passing through the axis A B. Suppose the impulse to be applied at
P, perpendicular to this plane, and at the distance P O from the
axis. The effect of the impulse being distributed through the mass will
cause the body to revolve on A B, with a uniform velocity. There
is a certain point G, at which, if the whole mass were concentrated,
it would receive from the impulse the same velocity round the axis.
The distance O G is called the _radius of gyration_ of the axis
A B, and the point G is called the _centre of gyration_ relatively
to that axis. The effect of the impulse upon the mass concentrated at
G is great in exactly the same proportion as O G is small. This
easily follows from the property of moments which has been already
explained; from whence it may be inferred, that the greater the radius
of gyration is, the less will be the velocity which the body will
receive from a given impulse.
(187.) Since the radius of gyration depends on the manner in which the
mass is arranged round the axis, it follows that for different axes
in the same body there will be different radii of gyration. Of all
axes taken in the same body parallel to each other, that which passes
through the centre of gravity has the least radius of gyration. If the
radius of gyration of any axis passing through the centre of gravity be
given, that of any parallel axis can be found; for the square of the
radius of gyration of any axis is equal to the square of the distance
of that axis from the centre of gravity added to the square of the
radius of gyration of the parallel axis through the centre of gravity.
(188.) The product of the numerical expressions for the mass of the
body and the square of the radius of gyration is a quantity much used
in mechanical science, and has been called the _moment of inertia_. The
moments of inertia, therefore, for different axes in the same body are
proportional to the squares of the corresponding radii of gyration; and
consequently increase as the distances of the axes from the centre of
gravity increase. (187.)
(189.) From what has been explained in (187.), it follows, that the
moment of inertia of any axis may be computed by common arithmetic, if
the moment of inertia of a parallel axis through the centre of gravity
be previously known. To determine this last, however, would require
analytical processes altogether unsuitable to the nature and objects of
the present treatise.
The velocity of rotation which a body receives from a given impulse
is great in exactly the same proportion as the moment of inertia is
small. Thus the moment of inertia may be considered in rotatory motion
analogous to the mass of the body in rectilinear motion.
From what has been explained in (187.) it follows that a given impulse
at a given distance from the axis will communicate the greatest
angular velocity when the axis passes through the centre of gravity,
and that the velocity which it will communicate round other axes
will be diminished in the same proportion as the squares of their
distances from the centre of gravity added to the square of the radius
of gyration for a parallel axis through the centre of gravity are
augmented.
(190.) If any point whatever be assumed in a body, and right lines
be conceived to diverge in all directions from that point, there are
generally two of these lines, which being taken as axes of rotation,
one has a greater and the other a less moment of inertia than any of
the others. It is a remarkable circumstance, that, whatever be the
nature of the body, whatever be its shape, and whatever be the position
of the point assumed, these two axes of greatest and least moment will
always be at right angles to each other.
These axes and a third through the same point, and at right angles to
both of them, are called the _principal axes_ of that point from which
they diverge. To form a distinct notion of their relative position,
let the axis of greatest moment be imagined to lie horizontally from
north to south, and the axis of least moment from east to west; then
the third principal axis will be presented perpendicularly upwards and
downwards. The first two being called the principal axes of greatest
and least moment, the third may be called the _intermediate principal
axis_.
(191.) Although the moments of the three principal axes be in general
unequal, yet bodies may be found having certain axes for which these
moments may be equal. In some cases the moment of the intermediate axis
is equal to that of the principal axis of greatest moment: in others it
is equal to that of the principal axis of least moment, and in others
the moments of all the three principal axes are equal to each other.
If the moments of any two of three principal axes be equal, the moments
of all axes through the same point and in their plane will also be
equal; and if the moments of the three principal axes through a point
be equal, the moments of all axes whatever, through the same point,
will be equal.
(192.) If the moments of the principal axes through the centre of
gravity be known, the moments for all other axes through that point may
be easily computed. To effect this it is only necessary to multiply
the moments of the principal axes by the squares of the co-sines of
the angles formed by them respectively with the axis whose moment is
sought. The products being added together will give the required moment.
(193.) By combining this result with that of (189.), it will be evident
that the moment of all axes whatever may be determined, if those of the
principal axes through the centre of gravity be known.
(194.) It is obvious that the principal axis of least moment through
the centre of gravity has a less moment of inertia than any other axis
whatever. For it has, by its definition (190.) a less moment of inertia
than any other axis through the centre of gravity, and every other
axis through the centre of gravity has a less moment of inertia than a
parallel axis through any other point (187.) and (189.)
(195.) If two of the principal axes through the centre of gravity have
equal moments of inertia, all axes in any plane parallel to the plane
of these axes, and passing through the point where a perpendicular from
the centre of gravity meets that plane, must have equal moments of
inertia. For by (191.) all axes in the plane of those two have equal
moments, and by (189.) the axes in the parallel plane have moments
which exceed these by the same quantity, being equally distant from
them. (187.)
Hence it is obvious that if the three principal axes through the centre
of gravity have equal moments, all axes situated in any given plane,
and passing through the point where the perpendicular from the centre
of gravity meets that plane, will have equal moments, being equally
distant from parallel axes through the centre of gravity.
(196.) If the three principal axes through the centre of gravity have
unequal moments, there is no point whatever for which all axes will
have equal moments; but if the principal axis of least moment and
the intermediate principal axis through the centre of gravity have
equal moments, then there will be two points on the principal axis
of greatest moment, equally distant at opposite sides of the centre
of gravity, at which all axes will have equal moments. If the three
principal axes through the centre of gravity have equal moments, no
other point of the body can have principal axes of equal moment.
(197.) When a body revolves on a fixed axis, the parts of its mass are
whirled in circles round the axis; and since they move with a common
angular velocity, they will have centrifugal forces proportional to
their distances from the axis. If the component parts of the mass were
not united together by cohesive forces of energies greater than these
centrifugal forces, they would be separated, and would fly off from
the axis; but their cohesion prevents this, and causes the effects of
the different centrifugal forces, which affect the different parts of
the mass, to be transmitted so as to modify each other, and finally
to produce one or more forces mechanically equivalent to the whole,
and which are exerted upon the axis and resisted by it. We propose
now to explain these effects, as far as it is possible to render them
intelligible without the aid of mathematical language.
It is obvious that any number of equal parts of the mass, which are
uniformly arranged in a circle round the axis, have equal centrifugal
forces acting from the centre of the circle in every direction. These
mutually neutralise each other, and therefore exert no force on the
axis. The same may be said of all parts of the mass which are regularly
and equally distributed on every side of the axis.
Also if equal masses be placed at equal distances on opposite sides
of the axis, their centrifugal forces will destroy each other. Hence
it appears that the pressure which the axis of rotation sustains from
the centrifugal forces of the revolving mass, arises from the unequal
distribution of the matter around it.
From this reasoning it will be easily perceived that in the following
examples the axis of rotation will sustain no pressure.
A globe revolving on any of its diameters, the density being the same
at equal distances from the centre.
A spheroid or a cylinder revolving on its axis, the density being equal
at equal distances from the axis.
A cube revolving on an axis which passes through the centre of two
opposite bases, being of uniform density.
A circular plate of uniform thickness and density revolving on one of
its diameters as an axis.
(198.) In all these examples it will be observed that the axis of
rotation passes through the centre of gravity. The general theorem, of
which they are only particular instances, is, “if a body revolve on a
principal axis, passing through the centre of gravity, the axis will
sustain no pressure from the centrifugal force of the revolving mass.”
This is a property in which the principal axes through the centre
of gravity are unique. There is no other axis on which a body could
revolve without pressure.
If two of the principal axes through the centre of gravity have equal
moments, every axis in their plane has the same moment, and is to be
considered equally as a principal axis. In this case the body would
revolve on any of these axes without pressure.
A homogeneous spheroid furnishes an example of this. If any of the
diameters of the earth’s equator were a fixed axis, the earth would
revolve on it without producing pressure.
If the three principal axes through the centre of gravity have equal
moments, all axes through the centre of gravity are to be considered as
principal axes. In this case the body would revolve without pressure on
any axis through the centre of gravity.
A globe, in which the density of the mass at equal distances from the
centre is the same, is an example of this. Such a body would revolve
without pressure on any axis through its centre.
(199.) Since no pressure is excited on the axis in these cases, the
state of the body will not be changed, if during its rotation the axis
cease to be fixed. The body will notwithstanding continue to revolve
round the axis, and the axis will maintain its position.
Thus a spinning-top of homogeneous material and symmetrical form will
revolve steadily in the same position, until the friction of its point
with the surface on which it rests deprives it of motion. This is a
phenomenon which can only be exhibited when the axis of rotation is a
principal axis through the centre of gravity.
(200.) If the body revolve round any axis through the centre of
gravity, which is not a principal axis, the centrifugal pressure is
represented by two forces, which are equal and parallel, but which act
in opposite directions on different points of the axis. The effect of
these forces is to produce a strain upon the axis, and give the body a
tendency to move round another axis at right angles to the former.
(201.) If the fixed axis on which a body revolves be a principal axis
through any point different from the centre of gravity, then a pressure
will be produced by the centrifugal force of the revolving mass, and
this pressure will act at right angles to the axis on the point to
which it is a principal axis, and in the plane through that axis and
the centre of gravity. The amount of the pressure will be proportional
to the mass of the body, the distance of the centre of gravity from the
axis, and the square of the velocity of rotation.
(202.) Since the whole pressure is in this case excited on a single
point, the stability of the axis will not be disturbed, provided that
point alone be fixed. So that even though the axis should be free to
turn on that point, no motion will ensue as long as no external forces
act upon the body.
(203.) If the axis of rotation be not a principal axis, the centrifugal
forces will produce an effect which cannot be represented by a single
force. The effect may be understood by conceiving two forces to act
on _different points_ of the axis at right angles to it and to each
other. The quantities of these pressures and their directions depend
on the figure and density of the mass and the position of the axis,
in a manner which cannot be explained without the aid of mathematical
language and principles.
(204.) The effects upon the axis which have been now explained are
those which arise from the motion of rotation, from whatever cause that
motion may have arisen. The forces which produce that motion, however,
are attended with effects on the axis which still remain to be noticed.
When these forces, whether they be of the nature of instantaneous
actions or continued forces, are entirely resisted by the axis, their
directions must severally be in a plane passing through the axis, or
they must, by the principles of the composition of force [(74.) et
seq.], be mechanically equivalent to forces in that plane. In every
other case the impressed forces _must_ produce motion, and, except in
certain cases, must also produce effects upon the axis.
By the rules for the composition of force it is possible in all cases
to resolve the impressed forces into others which are either in planes
through the axis, or in planes perpendicular to it, or, finally, some
in planes through it, and others in planes perpendicular to it. The
effect of those which are in planes through the axis has been already
explained; and we shall now confine our attention to those impelling
forces which act at right angles to the axis, and which produce motion.
It will be sufficient to consider the effect of a single force at right
angles to the axis; for whatever be the number of forces which act
either simultaneously or successively, the effect of the whole will
be decided by combining their separate effects. The effect which a
single force produces depends on two circumstances, 1. The position of
the axis with respect to the figure and mass of the body, and 2. The
quantity and direction of the force itself.
In general the shock which the axis sustains from the impact may be
represented by two impacts applied to it at different points, one
parallel to the impressed force, and the other perpendicular to it,
but both perpendicular to the axis. There are certain circumstances,
however, under which this effect will be modified.
If the impulse which the body receives be in a direction perpendicular
to a plane through the axis and the centre of gravity, and at a
distance from the axis which bears to the radius of gyration (186.)
the same proportion as that line bears to the distance of the centre
of gravity from the axis, there are certain cases in which the impulse
will produce no percussion. To characterise these cases generally would
require analytical formulæ which cannot conveniently be translated
into ordinary language. That point of the plane, however, where the
direction of the impressed force meets it, when no percussion on the
axis is produced, is called the _centre of percussion_.
If the axis of rotation be a principal axis, the centre of percussion
must be in the right line drawn through the centre of gravity,
intersecting the axis at right angles, and at the distance from the
axis already explained.
If the axis of rotation be parallel to a principal axis through the
centre of gravity, the centre of percussion will be determined in the
same manner.
(205.) There are many positions which the axis may have in which there
will be no centre of percussion; that is, there will be no direction in
which an impulse could be applied without producing a shock upon the
axis. One of these positions is when it is a principal axis through
the centre of gravity. This is the only case of rotation round an axis
in which no effect arises from the centrifugal force; and therefore it
follows that the only case in which the axis sustains no effect from
the motion produced, is one in which it must necessarily suffer an
effect from that which produces the motion.
If the body be acted upon by continued forces, their effect is at each
instant determined by the general principles for the composition of
force.
CHAP. XI.
ON THE PENDULUM.
(206.) When a body is placed on a horizontal axis which does not
pass through its centre of gravity, it will remain in permanent
equilibrium only when the centre of gravity is immediately below the
axis. If this point be placed in any other situation, the body will
oscillate from side to side, until the atmospherical resistance and the
friction of the axis destroy its motion. (159, 160.) Such a body is
called a _pendulum_. The swinging motion which it receives is called
_oscillation_ or _vibration_.
(207.) The use of the pendulum, not only for philosophical purposes,
but in the ordinary economy of life, renders it a subject of
considerable importance. It furnishes the most exact means of measuring
time, and of determining with precision various natural phenomena. By
its means the variation of the force of gravity in different latitudes
is discovered, and the law of that variation experimentally exhibited.
In the present chapter, we propose to explain the general principles
which regulate the oscillation of pendulums. Minute details concerning
their construction will be given in the twenty-first chapter of this
volume.
(208.) A simple pendulum is composed of a heavy molecule attached
to the end of a flexible thread, and suspended by a fixed point O,
_fig. 73._ When the pendulum is placed in the position O C,
the molecule being vertically below the point of suspension, it will
remain in equilibrium; but if it be drawn into the position O A
and there liberated, it will descend towards C, moving through the arc
A C with accelerated motion. Having arrived at C and acquired
a certain velocity, it will, by reason of its inertia, continue to
move in the same direction. It will therefore commence to ascend the
arc C A′ with the velocity so acquired. During its ascent, the
weight of the molecule retards its motion in exactly the same manner
as it had accelerated it in descending from A to C; and when the
molecule has ascended through the arc C A′ equal to C A,
its entire velocity will be destroyed, and it will cease to move in
that direction. It will thus be placed at A′ in the same manner as in
the first instance it had been placed at A, and consequently it will
descend from A′ to C with accelerated motion, in the same manner as
it first moved from A to C. It will then ascend from C to A, and so
on, continually. In this case the thread, by which the molecule is
suspended, is supposed to be perfectly flexible, inextensible, and
of inconsiderable weight. The point of suspension is supposed to be
without friction, and the atmosphere to offer no resistance to the
motion.
It is evident from what has been stated, that the times of moving from
A to A′ and from A′ to A are equal, and will continue to be equal so
long as the pendulum continues to vibrate. If the number of vibrations
performed by the pendulum were registered, and the time of each
vibration known, this instrument would become a chronometer.
The rate at which the motion of the pendulum is accelerated in its
descent towards its lowest position is not uniform, because the force
which impels it is continually decreasing, and altogether disappears
at the point C. The impelling force arises from the effect of gravity
on the suspended molecule, and this effect is always produced in the
vertical direction A V. The greater the angle O A V is,
the less efficient the force of gravity will be in accelerating the
molecule: this angle evidently increases as the molecule approaches
C, which will appear by inspecting _fig. 73._ At C, the force of
gravity acting in the direction C B is totally expended in giving
tension to the thread, and is inefficient in moving the molecule. It
follows, therefore, that the impelling force is greatest at A, and
continually diminishes from A to C, where it altogether vanishes. The
same observations will be applicable to the retarding force from C to
A′, and to the accelerating force from A′ to C, and so on.
When the length of the thread and the intensity of the force of
gravity are given, the time of vibration depends on the length of the
arc A C, or on the magnitude of the angle A O C. If,
however, this angle do not exceed a certain limit of magnitude, the
time of vibration will be subject to no sensible variation, however
that angle may vary. Thus the time of oscillation will be the same,
whether the angle A O C be 2°, or 1° 30′, or 1°, or any
lesser magnitude. This property of a pendulum is expressed by the word
_isochronism_. The strict demonstration of this property depends on
mathematical principles, the details of which would not be suitable
to the present treatise. It is not difficult, however, to explain
generally how it happens that the same pendulum will swing through
greater and smaller arcs of vibration in the same time. If it swing
from A, the force of gravity at the commencement of its motion impels
it with an effect depending on the obliquity of the lines O A and
A V. If it commence its motion from _a_, the impelling effect from
the force of gravity will be considerably less than at A; consequently,
the pendulum begins to move at a slower rate, when it swings from
_a_ than when it moves from A: the greater magnitude of the swing is
therefore compensated by the increased velocity, so that the greater
and the smaller arcs of vibration are moved through in the same time.
(209.) To establish this property experimentally, it is only necessary
to suspend a small ball of metal, or other heavy substance, by a
flexible thread, and to put it in a state of vibration, the entire
arc of vibration not exceeding 4° or 5°, the friction on the point
of suspension and other causes will gradually diminish the arc of
vibration, so that after the lapse of some hours it will be so small,
that the motion will scarcely be discerned without microscopic aid. If
the vibration of this pendulum be observed in reference to a correct
timekeeper, at the commencement, at the middle, and towards the end of
its motion, the rate will be found to suffer no sensible change.
This remarkable law of isochronism was one of the earliest discoveries
of Galileo. It is said, that when very young, he observed a chandelier
suspended from the roof of a church in Pisa swinging with a pendulous
motion, and was struck with the uniformity of the rate even when the
extent of the swing was subject to evident variation.
(210.) It has been stated in (117.) that the attraction of gravity
affects all bodies equally, and moves them with the same velocity,
whatever be the nature or quantity of the materials of which they are
composed. Since it is the force of gravity which moves the pendulum, we
should therefore expect that the circumstances of that motion should
not be affected either by the quantity or quality of the pendulous
body. And we find this, in fact, to be the case; for if small pieces
of different heavy substances such as lead, brass, ivory, &c., be
suspended by fine threads of equal length, they will vibrate in the
same time, provided their weights bear a considerable proportion to the
atmospherical resistance, or that they be suspended _in vacuo_.
(211.) Since the time of vibration of a pendulum, which oscillates in
small arcs, depends neither on the magnitude of the arc of vibration
nor on the quality or weight of the pendulous body, it will be
necessary to explain the circumstances on which the variation of this
time depends.