Chapter XVI: Part II: Kinetics (1)

§ 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.

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).

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.

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.

§ 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).

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.

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.

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.