the thread, but as the
square root of that length; that is to say, if the length of the thread
be increased in a four-fold proportion, the time of vibration will be
augmented in a two-fold proportion. If the thread be increased to nine
times its length, the time of vibration will be trebled, and so on.
This relation is exactly the same as that which was proved to subsist
between the spaces through which a body falls freely, and the times
of fall. In the table, page 89, if the figures representing the
height be understood to express the length of different pendulums, the
figures immediately above them will express the corresponding times of
vibration.
This law of the proportion of the lengths of pendulums to the squares
of the time of vibration may be experimentally established in the
following manner:--
Let A, B, C, _fig. 74._, be three small pieces of metal each
attached by threads to two points of suspension, and let them be placed
in the same vertical line under the point O; suppose them so adjusted
that the distances O A, O B, and O C shall be in the
proportion of the numbers 1, 4, and 9. Let them be removed from the
vertical in a direction at right angles to the plane of the paper, so
that the threads shall be in the same plane, and therefore the three
pendulums will have the same angle of vibration. Being now liberated,
the pendulum A will immediately gain upon B, and B upon C, so that A
will have completed one vibration before B or C. At the end of the
second vibration of A, the pendulum B will have arrived at the end of
its first vibration, so that the suspending threads of A and B will
then be separated by the whole angle of vibration; at the end of the
fourth vibration of A the suspending threads of A and B will return
to their first position, B having completed two vibrations; thus the
proportion of the times of vibration of B and A will be 2 to 1, the
proportion of their lengths being 4 to 1. At the end of the third
vibration of A, C will have completed one vibration, and the suspending
strings will coincide in the position distant by the whole angle of
vibration from their first position. So that three vibrations of A are
performed in the same time as one of C: the proportion of the time of
vibration of C and A are, therefore, 3 to 1, the proportion of their
lengths being 9 to 1, conformably to the law already explained.
(212.) In all the preceding observations we have assumed that the
material of the pendulous body is of inconsiderable magnitude, its
whole weight being conceived to be collected in a physical point.
This is generally called a simple pendulum; but since the conditions
of a suspending thread without weight, and a heavy molecule without
magnitude, cannot have practical existence, the simple pendulum must
be considered as imaginary, and merely used to establish hypothetical
theorems, which, though inapplicable in practice, are nevertheless the
means of investigating the laws which govern the real phenomena of
pendulous bodies.
A pendulous body being of determinate magnitude, its several parts
will be situated at different distances from the axis of suspension.
If each component part of such a body were separately connected with
the axis of suspension by a fine thread, it would, being unconnected
with the other particles, be an independent simple pendulum, and
would oscillate according to the laws already explained. It therefore
follows that those particles of the body which are nearest to the
axis of suspension would, if liberated from their connection with the
others, vibrate more rapidly than those which are more remote. The
connection, however, which the particles of the body have, by reason
of their solidity, compels them all to vibrate in the same time.
Consequently, those particles which are nearer the axis are retarded
by the slower motion of those which are more remote; while the more
remote particles, on the other hand, are urged forward by the greater
tendency of the nearer particles to rapid vibration. This will be more
readily comprehended, if we conceive two particles of matter A and B,
_fig. 75._, to be connected with the same axis O by an inflexible
wire O C, the weight of which may be neglected. If B were removed,
A would vibrate in a certain time depending upon the distance O A.
If A were removed, and B placed upon the wire at a distance B O
equal to four times A O, B would vibrate in twice the former time.
Now if both be placed on the wire at the distances just mentioned,
the tendency of A to vibrate more rapidly will be transmitted to B
by means of the wire, and will urge B forward more quickly than if
A were not present: on the other hand, the tendency of B to vibrate
more slowly will be transmitted by the wire to A, and will cause it to
move more slowly than if B were not present. The inflexible quality
of the connecting wire will in this case compel A and B to vibrate
simultaneously, the time of vibration being greater than that of A, and
less than that of B, if each vibrated unconnected with the other.
If, instead of supposing two particles of matter placed on the wire,
a greater number were supposed to be placed at various distances from
O, it is evident the same reasoning would be applicable. They would
mutually affect each other’s motion; those placed nearest to point
O accelerating the motion of those more remote, and being themselves
retarded by the latter. Among these particles one would be found
in which all these effects would be mutually neutralised, all the
particles nearer O being retarded in reference to that motion which
they would have if unconnected with the rest, and those more remote
being in the same respect accelerated. The point at which such a
particle is placed is called _the centre of oscillation_.
What has been here observed of the effects of particles of matter
placed upon rigid wire will be equally applicable to the particles of
a solid body. Those which are nearer to the axis are urged forward by
those which are more remote, and are in their turn retarded by them;
and as with the particles placed upon the wire, there is a certain
particle of the body at which the effects are mutually neutralised, and
which vibrates in the same time as it would if it were unconnected with
the other parts of the body, and simply connected by a fine thread to
the axis. By this centre of oscillation the calculations respecting the
vibration of a solid body are rendered as simple as those of a molecule
of inconsiderable magnitude. All the properties which have been
explained as belonging to a simple pendulum may thus be transferred
to a vibrating body of any magnitude and figure, by considering it as
equivalent to a single particle of matter vibrating at its centre of
oscillation.
(213.) It follows from this reasoning, that the virtual length of
a pendulum is to be estimated by the distance of its centre of
oscillation from the axis of suspension, and therefore that the times
of vibration of different pendulums are in the same proportion as the
square roots of the distances of their centres of oscillation from
their axes.
The investigation of the position of the centre of oscillation is, in
most cases, a subject of intricate mathematical calculation. It depends
on the magnitude and figure of the pendulous body, the manner in which
the mass is distributed through its volume, or the density of its
several parts, and the position of the axis on which it swings.
The place of the centre of oscillation may be determined when the
position of the centre of gravity and the centre of gyration are known;
for the distance of the centre of oscillation from the axis will always
be obtained by dividing the square of the radius of gyration (186.)
by the distance of the centre of gravity from the axis. Thus if 6 be
the radius of gyration, and 9 the distance of gravity from the axis,
36 divided by 9, which is 4, will be the distance of the centre of
oscillation from the axis. Hence it may be inferred generally, that
the greater the proportion which the radius of gyration bears to the
distance of the centre of gravity from the axis, the greater will be
the distance of the centre of oscillation.
It follows from this reasoning, that the length of a pendulum is not
limited by the dimensions of its volume. If the axis be so placed
that the centre of gravity is near it, and the centre of gyration
comparatively removed from it, the centre of oscillation may be placed
far beyond the limits of the pendulous body. Suppose the centre of
gravity is at a distance of one inch from the axis, and the centre
of gyration 12 inches, the centre of oscillation will then be at the
distance of 144 inches, or 12 feet. Such a pendulum may not in its
greatest dimensions exceed one foot, and yet its time of vibration
would be equal to that of a simple pendulum whose length is 12 feet.
By these means pendulums of small dimensions may be made to vibrate as
slowly as may be desired. The instruments called _metronomes_, used
for marking the time of musical performances, are constructed on this
principle.
(214.) The centre of oscillation is distinguished by a very remarkable
property in relation to the axis of suspension. If A, _fig. 76._,
be the point of suspension, and O the corresponding centre of
oscillation, the time of vibration of the pendulum will not be
changed if it be raised from its support, inverted, and suspended from
the point O. It follows, therefore, that if O be taken as the point
of suspension, A will be the corresponding centre of oscillation.
These two points are, therefore, convertible. This property may be
verified experimentally in the following manner. A pendulum being put
into a state of vibration, let a small heavy body be suspended by
a fine thread, the length of which is so adjusted that it vibrates
simultaneously with the pendulum. Let the distance from the point of
suspension to the centre of the vibrating body be measured, and take
this distance on the pendulum from the axis of suspension downwards;
the place of the centre of oscillation will thus be obtained, since
the distance so measured from the axis is the length of the equivalent
simple pendulum. If the pendulum be now raised from its support,
inverted, and suspended from the centre of oscillation thus obtained,
it will be found to vibrate simultaneously with the body suspended by
the thread.
(215.) This property of the interchangeable nature of the centres
of oscillation and suspension has been, at a late period, adopted
by Captain Kater, as an accurate means of determining the length of
a pendulum. Having ascertained with great accuracy two points of
suspension at which the same body will vibrate in the same time, the
distance between these points being accurately measured, is the length
of the equivalent simple pendulum. See Chapter XXI.
(216.) The manner in which the time of vibration of a pendulum
depends on its length being explained, we are next to consider how
this time is affected by the attraction of gravity. It is obvious
that, since the pendulum is moved by this attraction, the rapidity
of its motion will be increased, if the impelling force receive any
augmentation; but it still is to be decided, in what exact proportion
the time of oscillation will be diminished by any proposed increase
in the intensity of the earth’s attraction. It can be demonstrated
mathematically, that the time of one vibration of a pendulum has the
same proportion to the time of falling freely in the perpendicular
direction, through a height equal to half the length of the pendulum,
as the circumference of a circle has to its diameter. Since, therefore,
the times of vibration of pendulums are in a fixed proportion to the
times of falling freely through spaces equal to the halves of their
lengths, it follows that these times have the same relation to the
force of attraction as the times of falling freely through their
lengths have to that force. If the intensity of the force of gravity
were increased in a four-fold proportion, the time of falling through
a given height would be diminished in a two-fold proportion; if the
intensity were increased to a nine-fold proportion, the time of falling
through a given space would be diminished in a three-fold proportion,
and so on; the rate of diminution of the time being always as the
square root of the increased force. By what has been just stated this
law will also be applicable to the vibration of pendulums. Any increase
in the intensity of the force of gravity would cause a given pendulum
to vibrate more rapidly, and the increased rapidity of the vibration
would be in the same proportion as the square root of the increased
intensity of the force of gravity.
(217.) The laws which regulate the times of vibration of pendulums in
relation to one another being well understood, the whole theory of
these instruments will be completed, when the method of ascertaining
the actual time of vibration of any pendulum, in reference to its
length, has been explained. In such an investigation, the two elements
to be determined are, 1. the exact time of a single vibration, and,
2. the exact distance of the centre of oscillation from the point of
suspension.
The former is ascertained by putting a pendulum in motion in the
presence of a good chronometer, and observing precisely the number of
oscillations which are made in any proposed number of hours. The entire
time during which the pendulum swings, being divided by the number of
oscillations made during that time, the exact time of one oscillation
will be obtained.
The distance of the centre of oscillation from the point of suspension
may be rendered a matter of easy calculation, by giving a certain
uniform figure and material to the pendulous body.
(218.) The time of vibration of one pendulum of known length being
thus obtained, we shall be enabled immediately to solve either of the
following problems.
“To find the length of a pendulum which shall vibrate in a given time.”
“To find the time of vibration of a pendulum of a given length.”
The former is solved as follows: the time of vibration of the known
pendulum is to the time of vibration of the required pendulum, as the
square root of the length of the known pendulum is to the square root
of the length of the required pendulum. This length is therefore found
by the ordinary rules of arithmetic.
The latter may be solved as follows: the length of the known pendulum
is to the length of the proposed pendulum, as the square of the time
of vibration of the known pendulum is to the square of the time of
vibration of the proposed pendulum. The latter time may therefore be
found by arithmetic.
(219.) Since the rate of a pendulum has a known relation to the
intensity of the earth’s attraction, we are enabled, by this
instrument, not only to detect certain variations in that attraction in
various parts of the earth, but also to discover the actual amount of
the attraction at any given place.
The actual amount of the earth’s attraction at any given place is
estimated by the height through which a body would fall freely at that
place in any given time, as in one second. To determine this, let the
length of a pendulum which would vibrate in one second at that place
be found. As the circumference of a circle is to its diameter[2] (a
known proportion), so will one second be to the time of falling through
a height equal to half the length of this pendulum. This time is
therefore a matter of arithmetical calculation. It has been proved in
(120.), that the heights, through which a body falls freely, are in the
same proportion as the squares of the times; from whence it follows,
that the square of the time of falling through a height equal to half
the length of the pendulum is to one second as half the length of
that pendulum is to the height through which a body would fall in one
second. This height, therefore, may be immediately computed, and thus
the actual amount of the force of gravity at any given place may be
ascertained.
[2] This ratio is that of 31,416 to 10,000 very nearly.
(220.) To compare the force of gravity in different parts of the earth,
it is only necessary to swing the same pendulum in the places under
consideration, and to observe the rapidity of its vibrations. The
proportion of the force of gravity in the several places will be that
of the squares of the velocity of the vibration. Observations to this
effect have been made at several places, by Biot, Kater, Sabine, and
others.
The earth being a mass of matter of a form nearly spherical, revolving
with considerable velocity on an axis, its component parts are affected
by a centrifugal force; in virtue of which, they have a tendency to fly
off in a direction perpendicular to the axis. This tendency increases
in the same proportion as the distance of any part from the axis
increases, and consequently those parts of the earth which are near the
equator, are more strongly affected by this influence than those near
the pole. It has been already explained (145.) that the figure of the
earth is affected by this cause, and that it has acquired a spheroidal
form. The centrifugal force, acting in opposition to the earth’s
attraction, diminishes its effects; and consequently, where this force
is more efficient, a pendulum will vibrate more slowly. By these means
the rate of vibration of a pendulum becomes an indication of the amount
of the centrifugal force. But this latter varies in proportion to the
distance of the place from the earth’s axis; and thus the rate of a
pendulum indicates the relation of the distances of different parts of
the earth’s surface from its axis. The figure of the earth may be thus
ascertained, and that which theory assigns to it, it may be practically
proved to have.
This, however, is not the only method by which the figure of the earth
may be determined. The meridians being sections of the earth through
its axis, if their figure were exactly determined, that of the earth
would be known. Measurements of arcs of meridians on a large scale have
been executed, and are still being made in various parts of the earth,
with a view to determine the curvature of a meridian at different
latitudes. This method is independent of every hypothesis concerning
the density and internal structure of the earth, and is considered by
some to be susceptible of more accuracy than that which depends on the
observations of pendulums.
(221.) It has been stated that, when the arc of vibration of a pendulum
is not very small, a variation in its length will produce a sensible
effect on the time of vibration. To construct a pendulum such that the
time of vibration may be independent of the extent of the swing, was a
favourite speculation of geometers. This problem was solved by Huygens,
who showed that the curve called a _cycloid_, previously discovered and
described by Galileo, possessed the isochronal property; that is, that
a body moving in it by the force of gravity, would vibrate in the same
time, whatever be the length of the arc described.
Let O A, _fig. 77._, be a horizontal line, and let O B
be a circle placed below this line, and in contact with it. If this
circle be rolled upon the line from O towards A, a point upon its
circumference, which at the beginning of the motion is placed at O,
will during the motion trace the curve O C A. This curve is
called a _cycloid_. If the circle be supposed to roll in the opposite
direction towards A′, the same point will trace another cycloid
O C′ A′. The points C and C′ being the lowest points of the
curves, if the perpendiculars C D and C′ D′ be drawn, they
will respectively be equal to the diameter of the circle. By a known
property of this curve, the arcs O C and O C′ are equal to
twice the diameter of the circle. From the point O suppose a flexible
thread to be suspended, whose length is twice the diameter of the
circle, and which sustains a pendulous body P at its extremity. If
the curves O C and O C′, from the plane of the paper, be
raised so as to form surfaces to which the thread may be applied, the
extremity P will extend to the points C and C′, when the entire thread
has been applied to either of the curves. As the thread is deflected
on either side of its vertical position, it is applied to a greater
or lesser portion of either curve, according to the quantity of its
deflection from the vertical. If it be deflected on each side until
the point P reaches the points C and C′, the extremity would trace a
cycloid C P C′ precisely equal and similar to those already
mentioned. Availing himself of this property of the curve, Huygens
constructed his cycloidal pendulum. The time of vibration was subject
to no variation, however the arc of vibration might change, provided
only that the length of the string O P continued the same. If
small arcs of the cycloid be taken on either side of the point P, they
will not sensibly differ from arcs of a circle described with the
centre O and the radius O P; for, in slight deflections from the
vertical position, the effect of the curves O C and O C′ on
the thread O P is altogether inconsiderable. It is for this reason
that when the arcs of vibration of a circular pendulum are small, they
partake of the property of isochronism peculiar to those of a cycloid.
But when the deflection of P from the vertical is great, the effect of
the curves O C and O C′ on the thread produces a considerable
deviation of the point P from the arc of the circle whose centre is
O and whose radius is O P, and consequently the property of
isochronism will no longer be observed in the circular pendulum.
CHAP. XII.
OF SIMPLE MACHINES.
(222.) A MACHINE is an instrument by which force or motion may be
transmitted and modified as to its quantity and direction. There are
two ways in which a machine may be applied, and which give rise to
a division of mechanical science into parts denominated STATICS and
DYNAMICS; the one including the theory of equilibrium, and the other
the theory of motion. When a machine is considered statically, it is
viewed as an instrument by which forces of determinate quantities
and direction are made to balance other forces of other quantities
and other directions. If it be viewed dynamically, it is considered
as a means by which certain motions of determinate quantity and
direction may be made to produce other motions in other directions
and quantities. It will not be convenient, however, in the present
treatise, to follow this division of the subject. We shall, on the
other hand, as hitherto, consider the phenomena of equilibrium and
motion together.
The effects of machinery are too frequently described in such a manner
as to invest them with the appearance of paradox, and to excite
astonishment at what appears to contradict the results of the most
common experience. It will be our object here to take a different
course, and to attempt to show that those effects which have been held
up as matters of astonishment are the necessary, natural, and obvious
results of causes adapted to produce them in a manner analogous to the
objects of most familiar experience.
(223.) In the application of a machine there are three things to
be considered. 1. The force or resistance which is required to be
sustained, opposed, or overcome. 2. The force which is used to sustain,
support, or overcome that resistance. 3. The machine itself by which
the effect of this latter force is transmitted to the former. Of
whatever nature be the force or the resistance which is to be sustained
or overcome, it is technically called the _weight_, since, whatever it
be, a weight of equivalent effect may always be found. The force which
is employed to sustain or overcome it is technically called the _power_.