§ 12. _Rectilinear Motion._--Let x denote the distance OP of a moving
point P at time t from a fixed origin O on the line of motion, this
distance being reckoned positive or negative according as it lies to one
side or the other of O. At time t + [delta]t let the point be at Q, and
let OQ = x + [delta]x. The _mean velocity_ of the point in the interval
[delta]t is [delta]x/[delta]t. The limiting value of this when [delta]t
is infinitely small, viz. dx/dt, is adopted as the definition of the
_velocity_ at the instant t. Again, let u be the velocity at time t, u +
[delta]u that at time t + [delta]t. The mean rate of increase of
velocity, or the _mean acceleration_, in the interval [delta]t is then
[delta]u/[delta]t. The limiting value of this when [delta]t is
infinitely small, viz., du/dt, is adopted as the definition of the
_acceleration_ at the instant t. Since u = dx/dt, the acceleration is
also denoted by d²x/dt². It is often convenient to use the "fluxional"
notation for differential coefficients with respect to time; thus the
velocity may be represented by [.x] and the acceleration by [.u] or
[:x]. There is another formula for the acceleration, in which u is
regarded as a function of the position; thus du/dt = (du/dx)(dx/dt) =
u(du/dx). The relation between x and t in any particular case may be
illustrated by means of a curve constructed with t as abscissa and x as
ordinate. This is called the _curve of positions_ or _space-time curve_;
its gradient represents the velocity. Such curves are often traced
mechanically in acoustical and other experiments. A, curve with t as
abscissa and u as ordinate is called the _curve of velocities_ or
_velocity-time curve_. Its gradient represents the acceleration, and the
area ([int]udt) included between any two ordinates represents the space
described in the interval between the corresponding instants (see fig.
62).
So far nothing has been said about the measurement of time. From the
purely kinematic point of view, the t of our formulae may be any
continuous independent variable, suggested (it may be) by some physical
process. But from the dynamical standpoint it is obvious that equations
which represent the facts correctly on one system of time-measurement
might become seriously defective on another. It is found that for almost
all purposes a system of measurement based ultimately on the earth's
rotation is perfectly adequate. It is only when we come to consider such
delicate questions as the influence of tidal friction that other
standards become necessary.
The most important conception in kinetics is that of "inertia." It is a
matter of ordinary observation that different bodies acted on by the
same force, or what is judged to be the same force, undergo different
changes of velocity in equal times. In our ideal representation of
natural phenomena this is allowed for by endowing each material particle
with a suitable _mass_ or _inertia-coefficient_ m. The product _mu_ of
the mass into the velocity is called the _momentum_ or (in Newton's
phrase) the _quantity of motion_. On the Newtonian system the motion of
a particle entirely uninfluenced by other bodies, when referred to a
suitable base, would be rectilinear, with constant velocity. If the
velocity changes, this is attributed to the action of force; and if we
agree to measure the force (X) by the rate of change of momentum which
it produces, we have the equation
d
--- (mu) = X. (1)
dt
From this point of view the equation is a mere truism, its real
importance resting on the fact that by attributing suitable values to
the masses m, and by making simple assumptions as to the value of X in
each case, we are able to frame adequate representations of whole
classes of phenomena as they actually occur. The question remains, of
course, as to how far the measurement of force here implied is
practically consistent with the gravitational method usually adopted in
statics; this will be referred to presently.
The practical unit or standard of mass must, from the nature of the
case, be the mass of some particular body, e.g. the imperial pound, or
the kilogramme. In the "C.G.S." system a subdivision of the latter, viz.
the gramme, is adopted, and is associated with the centimetre as the
unit of length, and the mean solar second as the unit of time. The unit
of force implied in (1) is that which produces unit momentum in unit
time. On the C.G.S. system it is that force which acting on one gramme
for one second produces a velocity of one centimetre per second; this
unit is known as the _dyne_. Units of this kind are called _absolute_ on
account of their fundamental and invariable character as contrasted with
gravitational units, which (as we shall see presently) vary somewhat
with the locality at which the measurements are supposed to be made.
If we integrate the equation (1) with respect to t between the limits t,
t´ we obtain
_
/ t´
mu´- mu = | X dt. (2)
_/ t
The time-integral on the right hand is called the _impulse_ of the force
on the interval t´ - t. The statement that the increase of momentum is
equal to the impulse is (it maybe remarked) equivalent to Newton's own
formulation of his Second Law. The form (1) is deduced from it by
putting t´- t = [delta]t, and taking [delta]t to be infinitely small. In
problems of impact we have to deal with cases of practically
instantaneous impulse, where a very great and rapidly varying force
produces an appreciable change of momentum in an exceedingly minute
interval of time.
In the case of a constant force, the acceleration [.u] or [:x] is,
according to (1), constant, and we have
d²x
--- = [alpha], (3)
dt²
say, the general solution of which is
x = ½[alpha]t² + At + B. (4)
The "arbitrary constants" A, B enable us to represent the circumstances
of any particular case; thus if the velocity [.x] and the position x be
given for any one value of t, we have two conditions to determine A, B.
The curve of positions corresponding to (4) is a parabola, and that of
velocities is a straight line. We may take it as an experimental result,
although the best evidence is indirect, that a particle falling freely
under gravity experiences a constant acceleration which at the same
place is the same for all bodies. This acceleration is denoted by g; its
value at Greenwich is about 981 centimetre-second units, or 32.2 feet
per second. It increases somewhat with the latitude, the extreme
variation from the equator to the pole being about ½%. We infer that on
our reckoning the force of gravity on a mass m is to be measured by mg,
the momentum produced per second when this force acts alone. Since this
is proportional to the mass, the relative masses to be attributed to
various bodies can be determined practically by means of the balance. We
learn also that on account of the variation of g with the locality a
gravitational system of force-measurement is inapplicable when more than
a moderate degree of accuracy is desired.
[Illustration: FIG. 61.]
We take next the case of a particle attracted towards a fixed point O in
the line of motion with a force varying as the distance from that point.
If [mu] be the acceleration at unit distance, the equation of motion
becomes
d²x
--- = -[mu]x, (5)
dt²
the solution of which may be written in either of the forms
x = A cos [sigma]t + B sin [sigma]t, x = a cos ([sigma]t + [epsilon]), (6)
where [sigma]= [root][mu], and the two constants A, B or a, [epsilon]
are arbitrary. The particle oscillates between the two positions x = ±a,
and the same point is passed through in the same direction with the same
velocity at equal intervals of time 2[pi]/[sigma]. The type of motion
represented by (6) is of fundamental importance in the theory of
vibrations (§ 23); it is called a _simple-harmonic_ or (shortly) a
_simple_ vibration. If we imagine a point Q to describe a circle of
radius a with the angular velocity [sigma], its orthogonal projection P
on a fixed diameter AA´ will execute a vibration of this character. The
angle [sigma]t + [epsilon] (or AOQ) is called the _phase_; the arbitrary
elements a, [epsilon] are called the _amplitude_ and _epoch_ (or initial
phase), respectively. In the case of very rapid vibrations it is usual
to specify, not the _period_ (2[pi]/[sigma]), but its reciprocal the
_frequency_, i.e. the number of complete vibrations per unit time. Fig.
62 shows the curves of position and velocity; they both have the form of
the "curve of sines." The numbers correspond to an amplitude of 10
centimetres and a period of two seconds.
The vertical oscillations of a weight which hangs from a fixed point by
a spiral spring come under this case. If M be the mass, and x the
vertical displacement from the position of equilibrium, the equation of
motion is of the form
d²x
M --- = - Kx, (7)
dt²
provided the inertia of the spring itself be neglected. This becomes
identical with (5) if we put [mu] = K/M; and the period is therefore
2[pi][root](M/K), the same for all amplitudes. The period is increased
by an increase of the mass M, and diminished by an increase in the
stiffness (K) of the spring. If c be the statical increase of length
which is produced by the gravity of the mass M, we have Kc = Mg, and the
period is 2[pi][root](c/g).
[Illustration: FIG. 62.]
The small oscillations of a simple pendulum in a vertical plane also
come under equation (5). According to the principles of § 13, the
horizontal motion of the bob is affected only by the horizontal
component of the force acting upon it. If the inclination of the string
to the vertical does not exceed a few degrees, the vertical displacement
of the particle is of the second order, so that the vertical
acceleration may be neglected, and the tension of the string may be
equated to the gravity mg of the particle. Hence if l be the length of
the string, and x the horizontal displacement of the bob from the
equilibrium position, the horizontal component of gravity is mgx/l,
whence
d²x gx
--- = - ---, (8)
dt² l
The motion is therefore simple-harmonic, of period [tau] =
2[pi][root](l/g). This indicates an experimental method of determining g
with considerable accuracy, using the formula g = 4[pi]²l/[tau]².
In the case of a repulsive force varying as the distance from the
origin, the equation of motion is of the type
d²x
--- = [mu]x, (9)
dt²
the solution of which is
x = A e^(nt) + B e^(-nt), (10)
where n = [root][mu]. Unless the initial conditions be adjusted so as
to make A = 0 exactly, x will ultimately increase indefinitely with t.
The position x = 0 is one of equilibrium, but it is unstable. This
applies to the inverted pendulum, with [mu] = g/l, but the equation
(9) is then only approximate, and the solution therefore only serves
to represent the initial stages of a motion in the neighbourhood of
the position of unstable equilibrium.
In acoustics we meet with the case where a body is urged towards a fixed
point by a force varying as the distance, and is also acted upon by an
"extraneous" or "disturbing" force which is a given function of the
time. The most important case is where this function is simple-harmonic,
so that the equation (5) is replaced by
d²x
--- + [mu]x = f cos ([sigma]1t + [alpha]), (11)
dt²
where [sigma]1 is prescribed. A particular solution is
f
x = ---------------- cos ([sigma]1t + [alpha]). (12)
[mu] - [sigma]1²
This represents a _forced oscillation_ whose period 2[pi]/[sigma]1,
coincides with that of the disturbing force; and the phase agrees with
that of the force, or is opposed to it, according as [sigma]1² < or > [mu];
i.e. according as the imposed period is greater or less than the natural
period 2[pi]/[root][mu]. The solution fails when the two periods agree
exactly; the formula (12) is then replaced by
ft
x = ---------- sin ([sigma]1t + [alpha]), (13)
2 [sigma]1
which represents a vibration of continually increasing amplitude. Since
the equation (12) is in practice generally only an approximation (as in
the case of the pendulum), this solution can only be accepted as a
representation of the initial stages of the forced oscillation. To
obtain the complete solution of (11) we must of course superpose the
free vibration (6) with its arbitrary constants in order to obtain a
complete representation of the most general motion consequent on
arbitrary initial conditions.
[Illustration: FIG. 63.]
A simple mechanical illustration is afforded by the pendulum. If the
point of suspension have an imposed simple vibration [xi] = a cos
[sigma]t in a horizontal line, the equation of small motion of the bob
is
x - [xi]
m[:x] = -mg --------,
l
or
gx [xi]
[:x] + --- = ----. (14)
l l
This is the same as if the point of suspension were fixed, and a
horizontal disturbing force mg[xi]/l were to act on the bob. The
difference of phase of the forced vibration in the two cases is
illustrated and explained in the annexed fig. 63, where the pendulum
virtually oscillates about C as a fixed point of suspension. This
illustration was given by T. Young in connexion with the kinetic
theory of the tides, where the same point arises.
We may notice also the case of an attractive force varying inversely
as the square of the distance from the origin. If [mu] be the
acceleration at unit distance, we have
du [mu]
u --- = - ---- (15)
dx x²
whence
2[mu]
u² = ----- + C. (16)
x
In the case of a particle falling directly towards the earth from rest
at a very great distance we have C = 0 and, by Newton's Law of
Gravitation, [mu]/a² = g, where a is the earth's radius. The deviation
of the earth's figure from sphericity, and the variation of g with
latitude, are here ignored. We find that the velocity with which the
particle would arrive at the earth's surface (x = a) is [root](2ga).
If we take as rough values a = 21 × 10^6 feet, g = 32 foot-second
units, we get a velocity of 36,500 feet, or about seven miles, per
second. If the particles start from rest at a finite distance c, we
have in (16), C = - 2[mu]/c, and therefore
dx / / 2[mu](c - x) \
-- = u = - / ( ------------- ), (17)
dt \/ \ cx /
the minus sign indicating motion towards the origin. If we put x = c
cos² ½[phi], we find
c^(3/2)
t = ------------- ([phi] + sin [phi]), (18)
[root](8[mu])
no additive constant being necessary if t be reckoned from the instant
of starting, when [phi] = 0. The time t of reaching the origin ([phi]
= [pi]) is
[pi] c^(3/2)
t1 = -------------. (19)
[root](8[mu])
This may be compared with the period of revolution in a circular orbit
of radius c about the same centre of force, viz.
2[pi]c^(3/2)/[root][mu](§ 14). We learn that if the orbital motion of
a planet, or a satellite, were arrested, the body would fall into the
sun, or into its primary, in the fraction 0.1768 of its actual
periodic time. Thus the moon would reach the earth in about five days.
It may be noticed that if the scales of x and t be properly adjusted,
the curve of positions in the present problem is the portion of a
cycloid extending from a vertex to a cusp.
In any case of rectilinear motion, if we integrate both sides of the
equation
du
mu -- = X, (20)
dx
which is equivalent to (1), with respect to x between the limits x0, x1,
we obtain
_
/ x1
½ mu1² - ½ mu0² = | X dx. (21)
_/ x0
We recognize the right-hand member as the _work_ done by the force X on
the particle as the latter moves from the position x0 to the position
x1. If we construct a curve with x as abscissa and X as ordinate, this
work is represented, as in J. Watt's "indicator-diagram," by the area
cut off by the ordinates x = x0, x = x1. The product ½mu² is called the
_kinetic energy_ of the particle, and the equation (21) is therefore
equivalent to the statement that the increment of the kinetic energy is
equal to the work done on the particle. If the force X be always the
same in the same position, the particle may be regarded as moving in a
certain invariable "field of force." The work which would have to be
supplied by other forces, extraneous to the field, in order to bring the
particle from rest in some standard position P0 to rest in any assigned
position P, will depend only on the position of P; it is called the
_statical_ or _potential energy_ of the particle with respect to the
field, in the position P. Denoting this by V, we have [delta]V -
X[delta]x = 0, whence
dV
X = - --, (22)
dx
The equation (21) may now be written
½ mu1² + V1 = ½ mu0² + V0, (23)
which asserts that when no extraneous forces act the sum of the kinetic
and potential energies is constant. Thus in the case of a weight hanging
by a spiral spring the work required to increase the length by x is V =
[int 0 to x] Kxdx = ½Kx², whence ½Mu² + ½Kx² = const., as is easily
verified from preceding results. It is easily seen that the effect of
extraneous forces will be to increase the sum of the kinetic and
potential energies by an amount equal to the work done by them. If this
amount be negative the sum in question is diminished by a corresponding
amount. It appears then that this sum is a measure of the total capacity
for doing work against extraneous resistances which the particle
possesses in virtue of its motion and its position; this is in fact the
origin of the term "energy." The product mv² had been called by G. W.
Leibnitz the "vis viva"; the name "energy" was substituted by T. Young;
finally the name "actual energy" was appropriated to the expression ½mv²
by W. J. M. Rankine.
The laws which regulate the resistance of a medium such as air to the
motion of bodies through it are only imperfectly known. We may briefly
notice the case of resistance varying as the square of the velocity,
which is mathematically simple. If the positive direction of x be
downwards, the equation of motion of a falling particle will be of the
form
du
-- = g - ku²; (24)
dt
this shows that the velocity u will send asymptotically to a certain
limit V (called the _terminal velocity_) such that kV² = g. The
solution is
gt V² gt
u = V tanh ---, x = --- log cosh ---, (25)
V g V
if the particle start from rest in the position x = 0 at the instant t
= 0. In the case of a particle projected vertically upwards we have
du
-- = -g - ku², (26)
dt
the positive direction being now upwards. This leads to
u u0 gt V² V² + u0²
tan^-1 --- = tan^-1 --- - ---, x = --- log --------, (27)
V V V 2g V² + u²
where u0 is the velocity of projection. The particle comes to rest
when
V u0 V² / u0² \
t = --- tan^-1 ---, x = --- log ( 1 + --- ). (28)
g V 2g \ V² /
For small velocities the resistance of the air is more nearly
proportional to the first power of the velocity. The effect of forces
of this type on small vibratory motions may be investigated as
follows. The equation (5) when modified by the introduction of a
frictional term becomes
[:x] = -[mu]x - k [.x]. (29)
If k² < 4[mu] the solution is
x = a e^{-t/[tau]} cos ([sigma]t + [epsilon]), (30)
where
[tau] = 2/k, [sigma] = [root]([mu] - ¼k²), (31)
and the constants a, [epsilon] are arbitrary. This may be described as
a simple harmonic oscillation whose amplitude diminishes
asymptotically to zero according to the law e^(-t/[tau]). The constant
[tau] is called the _modulus of decay_ of the oscillations; if it is
large compared with 2[pi]/[sigma] the effect of friction on the period
is of the second order of small quantities and may in general be
ignored. We have seen that a true simple-harmonic vibration may be
regarded as the orthogonal projection of uniform circular motion; it
was pointed out by P. G. Tait that a similar representation of the
type (30) is obtained if we replace the circle by an equiangular
spiral described, with a constant angular velocity about the pole, in
the direction of diminishing radius vector. When k² > 4[mu], the
solution of (29) is, in real form,
x = a1 e^(-t/[tau]1) + a2 e^(-t/[tau]2), (32)
where
1/[tau]1, 1/[tau]2 = ½k ± [root](¼k² - [mu]). (33)
The body now passes once (at most) through its equilibrium position,
and the vibration is therefore styled _aperiodic_.
To find the forced oscillation due to a periodic force we have
[:x] + k[.x] + [mu]x = f cos ([sigma]1t + [epsilon]). (34)
The solution is
f
x = --- cos ([sigma]1t + [epsilon] - [epsilon]1), (35)
R
provided
k[sigma]1
R = {([mu] - [sigma]1²)² + k²[sigma]1²}^½, tan[epsilon]1 = ----------------. (36)
[mu] - [sigma]1²
Hence the phase of the vibration lags behind that of the force by the
amount [epsilon]1, which lies between 0 and ½[pi] or between ½[pi] and
[pi], according as [sigma]1² <> [mu]. If the friction be comparatively
slight the amplitude is greatest when the imposed period coincides
with the free period, being then equal to f/k[sigma]1, and therefore
very great compared with that due to a slowly varying force of the
same average intensity. We have here, in principle, the explanation of
the phenomenon of "resonance" in acoustics. The abnormal amplitude is
greater, and is restricted to a narrower range of frequency, the
smaller the friction. For a complete solution of (34) we must of
course superpose the free vibration (30); but owing to the factor
e^(-t/[tau]) the influence of the initial conditions gradually
disappears.
For purposes of mathematical treatment a force which produces a finite
change of velocity in a time too short to be appreciated is regarded as
infinitely great, and the time of action as infinitely short. The whole
effect is summed up in the value of the instantaneous impulse, which is
the time-integral of the force. Thus if an instantaneous impulse [xi]
changes the velocity of a mass m from u to u´ we have
mu´- mu = [xi]. (37)
The effect of ordinary finite forces during the infinitely short
duration of this impulse is of course ignored.
We may apply this to the theory of impact. If two masses m1, m2 moving
in the same straight line impinge, with the result that the velocities
are changed from u1, u2, to u1´, u2´, then, since the impulses on the
two bodies must be equal and opposite, the total momentum is unchanged,
i.e.
m1u1´ + m2u2´ = m1u1 + m2u2. (38)
The complete determination of the result of a collision under given
circumstances is not a matter of abstract dynamics alone, but requires
some auxiliary assumption. If we assume that there is no loss of
apparent kinetic energy we have also
m1u1² + m2u2´² = m1u1² + m2u2². (39)
Hence, and from (38),
u2´ - u1´ = -(u2 - u1), (40)
i.e. the relative velocity of the two bodies is reversed in direction,
but unaltered in magnitude. This appears to be the case very
approximately with steel or glass balls; generally, however, there is
some appreciable loss of apparent energy; this is accounted for by
vibrations produced in the balls and imperfect elasticity of the
materials. The usual empirical assumption is that
u2´ - u1´ = -e(u2 - u1), (41)
where e is a proper fraction which is constant for the same two bodies.
It follows from the formula § 15 (10) for the internal kinetic energy of
a system of particles that as a result of the impact this energy is
diminished by the amount
m1m2
½(1 - e²) ------- (u1 - u2)². (42)
m1 + m2
The further theoretical discussion of the subject belongs to ELASTICITY.
This is perhaps the most suitable place for a few remarks on the theory
of "dimensions." (See also UNITS, DIMENSIONS OF.) In any absolute system
of dynamical measurement the fundamental units are those of mass, length
and time; we may denote them by the symbols M, L, T, respectively. They
may be chosen quite arbitrarily, e.g. on the C.G.S. system they are the
gramme, centimetre and second. All other units are derived from these.
Thus the unit of velocity is that of a point describing the unit of
length in the unit of time; it may be denoted by LT^-1, this symbol
indicating that the magnitude of the unit in question varies directly as
the unit of length and inversely as the unit of time. The unit of
acceleration is the acceleration of a point which gains unit velocity in
unit time; it is accordingly denoted by LT^-2. The unit of momentum is
MLT^-1; the unit force generates unit momentum in unit time and is
therefore denoted by MLT^-2. The unit of work on the same principles is
ML²T^-2, and it is to be noticed that this is identical with the unit of
kinetic energy. Some of these derivative units have special names
assigned to them; thus on the C.G.S. system the unit of force is called
the _dyne_, and the unit of work or energy the _erg_. The number which
expresses a physical quantity of any particular kind will of course vary
inversely as the magnitude of the corresponding unit. In any general
dynamical equation the dimensions of each term in the fundamental units
must be the same, for a change of units would otherwise alter the
various terms in different ratios. This principle is often useful as a
check on the accuracy of an equation.
The theory of dimensions often enables us to forecast, to some extent,
the manner in which the magnitudes involved in any particular problem
will enter into the result. Thus, assuming that the period of a small
oscillation of a given pendulum at a given place is a definite
quantity, we see that it must vary as [root](l/g). For it can only
depend on the mass m of the bob, the length l of the string, and the
value of g at the place in question; and the above expression is the
only combination of these symbols whose dimensions are those of a
time, simply. Again, the time of falling from a distance a into a
given centre of force varying inversely as the square of the distance
will depend only on a and on the constant [mu] of equation (15). The
dimensions of [mu]/x² are those of an acceleration; hence the
dimensions of [mu] are L³T^-2. Assuming that the time in question
varies as a^x[mu]^y, whose dimensions are L^(x + 3y)T^(-2y), we must
have x + 3y = 0, -2y = 1, so that the time of falling will vary as
a^(3/2)/[root][mu], in agreement with (19).
The argument appears in a more demonstrative form in the theory of
"similar" systems, or (more precisely) of the similar motion of
similar systems. Thus, considering the equations
d²x [mu] d²x´ [mu]´
--- = - ----, ---- = - -----, (43)
dt² x² dt´² x´²
which refer to two particles falling independently into two distinct
centres of force, it is obvious that it is possible to have x in a
constant ratio to x´, and t in a constant ratio to t´, provided that
x x´ [mu] [mu]´
--- : --- = ---- : -----, (44)
t² t´² x² x´²
and that there is a suitable correspondence between the initial
conditions. The relation (44) is equivalent to
x^(3/2) x´^(3/2)
t : t´ = ------- : --------, (45)
[mu]^½ [mu]´^½
where x, x´ are any two corresponding distances; e.g. they may be the
initial distances, both particles being supposed to start from rest.
The consideration of dimensions was introduced by J. B. Fourier (1822)
in connexion with the conduction of heat.
[Illustration: FIG. 64.]
§ 13. _General Motion of a Particle._--Let P, Q be the positions of a
moving point at times t, t + [delta]t respectively. A vector [->OU]
drawn parallel to PQ, of length proportional to PQ/[delta]t on any
convenient scale, will represent the _mean velocity_ in the interval
[delta]t, i.e. a point moving with a constant velocity having the
magnitude and direction indicated by this vector would experience the
same resultant displacement [->PQ] in the same time. As [delta]t is
indefinitely diminished, the vector [->OU] will tend to a definite limit
[->OV]; this is adopted as the definition of the _velocity_ of the
moving point at the instant t. Obviously [->OV] is parallel to the
tangent to the path at P, and its magnitude is ds/dt, where s is the
arc. If we project [->OV] on the co-ordinate axes (rectangular or
oblique) in the usual manner, the projections u, v, w are called the
_component velocities_ parallel to the axes. If x, y, z be the
co-ordinates of P it is easily proved that
dx dy dz
u = --, v = --, w = --. (1)
dt dt dt
The momentum of a particle is the vector obtained by multiplying the
velocity by the mass m. The _impulse_ of a force in any infinitely small
interval of time [delta]t is the product of the force into [delta]t; it
is to be regarded as a vector. The total impulse in any finite interval
of time is the integral of the impulses corresponding to the
infinitesimal elements [delta]t into which the interval may be
subdivided; the summation of which the integral is the limit is of
course to be understood in the vectorial sense.
Newton's Second Law asserts that change of momentum is equal to the
impulse; this is a statement as to equality of vectors and so implies
identity of direction as well as of magnitude. If X, Y, Z are the
components of force, then considering the changes in an infinitely short
time [delta]t we have, by projection on the co-ordinate axes,
[delta](mu) = X[delta]t, and so on, or
du dv dw
m -- = X, m -- = Y, m -- = Z. (2)
dt dt dt
For example, the path of a particle projected anyhow under gravity will
obviously be confined to the vertical plane through the initial
direction of motion. Taking this as the plane xy, with the axis of x
drawn horizontally, and that of y vertically upwards, we have X = 0, Y =
-mg; so that
d²x d²y
--- = 0, --- = -g. (3)
dt² dt²
The solution is
x = At + B, y = -½ gt² + Ct + D. (4)
If the initial values of x, y, [.x], [.y] are given, we have four
conditions to determine the four arbitrary constants A, B, C, D. Thus if
the particle start at time t = 0 from the origin, with the component
velocities u0, v0, we have
x = u0t, y = v0t - ½ gt². (5)
Eliminating t we have the equation of the path, viz.
v0 gx²
y = --- x - ---. (6)
u0 2u²
This is a parabola with vertical axis, of latus-rectum 2u0²/g. The range
on a horizontal plane through O is got by putting y = 0, viz. it is
2u0v0/g. we denote the resultant velocity at any instant by [.s] we have
[.s]² = [.x]² + [.y]² = [.s]0² - 2gy. (7)
Another important example is that of a particle subject to an
acceleration which is directed always towards a fixed point O and is
proportional to the distance from O. The motion will evidently be in one
plane, which we take as the plane z = 0. If [mu] be the acceleration at
unit distance, the component accelerations parallel to axes of x and y
through O as origin will be -[mu]x, -[mu]y, whence
d²x d²y
--- = -[mu]x, --- = - [mu]y. (8)
dt² dt²
The solution is
x = A cos nt + B sin nt, y = C cos nt + D sin nt, (9)
where n = [root][mu]. If P be the initial position of the particle, we
may conveniently take OP as axis of x, and draw Oy parallel to the
direction of motion at P. If OP = a, and [.s]0 be the velocity at P, we
have, initially, x = a, y = 0, [.x] = 0, [.y] = [.s]0 whence
x = a cos nt, y = b sin nt, (10)
if b = [.s]0/n. The path is therefore an ellipse of which a, b are
conjugate semi-diameters, and is described in the period
2[pi]/[root][mu]; moreover, the velocity at any point P is equal to
[root][mu]·OD, where OD is the semi-diameter conjugate to OP. This type
of motion is called _elliptic harmonic_. If the co-ordinate axes are the
principal axes of the ellipse, the angle nt in (10) is identical with
the "excentric angle." The motion of the bob of a "spherical pendulum,"
i.e. a simple pendulum whose oscillations are not confined to one
vertical plane, is of this character, provided the extreme inclination
of the string to the vertical be small. The acceleration is towards the
vertical through the point of suspension, and is equal to gr/l,
approximately, if r denote distance from this vertical. Hence the path
is approximately an ellipse, and the period is 2[pi] [root](l/g).
[Illustration: FIG. 65.]
The above problem is identical with that of the oscillation of a
particle in a smooth spherical bowl, in the neighbourhood of the
lowest point. If the bowl has any other shape, the axes Ox, Oy may be
taken tangential to the lines of curvature at the lowest point O; the
equations of small motion then are
d²x x d²y y
--- = -g ------, --- = -g ------, (11)
dt² [rho]1 dt² [rho]2
where [rho]1, [rho]2, are the principal radii of curvature at O. The
motion is therefore the resultant of two simple vibrations in
perpendicular directions, of periods 2[pi] [root]([rho]1/g),
2[pi] [root]([rho]2/g). The circumstances are realized in "Blackburn's
pendulum," which consists of a weight P hanging from a point C of a
string ACB whose ends A, B are fixed. If E be the point in which the
line of the string meets AB, we have [rho]1 = CP, [rho]2 = EP. Many
contrivances for actually drawing the resulting curves have been
devised.
[Illustration: FIG. 66.]
It is sometimes convenient to resolve the accelerations in directions
having a more intrinsic relation to the path. Thus, in a plane path, let
P, Q be two consecutive positions, corresponding to the times t, t +
[delta]t; and let the normals at P, Q meet in C, making an angle
[delta][psi]. Let v (= [.s]) be the velocity at P, v + [delta]v that at
Q. In the time [delta]t the velocity parallel to the tangent at P
changes from v to v + [delta]v, ultimately, and the tangential
acceleration at P is therefore dv/dt or [:s]. Again, the velocity
parallel to the normal at P changes from 0 to v[delta][psi], ultimately,
so that the normal acceleration is v d[psi]/dt. Since
dv dv ds dv d[psi] d[psi] ds v²
-- = -- -- = v --, v ------ = v ------ -- = -----, (12)
dt ds dt ds dt ds dt [rho]
where [rho] is the radius of curvature of the path at P, the tangential
and normal accelerations are also expressed by v dv/ds and v²/[rho],
respectively. Take, for example, the case of a particle moving on a
smooth curve in a vertical plane, under the action of gravity and the
pressure R of the curve. If the axes of x and y be drawn horizontal and
vertical (upwards), and if [psi] be the inclination of the tangent to
the horizontal, we have
dv dy mv²
mv -- = - mg sin [psi] = - mg --, ----- = - mg cos [psi] + R. (13)
ds ds [rho]
The former equation gives
v² = C - 2gy, (14)
and the latter then determines R.
In the case of the pendulum the tension of the string takes the place
of the pressure of the curve. If l be the length of the string, [psi]
its inclination to the downward vertical, we have [delta]s =
l[delta][psi], so that v = ld[psi]/dt. The tangential resolution then
gives
d²[psi]
l ------- = - g sin [psi]. (15)
dt²
If we multiply by 2d[psi]/dt and integrate, we obtain
/ d[psi]\² 2g
( ------ ) = --- cos [psi] + const., (16)
\ dt / l
which is seen to be equivalent to (14). If the pendulum oscillate
between the limits [psi] = ±[alpha], we have
/[delta][psi]\² 2g 4g
( ------------ ) = --- (cos [psi] - cos [alpha]) = --- (sin² ½[alpha] - sin² ½[psi]); (17)
\ dt / l l
and, putting sin ½[psi] = sin ½[alpha]. sin [phi], we find for the
period ([tau]) of a complete oscillation
_½[pi] _½[pi]
/ dt / l / d[phi]
[tau] = 4 | ------ d[phi] = 4 / --- · | ------------------------------------
_/0 d[phi] \/ g _/0 [root](1 - sin² ½[alpha]·sin² [phi])
/ l
= 4 / ---·F1(sin ½[alpha]), (18)
\/ g
in the notation of elliptic integrals. The function F1 (sin [beta])
was tabulated by A. M. Legendre for values of [beta] ranging from 0°
to 90°. The following table gives the period, for various amplitudes
[alpha], in terms of that of oscillation in an infinitely small arc
[viz. 2[pi] [root](l/g)] as unit.
+--------------+----------++--------------+----------+
| [alpha]/[pi] | [tau] || [alpha]/[pi] | [tau] |
+--------------+----------++--------------+----------+
| .1 | 1.0062 || .6 | 1.2817 |
| .2 | 1.0253 || .7 | 1.4283 |
| .3 | 1.0585 || .8 | 1.6551 |
| .4 | 1.1087 || .9 | 2.0724 |
| .5 | 1.1804 || 1.0 | [oo] |
+--------------+----------++--------------+----------+
The value of [tau] can also be obtained as an infinite series, by
expanding the integrand in (18) by the binomial theorem, and
integrating term by term. Thus
/ l / 1² 1²·3² \
[tau] = 2[pi] / --- · ( 1 + --- sin² ½[alpha] + ----- sin^4 ½[alpha] + ... ). (19)
\/ g \ 2² 2²·4² /
If [alpha] be small, an approximation (usually sufficient) is
[tau] = 2[pi] [root](l/g)·(1 + (1/16)[alpha]²).
In the extreme case of [alpha] = [pi], the equation (17) is
immediately integrable; thus the time from the lowest position is
t = [root](l/g)·log tan (¼[pi] + ¼[psi]). (20)
This becomes infinite for [psi] = [pi], showing that the pendulum only
tends asymptotically to the highest position.
[Illustration: FIG. 67.]
The variation of period with amplitude was at one time a hindrance to
the accurate performance of pendulum clocks, since the errors produced
are cumulative. It was therefore sought to replace the circular
pendulum by some other contrivance free from this defect. The equation
of motion of a particle in any smooth path is
d²s
--- = -g sin [psi], (21)
dt²
where [psi] is the inclination of the tangent to the horizontal. If
sin [psi] were accurately and not merely approximately proportional to
the arc s, say
s = k sin [psi], (22)
the equation (21) would assume the same form as § 12 (5). The motion
along the arc would then be accurately simple-harmonic, and the period
2[pi][root](k/g) would be the same for all amplitudes. Now equation
(22) is the intrinsic equation of a cycloid; viz. the curve is that
traced by a point on the circumference of a circle of radius ¼k which
rolls on the under side of a horizontal straight line. Since the
evolute of a cycloid is an equal cycloid the object is attained by
means of two metal cheeks, having the form of the evolute near the
cusp, on which the string wraps itself alternately as the pendulum
swings. The device has long been abandoned, the difficulty being met
in other ways, but the problem, originally investigated by C. Huygens,
is important in the history of mathematics.
The component accelerations of a point describing a tortuous curve, in
the directions of the tangent, the principal normal, and the binormal,
respectively, are found as follows. If [->OV], [->OV´] be vectors
representing the velocities at two consecutive points P, P´ of the path,
the plane VOV´ is ultimately parallel to the osculating plane of the
path at P; the resultant acceleration is therefore in the osculating
plane. Also, the projections of [->VV´] on OV and on a perpendicular to
OV in the plane VOV´ are [delta]v and v[delta][epsilon], where
[delta][epsilon] is the angle between the directions of the tangents at
P, P´. Since [delta][epsilon] = [delta]s/[rho], where [delta]s = PP´ =
v[delta]t and [rho] is the radius of principal curvature at P, the
component accelerations along the tangent and principal normal are dv/dt
and vd[epsilon]/dt, respectively, or vdv/ds and v²/[rho]. For example,
if a particle moves on a smooth surface, under no forces except the
reaction of the surface, v is constant, and the principal normal to the
path will coincide with the normal to the surface. Hence the path is a
"geodesic" on the surface.
If we resolve along the tangent to the path (whether plane or tortuous),
the equation of motion of a particle may be written
dv
mv -- = [T], (23)
ds
where [T] is the tangential component of the force. Integrating with
respect to s we find
_
/ s1
½ mv1² - ½ mv0² = | [T] ds; (24)
_/ s0
i.e. the increase of kinetic energy between any two positions is equal
to the work done by the forces. The result follows also from the
Cartesian equations (2); viz. we have
m([.x][:x] + [.y][:y] + [.z][:z]) = X[.x] + Y[.y] + Z[.z], (25)
whence, on integration with respect to t,
_
/
½m([.x]² + [.y]² + [.z]²) = |(X[.x] + Y[.y] + Z[.z]) dt + const.
_/
_
/
= |(X dx + Y dy + Z dz) + const. (26)
_/
If the axes be rectangular, this has the same interpretation as (24).
Suppose now that we have a constant field of force; i.e. the force
acting on the particle is always the same at the same place. The work
which must be done by forces extraneous to the field in order to bring
the particle from rest in some standard position A to rest in any other
position P will not necessarily be the same for all paths between A and
P. If it is different for different paths, then by bringing the particle
from A to P by one path, and back again from P to A by another, we might
secure a gain of work, and the process could be repeated indefinitely.
If the work required is the same for all paths between A and P, and
therefore zero for a closed circuit, the field is said to be
_conservative_. In this case the work required to bring the particle
from rest at A to rest at P is called the _potential energy_ of the
particle in the position P; we denote it by V. If PP´ be a linear
element [delta]s drawn in any direction from P, and S be the force due
to the field, resolved in the direction PP´, we have [delta]V =
-S[delta]s or
[dP]V
S = -----. (27)
[dP]s
In particular, by taking PP´ parallel to each of the (rectangular)
co-ordinate axes in succession, we find
[dP]V [dP]V [dP]V
X = -----, Y = -----, Z = -----. (28)
[dP]x [dP]y [dP]z
The equation (24) or (26) now gives
½ mv1² + V1 = ½ mv0² + V0; (29)
i.e. the sum of the kinetic and potential energies is constant when no
work is done by extraneous forces. For example, if the field be that due
to gravity we have V = fmgdy = mgy + const., if the axis of y be drawn
vertically upwards; hence
½ mv² + mgy = const. (30)
This applies to motion on a smooth curve, as well as to the free motion
of a projectile; cf. (7), (14). Again, in the case of a force Kr towards
O, where r denotes distance from O we have V = [int] Kr dr = ½Kr² +
const., whence
½ mv² + ½ Kr² = const. (31)
It has been seen that the orbit is in this case an ellipse; also that if
we put [mu] = K/m the velocity at any point P is v = [root][mu]·OD,
where OD is the semi-diameter conjugate to OP. Hence (31) is consistent
with the known property of the ellipse that OP² + OD² is constant.
The forms assumed by the dynamical equations when the axes of
reference are themselves in motion will be considered in § 21. At
present we take only the case where the rectangular axes Ox, Oy rotate
in their own plane, with angular velocity [omega] about Oz, which is
fixed. In the interval [delta]t the projections of the line joining
the origin to any point (x, y, z) on the directions of the co-ordinate
axes at time t are changed from x, y, z to (x + [delta]x) cos
[omega][delta]t - (y + [delta]y) sin [omega][delta]t, (x + [delta]x)
sin [omega][delta]t + (y + [delta]y) cos [omega][delta]t, z
respectively. Hence the component velocities parallel to the
instantaneous positions of the co-ordinate axes at time t are
u = [.x] - [omega]y, v = [.y] + [omega]z, [omega] = [.z]. (32)
In the same way we find that the component accelerations are
[.u] - [omega]v, [.v] + [omega]u, [.omega]. (33)
Hence if [omega] be constant the equations of motion take the forms
m([:x] - 2[omega][.y] - [omega]²[.x]) = X, m([:y] + 2[omega][.x] - [omega]²y) = Y, m[:z] = Z. (34)
These become identical with the equations of motion relative to fixed
axes provided we introduce a fictitious force m[omega]²r acting
outwards from the axis of z, where r = [root](x² + y²), and a second
fictitious force 2m[omega]v at right angles to the path, where v is
the component of the relative velocity parallel to the plane xy. The
former force is called by French writers the _force centrifuge
ordinaire_, and the latter the _force centrifuge composée_, or _force
de Coriolis_. As an application of (34) we may take the case of a
symmetrical Blackburn's pendulum hanging from a horizontal bar which
is made to rotate about a vertical axis half-way between the points
of attachment of the upper string. The equations of small motion are
then of the type
[:x] - 2[omega][.y] - [omega]²x = -p²x, [:y] + 2[omega][.x] - [omega]²y = -q²y. (35)
This is satisfied by
[:x] = A cos ([sigma]t + [epsilon]), y = B sin ([sigma]t + [epsilon]), (36)
provided
([sigma]² + [omega]² - p²)A + 2[sigma][omega]B = 0, \ (37)
2[sigma][omega]A + ([sigma]² + [omega]² - q²)B = 0. /
Eliminating the ratio A : B we have
([sigma]² + [omega]² - p²)([sigma]² + [omega]² - q²) - 4[sigma]²[omega]² = 0. (38)
It is easily proved that the roots of this quadratic in [sigma]² are
always real, and that they are moreover both positive unless [omega]²
lies between p² and q². The ratio B/A is determined in each case by
either of the equations (37); hence each root of the quadratic gives a
solution of the type (36), with two arbitrary constants A, [epsilon].
Since the equations (35) are linear, these two solutions are to be
superposed. If the quadratic (38) has a negative root, the
trigonometrical functions in (36) are to be replaced by real
exponentials, and the position x = 0, y = 0 is unstable. This occurs
only when the period (2[pi]/[omega]) of revolution of the arm lies
between the two periods (2[pi]/p, 2[pi]/q) of oscillation when the arm
is fixed.
§ 14. _Central Forces. Hodograph._--The motion of a particle subject to
a force which passes always through a fixed point O is necessarily in a
plane orbit. For its investigation we require two equations; these may
be obtained in a variety of forms.
Since the impulse of the force in any element of time [delta]t has zero
moment about O, the same will be true of the additional momentum
generated. Hence the moment of the momentum (considered as a localized
vector) about O will be constant. In symbols, if v be the velocity and p
the perpendicular from O to the tangent to the path,
pv = h, (1)
where h is a constant. If [delta]s be an element of the path, p[delta]s
is twice the area enclosed by [delta]s and the radii drawn to its
extremities from O. Hence if [delta]A be this area, we have [delta]A = ½
p[delta]s = ½ h[delta]t, or
dA
-- = ½h. (2)
dt
Hence equal areas are swept over by the radius vector in equal times.
If P be the acceleration towards O, we have
dv dr
v -- = -P --, (3)
ds ds
since dr/ds is the cosine of the angle between the directions of r and
[delta]s. We will suppose that P is a function of r only; then
integrating (3) we find
_
/
½ v² = - | P dr + const., (4)
_/
which is recognized as the equation of energy. Combining this with (1)
we have
_
h² /
-- = C - 2 | P dr, (5)
p² _/
which completely determines the path except as to its orientation with
respect to O.
If the law of attraction be that of the inverse square of the distance,
we have P = [mu]/r², and
h² 2[mu]
-- = C + -----. (6)
p² [tau]
Now in a conic whose focus is at O we have
l 2 1
--- = -- ± ---, (7)
p² r a
where l is half the latus-rectum, a is half the major axis, and the
upper or lower sign is to be taken according as the conic is an ellipse
or hyperbola. In the intermediate case of the parabola we have a = [oo]
and the last term disappears. The equations (6) and (7) are identified
by putting
l = h²/[mu], a = ± [mu]/C. (8)
Since
h² / 2 1 \
v² = -- = [mu]( --- ± --- ), (9)
p² \ r a /
it appears that the orbit is an ellipse, parabola or hyperbola,
according as v² is less than, equal to, or greater than 2[mu]/r. Now it
appears from (6) that 2[mu]/r is the square of the velocity which would
be acquired by a particle falling from rest at infinity to the distance
r. Hence the character of the orbit depends on whether the velocity at
any point is less than, equal to, or greater than the _velocity from
infinity_, as it is called. In an elliptic orbit the area [pi]ab is
swept over in the time
[pi]ab 2[pi]a^(3/2)
r = ------ = ------------, (10)
½h [root][mu]
since h = [mu]^½ l^½ = [mu]^½ ba^-½ by (8).
The converse problem, to determine the law of force under which a
given orbit can be described about a given pole, is solved by
differentiating (5) with respect to r; thus
h² dp
P = -----. (11)
p³ dr
In the case of an ellipse described about the centre as pole we have
a²b²
---- = a² + b² - r²; (12)
p²
hence P = [mu]r, if [mu] = h²/a²b². This merely shows that a
particular ellipse may be described under the law of the direct
distance provided the circumstances of projection be suitably
adjusted. But since an ellipse can always be constructed with a given
centre so as to touch a given line at a given point, and to have a
given value of ab (= h/[root][mu]) we infer that the orbit will be
elliptic whatever the initial circumstances. Also the period is
2[pi]ab/h = 2[pi]/[root][mu], as previously found.
Again, in the equiangular spiral we have p = r sin[alpha], and
therefore P = [mu]/r³, if [mu] = h²/sin²[alpha]. But since an
equiangular spiral having a given pole is completely determined by a
given point and a given tangent, this type of orbit is not a general
one for the law of the inverse cube. In order that the spiral may be
described it is necessary that the velocity of projection should be
adjusted to make h = [root][mu]·sin[alpha]. Similarly, in the case of
a circle with the pole on the circumference we have p² = r²/2a, P =
[mu]/r^5, if [mu] = 8h²a²; but this orbit is not a general one for the
law of the inverse fifth power.
[Illustration: FIG. 68.]
In astronomical and other investigations relating to central forces it
is often convenient to use polar co-ordinates with the centre of force
as pole. Let P, Q be the positions of a moving point at times t, t +
[delta]t, and write OP = r, OQ = r + [delta]r, [angle]POQ =
[delta][theta], O being any fixed origin. If u, v be the component
velocities at P along and perpendicular to OP (in the direction of
[theta] increasing), we have
[delta]r dr r[delta][theta] d[theta]
u = lim.-------- = --, v = lim. --------------- = r --------. (13)
[delta]t dt [delta]t dt
Again, the velocities parallel and perpendicular to OP change in the
time [delta]t from u, v to u - v[delta][theta], v + u[delta][theta],
ultimately. The component accelerations at P in these directions are
therefore
du d[theta] d²r /d[theta]\² \
-- - v -------- = --- - r ( -------- ), |
dt dt dt² \ dt / |
> (14)
dv d[theta] 1 d / d[theta]\ |
-- + u -------- = --- --- ( r² -------- ), |
dt dt r dt \ dt / /
respectively.
In the case of a central force, with O as pole, the transverse
acceleration vanishes, so that
r²d[theta]/dt = h, (15)
where h is constant; this shows (again) that the radius vector sweeps
over equal areas in equal times. The radial resolution gives
d²r /d[theta]\²
--- - r ( -------- ) = -P, (16)
dt² \ dt /
where P, as before, denotes the acceleration towards O. If in this we
put r = 1/u, and eliminate t by means of (15), we obtain the general
differential equation of central orbits, viz.
d²u P
--------- + u = ----. (17)
d[theta]² h²u²
If, for example, the law be that of the inverse square, we have P =
[mu]u², and the solution is of the form
[mu]
u = ------ {1 + e cos ([theta] - [alpha])}, (18)
h²
where e, [alpha] are arbitrary constants. This is recognized as the
polar equation of a conic referred to the focus, the half latus-rectum
being h²/[mu].
The law of the inverse cube P = [mu]u³ is interesting by way of
contrast. The orbits may be divided into two classes according as h²
<> [mu], i.e. according as the transverse velocity (hu) is greater or
less than the velocity [root]([mu]·u) appropriate to a circular orbit
at the same distance. In the former case the equation (17) takes the
form
d²u
-------- + m²u = 0, (19)
d[theta]²
the solution of which is
au = sin m ([theta] - [alpha]). (20)
The orbit has therefore two asymptotes, inclined at an angle [pi]/m.
In the latter case the differential equation is of the form
d²u
--------- = m²u, (21)
d[theta]²
so that
u = A e^(m[theta]) + B e^(-m[theta]) (22)
If A, B have the same sign, this is equivalent to
au = cosh m[theta], (23)
if the origin of [theta] be suitably adjusted; hence r has a maximum
value [alpha], and the particle ultimately approaches the pole
asymptotically by an infinite number of convolutions. If A, B have
opposite signs the form is
au = sinh m[theta], (24)
this has an asymptote parallel to [theta] = 0, but the path near the
origin has the same general form as in the case of (23). If A or B
vanish we have an equiangular spiral, and the velocity at infinity is
zero. In the critical case of h² = [mu], we have d²u/d[theta]² = 0,
and
u = A[theta] + B; (25)
the orbit is therefore a "reciprocal spiral," except in the special
case of A = 0, when it is a circle. It will be seen that unless the
conditions be exactly adjusted for a circular orbit the particle will
either recede to infinity or approach the pole asymptotically. This
problem was investigated by R. Cotes (1682-1716), and the various
curves obtained arc known as _Coles's spirals_.
A point on a central orbit where the radial velocity (dr/dt) vanishes is
called an _apse_, and the corresponding radius is called an _apse-line_.
If the force is always the same at the same distance any apse-line will
divide the orbit symmetrically, as is seen by imagining the velocity at
the apse to be reversed. It follows that the angle between successive
apse-lines is constant; it is called the _apsidal angle_ of the orbit.
If in a central orbit the velocity is equal to the velocity from
infinity, we have, from (5),
_
h² / [oo]
-- = 2 | P dr; (26)
p² _/ r
this determines the form of the critical orbit, as it is called. If P =
[mu]/r^[n], its polar equation is
r^m cos m[theta] = a^m, (27)
where m = ½(3 - n), except in the case n = 3, when the orbit is an
equiangular spiral. The case n = 2 gives the parabola as before.
If we eliminate d[theta]/dt between (15) and (16) we obtain
d²r h²
--- - -- = -P = -f(r),
dt² r³
say. We may apply this to the investigation of the stability of a
circular orbit. Assuming that r = a + x, where x is small, we have,
approximately,
d²x h² / 3x\
--- - -- ( 1 - -- ) = -f(a) - xf´(a).
dt² r³ \ a /
Hence if h and a be connected by the relation h² = a³f(a) proper to a
circular orbit, we have
_ _
d²x | 3 |
--- + | f´(a) + --- f(a)| x = 0. (28)
dt² |_ a _|
If the coefficient of x be positive the variations of x are
simple-harmonic, and x can remain permanently small; the circular
orbit is then said to be stable. The condition for this may be written
_ _
d | |
-- | a³f(a) | > 0, (29)
da |_ _|
i.e. the intensity of the force in the region for which r = a, nearly,
must diminish with increasing distance less rapidly than according to
the law of the inverse cube. Again, the half-period of x is
[pi]/sqrt[f´(a) + 3^{-1}f(a)], and since the angular velocity in the
orbit is h/a², approximately, the apsidal angle is, ultimately,
_ _
/ | f(a) |
[pi] / | --------------- |, (30)
\/ |_ af´(a) + 3f(a) _|
or, in the case of f(a) = [mu]/r^n, [pi]/[root](3 - n). This is in
agreement with the known results for n = 2, n = -1.
We have seen that under the law of the inverse square all finite
orbits are elliptical. The question presents itself whether there
then is any other law of force, giving a finite velocity from
infinity, under which all finite orbits are necessarily closed curves.
If this is the case, the apsidal angle must evidently be commensurable
with [pi], and since it cannot vary discontinuously the apsidal angle
in a nearly circular orbit must be constant. Equating the expression
(30) to [pi]/m, we find that f(a) = C/a^n, where n = 3 - m². The
force must therefore vary as a power of the distance, and n must be
less than 3. Moreover, the case n = 2 is the only one in which the
critical orbit (27) can be regarded as the limiting form of a closed
curve. Hence the only law of force which satisfies the conditions is
that of the inverse square.
At the beginning of § 13 the velocity of a moving point P was
represented by a vector [->OV] drawn from a fixed origin O. The locus of
the point V is called the _hodograph_ (q.v.); and it appears that the
velocity of the point V along the hodograph represents in magnitude and
in direction the acceleration in the original orbit. Thus in the case of
a plane orbit, if v be the velocity of P, [psi] the inclination of the
direction of motion to some fixed direction, the polar co-ordinates of V
may be taken to be v, [psi]; hence the velocities of V along and
perpendicular to OV will be dv/dt and vd[psi]/dt. These expressions
therefore give the tangential and normal accelerations of P; cf. § 13
(12).
[Illustration: FIG. 69.]
In the motion of a projectile under gravity the hodograph is a
vertical line described with constant velocity. In elliptic harmonic
motion the velocity of P is parallel and proportional to the
semi-diameter CD which is conjugate to the radius CP; the hodograph is
therefore an ellipse similar to the actual orbit. In the case of a
central orbit described under the law of the inverse square we have v
= h/SY = h. SZ/b², where S is the centre of force, SY is the
perpendicular to the tangent at P, and Z is the point where YS meets
the auxiliary circle again. Hence the hodograph is similar and
similarly situated to the locus of Z (the auxiliary circle) turned
about S through a right angle. This applies to an elliptic or
hyperbolic orbit; the case of the parabolic orbit may be examined
separately or treated as a limiting case. The annexed fig. 70 exhibits
the various cases, with the hodograph in its proper orientation. The
pole O of the hodograph is inside on or outside the circle, according
as the orbit is an ellipse, parabola or hyperbola. In any case of a
central orbit the hodograph (when turned through a right angle) is
similar and similarly situated to the "reciprocal polar" of the orbit
with respect to the centre of force. Thus for a circular orbit with
the centre of force at an excentric point, the hodograph is a conic
with the pole as focus. In the case of a particle oscillating under
gravity on a smooth cycloid from rest at the cusp the hodograph is a
circle through the pole, described with constant velocity.
§ 15. _Kinetics of a System of Discrete Particles._--The momenta of the
several particles constitute a system of localized vectors which, for
purposes of resolving and taking moments, may be reduced like a system
of forces in statics (§ 8). Thus taking any point O as base, we have
first a _linear momentum_ whose components referred to rectangular axes
through O are
[Sigma](m[.x]), [Sigma](m[.y]), [Sigma](m[.z]); (1)
its representative vector is the same whatever point O be chosen.
Secondly, we have an _angular momentum_ whose components are
[Sigma]{m(y[.z] - z[.y])}, [Sigma]{m(z[.x] - xz[.z])}, [Sigma]{m(x[.y] - y[.x])}, (2)
these being the sums of the moments of the momenta of the several
particles about the respective axes. This is subject to the same
relations as a couple in statics; it may be represented by a vector
which will, however, in general vary with the position of O.
The linear momentum is the same as if the whole mass were concentrated
at the centre of mass G, and endowed with the velocity of this point.
This follows at once from equation (8) of § 11, if we imagine the two
configurations of the system there referred to to be those corresponding
to the instants t, t + [delta]t. Thus
__ / [->PP] \ __ [->GG´]
\ ( m·-------- ) = \ (m)·--------. (3)
/__ \ [delta]t / /__ [delta]t
Analytically we have
d d[|x]
[Sigma](m[.x]) = --- [Sigma](mx) = [Sigma](m)·-----. (4)
dt dt
with two similar formulae.
[Illustration: FIG. 70.]
Again, if the instantaneous position of G be taken as base, the angular
momentum of the absolute motion is the same as the angular momentum of
the motion relative to G. For the velocity of a particle m at P may be
replaced by two components one of which (v) is identical in magnitude
and direction with the velocity of G, whilst the other (v) is the
velocity relative to G. The aggregate of the components mv of momentum
is equivalent to a single localized vector [Sigma](m)·v in a line
through G, and has therefore zero moment about any axis through G; hence
in taking moments about such an axis we need only regard the velocities
relative to G. In symbols, we have
/ d[|z] d[|y]\
[Sigma]{m(y[.z] - z[.y])} = [Sigma](m)·( y ----- - z ----- ) + [Sigma]{m([eta][zeta] - [.zeta][eta])}. (5)
\ dt dt /
since [Sigma](m[xi]) = 0, [Sigma](m[xi]) = 0, and so on, the notation
being as in § 11. This expresses that the moment of momentum about any
fixed axis (e.g. Ox) is equal to the moment of momentum of the motion
relative to G about a parallel axis through G, together with the moment
of momentum of the whole mass supposed concentrated at G and moving with
this point. If in (5) we make O coincide with the instantaneous position
of G, we have [|x], [|y], [|z] = 0, and the theorem follows.
[Illustration: FIG. 71.]
Finally, the rates of change of the components of the angular momentum
of the motion relative to G referred to G as a moving base, are equal to
the rates of change of the corresponding components of angular momentum
relative to a fixed base coincident with the instantaneous position of
G. For let G´ be a consecutive position of G. At the instant t +
[delta]t the momenta of the system are equivalent to a linear momentum
represented by a localized vector [Sigma](m)·(v + [delta]v) in a line
through G´ tangential to the path of G´, together with a certain angular
momentum. Now the moment of this localized vector with respect to any
axis through G is zero, to the first order of [delta]t, since the
perpendicular distance of G from the tangent line at G´ is of the order
([delta]t)². Analytically we have from (5),
d / d[|z]² d²[|y] \ d
--- [Sigma] {m (y[.z] - z[.y])} = [Sigma](m)·( y ------ - z ------- ) + --- [Sigma] {m([eta][zeta - [zeta][.eta])} (6)
dt \ dt² dt² / dt
If we put x, y, z = 0, the theorem is proved as regards axes parallel to
Ox.
Next consider the kinetic energy of the system. If from a fixed point O
we draw vectors [->OV1], [->OV2] to represent the velocities of the
several particles m1, m2 ..., and if we construct the vector
[Sigma](m·[->OV])
[->OK] = ----------------- (7)
[Sigma](m)
this will represent the velocity of the mass-centre, by (3). We find,
exactly as in the proof of Lagrange's First Theorem (§ 11), that
½[Sigma](m·OV²) = ½[Sigma](m)·OK² + ½[Sigma](m·KV²); (8)
i.e. the total kinetic energy is equal to the kinetic energy of the
whole mass supposed concentrated at G and moving with this point,
together with the kinetic energy of the motion relative to G. The latter
may be called the _internal kinetic energy_ of the system. Analytically
we have
_ _
| /d[|x]\² /d[|y]\² /d[|z]\ |
½[Sigma]{m([.x]² + [.y]² + [.z]²)} = ½[Sigma](m)·| ( ----- ) + ( ----- ) + ( ----- ) |
|_ \ dt / \ dt / \ dt / _|
+ ½[Sigma] {m([zeta]² + [.eta]² + [zeta]²)}. (9)
There is also an analogue to Lagrange's Second Theorem, viz.
[Sigma][Sigma] (m_p m_q·V_p V_q²)
½[Sigma](m·KV²) = ½ --------------------------------- (10)
[Sigma]m
which expresses the internal kinetic energy in terms of the relative
velocities of the several pairs of particles. This formula is due to
Möbius.
The preceding theorems are purely kinematical. We have now to consider
the effect of the forces acting on the particles. These may be divided
into two categories; we have first, the _extraneous forces_ exerted on
the various particles from without, and, secondly, the mutual or
_internal forces_ between the various pairs of particles. It is assumed
that these latter are subject to the law of equality of action and
reaction. If the equations of motion of each particle be formed
separately, each such internal force will appear twice over, with
opposite signs for its components, viz. as affecting the motion of each
of the two particles between which it acts. The full working out is in
general difficult, the comparatively simple problem of "three bodies,"
for instance, in gravitational astronomy being still unsolved, but some
general theorems can be formulated.
The first of these may be called the _Principle of Linear Momentum_. If
there are no extraneous forces, the resultant linear momentum is
constant in every respect. For consider any two particles at P and Q,
acting on one another with equal and opposite forces in the line PQ. In
the time [delta]t a certain impulse is given to the first particle in
the direction (say) from P to Q, whilst an equal and opposite impulse is
given to the second in the direction from Q to P. Since these impulses
produce equal and opposite momenta in the two particles, the resultant
linear momentum of the system is unaltered. If extraneous forces act, it
is seen in like manner that the resultant linear momentum of the system
is in any given time modified by the geometric addition of the total
impulse of the extraneous forces. It follows, by the preceding kinematic
theory, that the mass-centre G of the system will move exactly as if the
whole mass were concentrated there and were acted on by the extraneous
forces applied parallel to their original directions. For example, the
mass-centre of a system free from extraneous force will describe a
straight line with constant velocity. Again, the mass-centre of a chain
of particles connected by strings, projected anyhow under gravity, will
describe a parabola.
The second general result is the _Principle of Angular Momentum_. If
there are no extraneous forces, the moment of momentum about any fixed
axis is constant. For in time [delta]t the mutual action between two
particles at P and Q produces equal and opposite momenta in the line PQ,
and these will have equal and opposite moments about the fixed axis. If
extraneous forces act, the total angular momentum about any fixed axis
is in time [delta]t increased by the total extraneous impulse about that
axis. The kinematical relations above explained now lead to the
conclusion that in calculating the effect of extraneous forces in an
infinitely short time [delta]t we may take moments about an axis passing
through the instantaneous position of G exactly as if G were fixed;
moreover, the result will be the same whether in this process we employ
the true velocities of the particles or merely their velocities relative
to G. If there are no extraneous forces, or if the extraneous forces
have zero moment about any axis through G, the vector which represents
the resultant angular momentum relative to G is constant in every
respect. A plane through G perpendicular to this vector has a fixed
direction in space, and is called the _invariable plane_; it may
sometimes be conveniently used as a plane of reference.
For example, if we have two particles connected by a string, the
invariable plane passes through the string, and if [omega] be the
angular velocity in this plane, the angular momentum relative to G is
m1[omega]1r1·r1 + m2[omega]r2·r2 = (m1r1² + m2r2²)[omega],
where r1, r2 are the distances of m1, m2 from their mass-centre G.
Hence if the extraneous forces (e.g. gravity) have zero moment about
G, [omega] will be constant. Again, the tension R of the string is
given by
m1m2
R = m1[omega]²r1 = ------- [omega]²a,
m1 + m2
where a = r1 + r2. Also by (10) the internal kinetic energy is
m1m2
½ ------- [omega]²a².
m1 + m2
The increase of the kinetic energy of the system in any interval of time
will of course be equal to the total work done by all the forces acting
on the particles. In many questions relating to systems of discrete
particles the internal force R_pq (which we will reckon positive when
attractive) between any two particles m_p, m_q is a function only of the
distance r_pq between them. In this case the work done by the internal
forces will be represented by
_
/
-[Sigma] | R_(pg) dr_(pq),
_/
when the summation includes every pair of particles, and each integral
is to be taken between the proper limits. If we write
_
/
V = [Sigma] | R_(pq) dr_(pq), (11)
_/
when r_pq ranges from its value in some standard configuration A of the
system to its value in any other configuration P, it is plain that V
represents the work which would have to be done in order to bring the
system from rest in the configuration A to rest in the configuration P.
Hence V is a definite function of the configuration P; it is called the
_internal potential energy_. If T denote the kinetic energy, we may say
then that the sum T + V is in any interval of time increased by an
amount equal to the work done by the extraneous forces. In particular,
if there are no extraneous forces T + V is constant. Again, if some of
the extraneous forces are due to a conservative field of force, the work
which they do may be reckoned as a diminution of the potential energy
relative to the field as in § 13.
§ 16. _Kinetics of a Rigid Body. Fundamental Principles._--When we pass
from the consideration of discrete particles to that of continuous
distributions of matter, we require some physical postulate over and
above what is contained in the Laws of Motion, in their original
formulation. This additional postulate may be introduced under various
forms. One plan is to assume that any body whatever may be treated as if
it were composed of material particles, i.e. mathematical points endowed
with inertia coefficients, separated by finite intervals, and acting on
one another with forces in the lines joining them subject to the law of
equality of action and reaction. In the case of a rigid body we must
suppose that those forces adjust themselves so as to preserve the mutual
distances of the various particles unaltered. On this basis we can
predicate the principles of linear and angular momentum, as in § 15.
An alternative procedure is to adopt the principle first formally
enunciated by J. Le R. d'Alembert and since known by his name. If x, y,
z be the rectangular co-ordinates of a mass-element m, the expressions
m[:x], m[:y], m[:z] must be equal to the components of the total force
on m, these forces being partly extraneous and partly forces exerted on
m by other mass-elements of the system. Hence (m[:x], m[:y], m[:z]) is
called the actual or _effective_ force on m. According to d'Alembert's
formulation, the extraneous forces together with the _effective forces
reversed_ fulfil the statical conditions of equilibrium. In other words,
the whole assemblage of effective forces is statically equivalent to the
extraneous forces. This leads, by the principles of § 8, to the
equations
[Sigma](m[:x]) = X, [Sigma](m[:y]) = Y, [Sigma](m[:z]) = Z, \
> (1)
[Sigma]{m(y[:z] - z[:y]) = L, [Sigma]{m(z[:x] - x[:z]) = M, [Sigma]{m(x[:y] - y[:x]) = N, /
where (X, Y, Z) and (L, M, N) are the force--and couple--constituents of
the system of extraneous forces, referred to O as base, and the
summations extend over all the mass-elements of the system. These
equations may be written
d d d
--- [Sigma](m[.x]) = X, --- [Sigma](m[.y]) = Y, --- [Sigma](m[.z]) = Z, \
dt dt dt | } (2)
> (2)
d d d |
--- [Sigma]{m(y[.z] - z[.y]) = L, --- [Sigma]{m(z[.x]-x[.z]) = M, --- [Sigma]{m(x[.y] - y[.x]) = N, /
dt dt dt
and so express that the rate of change of the linear momentum in any
fixed direction (e.g. that of Ox) is equal to the total extraneous force
in that direction, and that the rate of change of the angular momentum
about any fixed axis is equal to the moment of the extraneous forces
about that axis. If we integrate with respect to t between fixed limits,
we obtain the principles of linear and angular momentum in the form
previously given. Hence, whichever form of postulate we adopt, we are
led to the principles of linear and angular momentum, which form in fact
the basis of all our subsequent work. It is to be noticed that the
preceding statements are not intended to be restricted to rigid bodies;
they are assumed to hold for all material systems whatever. The peculiar
status of rigid bodies is that the principles in question are in most
cases sufficient for the complete determination of the motion, the
dynamical equations (1 or 2) being equal in number to the degrees of
freedom (six) of a rigid solid, whereas in cases where the freedom is
greater we have to invoke the aid of other supplementary physical
hypotheses (cf. ELASTICITY; HYDROMECHANICS).
The increase of the kinetic energy of a rigid body in any interval of
time is equal to the work done by the extraneous forces acting on the
body. This is an immediate consequence of the fundamental postulate, in
either of the forms above stated, since the internal forces do on the
whole no work. The statement may be extended to a system of rigid
bodies, provided the mutual reactions consist of the stresses in
inextensible links, or the pressures between smooth surfaces, or the
reactions at rolling contacts (§ 9).
§ 17. _Two-dimensional Problems._--In the case of rotation about a fixed
axis, the principles take a very simple form. The position of the body
is specified by a single co-ordinate, viz. the angle [theta] through
which some plane passing through the axis and fixed in the body has
turned from a standard position in space. Then d[theta]/dt, = [omega]
say, is the _angular velocity_ of the body. The angular momentum of a
particle m at a distance r from the axis is m[omega]r·r, and the total
angular momentum is [Sigma](mr²)·[omega], or I[omega], if I denote the
moment of inertia (§ 11) about the axis. Hence if N be the moment of the
extraneous forces about the axis, we have
d
--- (I[omega]) = N. (1)
dt
This may be compared with the equation of rectilinear motion of a
particle, viz. d/dt·(Mu) = X; it shows that I measures the inertia of
the body as regards rotation, just as M measures its inertia as regards
translation. If N = 0, [omega] is constant.
[Illustration: FIG. 72.]
[Illustration: FIG. 73.]
As a first example, suppose we have a flywheel free to rotate about a
horizontal axis, and that a weight m hangs by a vertical string from
the circumferences of an axle of radius b (fig. 72). Neglecting
frictional resistance we have, if R be the tension of the string,
I[.omega] = Rb, m[.u] = mg - R,
whence
mb²
b[.omega] = ------- (2)
1 + mb²
This gives the acceleration of m as modified by the inertia of the
wheel.
A "compound pendulum" is a body of any form which is free to rotate
about a fixed horizontal axis, the only extraneous force (other than
the pressures of the axis) being that of gravity. If M be the total
mass, k the radius of gyration (§ 11) about the axis, we have
d / d[theta]\
--- ( Mk² -------- ) = -Mgh sin [theta], (3)
dt \ dt /
where [theta] is the angle which the plane containing the axis and the
centre of gravity G makes with the vertical, and h is the distance of
G from the axis. This coincides with the equation of motion of a
simple pendulum [§ 13 (15)] of length l, provided l = k²/h. The plane
of the diagram (fig. 73) is supposed to be a plane through G
perpendicular to the axis, which it meets in O. If we produce OG to P,
making OP = l, the point P is called the _centre of oscillation_; the
bob of a simple pendulum of length OP suspended from O will keep step
with the motion of P, if properly started. If [kappa] be the radius of
gyration about a parallel axis through G, we have k² = [kappa]² + h²
by § 11 (16), and therefore l = h + [kappa]²/h, whence
GO·GP = [kappa]². (4)
This shows that if the body were swung from a parallel axis through P
the new centre of oscillation would be at O. For different parallel
axes, the period of a small oscillation varies as [root]l, or
[root](GO + OP); this is least, subject to the condition (4), when GO
= GP = [kappa]. The reciprocal relation between the centres of
suspension and oscillation is the basis of Kater's method of
determining g experimentally. A pendulum is constructed with two
parallel knife-edges as nearly as possible in the same plane with G,
the position of one of them being adjustable. If it could be arranged
that the period of a small oscillation should be exactly the same
about either edge, the two knife-edges would in general occupy the
positions of conjugate centres of suspension and oscillation; and the
distances between them would be the length l of the equivalent simple
pendulum. For if h1 + [kappa]²/h1 = h2 + [kappa]²/h2, then unless h1 =
h2, we must have [kappa]² = h1h2, l = h1 + h2. Exact equality of the
two observed periods ([tau]1, [tau]2, say) cannot of course be secured
in practice, and a modification is necessary. If we write l1 = h1 +
[kappa]²/h1, l2 = h2 + [kappa]²/h2, we find, on elimination of
[kappa],
l1 + l2 l1 - l2
½ ------- + ½ ------- = 1,
h1 + h2 h1 - h2
whence
4[pi]² ½ ([tau]1² + [tau]2²) ½ ([tau]1² - [tau]2²)
------ = --------------------- + --------------------- (5)
g h1 + h2 h1 - h2
The distance h1 + h2, which occurs in the first term on the right hand
can be measured directly. For the second term we require the values of
h1, h2 separately, but if [tau]1, [tau]2 are nearly equal whilst h1,
h2 are distinctly unequal this term will be relatively small, so that
an approximate knowledge of h1, h2 is sufficient.
As a final example we may note the arrangement, often employed in
physical measurements, where a body performs small oscillations about
a vertical axis through its mass-centre G, under the influence of a
couple whose moment varies as the angle of rotation from the
equilibrium position. The equation of motion is of the type
I[:theta] = -K[theta], (6)
and the period is therefore [tau] = 2[pi][root](I/K). If by the
attachment of another body of known moment of inertia I´, the period
is altered from [tau] to [tau]´, we have [tau]´ = 2[pi][root][(I +
I´)/K]. We are thus enabled to determine both I and K, viz.
I/I´ = [tau]²/([tau]´² - [tau]²), K = 4[pi]²[tau]²I/([tau]´² - [tau]²). (7)
The couple may be due to the earth's magnetism, or to the torsion of
a suspending wire, or to a "bifilar" suspension. In the latter case,
the body hangs by two vertical threads of equal length l in a plane
through G. The motion being assumed to be small, the tensions of the
two strings may be taken to have their statical values Mgb/(a + b),
Mga/(a + b), where a, b are the distances of G from the two threads.
When the body is twisted through an angle [theta] the threads make
angles a[theta]/l, b[theta]/l with the vertical, and the moment of the
tensions about the vertical through G is accordingly -K[theta], where
K = M gab/l.
For the determination of the motion it has only been necessary to use
one of the dynamical equations. The remaining equations serve to
determine the reactions of the rotating body on its bearings. Suppose,
for example, that there are no extraneous forces. Take rectangular axes,
of which Oz coincides with the axis of rotation. The angular velocity
being constant, the effective force on a particle m at a distance r from
Oz is m[omega]²r towards this axis, and its components are accordingly
-[omega]²mx, -[omega]²my, O. Since the reactions on the bearings must be
statically equivalent to the whole system of effective forces, they will
reduce to a force (X Y Z) at O and a couple (L M N) given by
X = -[omega]²[Sigma](mx) = -[omega]²[Sigma](m)[|x], Y = -[omega]²[Sigma](my) = -[omega]²[Sigma](m)[|y], Z = 0,
L = [omega]²[Sigma](myz), M = -[omega]²[Sigma](mzx), N = 0, (8)
where [|x], [|y] refer to the mass-centre G. The reactions do not
therefore reduce to a single force at O unless [Sigma](myz) = 0,
[Sigma](msx) = 0, i.e. unless the axis of rotation be a principal axis
of inertia (§ 11) at O. In order that the force may vanish we must also
have x, y = 0, i.e. the mass-centre must lie in the axis of rotation.
These considerations are important in the "balancing" of machinery. We
note further that if a body be free to turn about a fixed point O, there
are three mutually perpendicular lines through this point about which it
can rotate steadily, without further constraint. The theory of principal
or "permanent" axes was first investigated from this point of view by J.
A. Segner (1755). The origin of the name "deviation moment" sometimes
applied to a product of inertia is also now apparent.
[Illustration: FIG. 74.]
Proceeding to the general motion of a rigid body in two dimensions we
may take as the three co-ordinates of the body the rectangular Cartesian
co-ordinates x, y of the mass-centre G and the angle [theta] through
which the body has turned from some standard position. The components of
linear momentum are then M[.x], M[.y], and the angular momentum relative
to G as base is I[.theta], where M is the mass and I the moment of
inertia about G. If the extraneous forces be reduced to a force (X, Y)
at G and a couple N, we have
M[:x] = X, M[:y] = Y, I[:theta] = N. (9)