ELASTICITY.
This is perhaps the most suitable place for a few remarks on the theory of " dimensions." (See also UNITS, DIMENSIONS or.) 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 LT1, 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 LT2. The unit of momentum is MLT1; the unit force generates unit momentum in unit time and is therefore denoted by MLT2. The unit of work on the same principles is ML2T2, 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 sl (l/g). For it can only depend on the mass m of the bob, the length 1 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 µ of equation (15). The dimensions of A/x2 are those of an acceleration; hence the dimensions of g are L2T2. Assuming that the time in question varies as asµv, whose dimensions are L2+3vT_2y, we must have x+3Y=o, 2y=1,
so that the time of falling will vary as ai/AI,2, 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
xtz = —zz' 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 tin a constant ratio to t', provided that
x x'
t2' t2x2. x2,
and that there is a suitable correspondence between the initial conditions. The relation (44) is equivalent to
x x'3
Al u 2'
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+St respectively. A vector OU drawn parallel to PQ, of length proportional to PQ/St
on any convenient scale, will represent the mean velocity in the interval it, i.e. a point moving with a constant velocity having the magnitude and direction indicated by this vector would 4 u
experience the same resultant displacement PQ in the same time. As St is indefinitely diminished, the vector OU will tend to a definite limit OV; this is adopted as the definition
provided
(43)
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 coordinate
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 coordinates of P it is easily proved that
dx dz
u=at,°=dt'w=at'
The momentum of a particle is the vector obtained by multiplying the velocity by the mass in. The impulse of a force in any infinitely small interval of time St is the product of the force into at; 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 St 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 St we have, by projection on the coordinate axes, b(mu) =XSt, and so on, or
mau=X may=Y mdw=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 = o, Y= mg; so that
d'x
ate =o, dt2 = g•
The solution is
x=At+B, y= I,gt2+Ct+D. (4) If the initial values of x, y, z, 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=o from the origin, with the component velocities us, vs, we have
x=uot, y=vat'lgt2.
Eliminating t we have the equation of the path, viz. vs g .
y=uox2u2
This is a parabola with vertical axis, of latusrectum zuo2/g. The range on a horizontal plane through 0 is got by putting y=o, viz. it is 2uovo/g. If we denote the resultant velocity at any instant h y s we have
$2 x2+3,2 =sot 2gy. .(7)
Another important example is that of a particle subject to an acceleration which is directed always towards a fixed point 0 and is proportional to the distance from O. The motion will evidently be in one plane, which we take as the plane z=o. Ifµ be the acceleration at unit distance, the component accelerations parallel to axes of x and y through 0 as origin will be
µx, Ay, whence
dzx d2
dt2 = µx, dt = µy. (8)
The solution is
x=A cos nl+B sin nt, y=C cos nt+D sin nt, (9)
where n= Jµ. 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 so be the velocity at P, we have, initially, x=a, y=o, x=o, jy=so; whence
x=a cos nt, y=b sin nt, (to) if b.= so/n. The path is therefore an ellipse of which a, b are conjugate semidiameters, and is described in the period 24dµ; moreover, the velocity at any point P is equal to ,/µ • OD, where OD is the semidiameter conjugate to OP. ,This type of motion is called elliptic harmonic. If the coordinate axes are the principal axes of the ellipse, the angle nt in (so) is identicalwith 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 27r ,/(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 0; the equations of small A motion then are
d"x x d2y o ate = gp,= (lt2 = g ,
where p,, p2, are the principal radii of curvature at O. The motion is therefore the resultant of two simple vibrations in perpendicular directions, of periods 2,r 3I (pug), 21r sl (p2/g). The circumstances are realized in Blackburn's P
pendulum," which consists of a weight P FIG. 65. 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 pl=CP, p2 =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 C times t, t + St; and let the normals at P, Q meet in C, making an angle &'. Let v (=s) be the velocity at P, v+Sv that at Q. In the time St the velocity parallel to the tangent at ~P
P changes from v to v+Sv, ulti YIG.66.
mately, and the tangential accelera
tion at P is therefore dv/dt or N. Again, the velocity parallel to the normal at P changes from o to vSl/i, ultimately, so that the normal acceleration is vdil,ldt. Since
dv dv ds dv d¢ d4i ds v2
dl ds dt=v, vat =Td dtp, (12)
where p is the radius of curvature of the path at P, the tangential and normal accelerations are also expressed by v dv/ds and v2/p, 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 ¢ be the inclination of the tangent to the horizontal, we have
2
mvas = mg sin ¢ = mgrs, pv = mg cos ¢+R. (13)
The former equation gives
v2 = C  2gy,
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, 4' its inclination to the downward vertical, we have Ss=104', so that v=ldt/,/dt. The tangential resolution then gives
z
ldis = g sin 4'.
If we multiply by 24/dt and integrate, we obtain
(&P) T cos ¢+const., (16)
&P/ l
which is seen to be equivalent to (14). If the pendulum oscillate between the limits 4'= =a, we have 2=M22Zpp (cos 4'—cos a) = (sing la—sing ilk) ; (17)
and, putting sin a¢ =sin 2a. sin 42, we find for the period (T) of a complete oscillation
T = ird—u•ay = I l f2l
4f d4, 4 g • Jo (1—sine Za.sin24')
=4'~ l • FI(sin 2la), (i8)
g
(I)
(3)
(5) (6)
v+Ev
(14)
(15)
in the notation of elliptic integrals. The function Fl (sin /3) was tabulated by A. M. Legendre for values of t3 ranging from 0° to 90°. The following table gives the period, for various amplitudes a, in terms of that of oscillation in an infinitely small arc [viz. 2*' / (l/g] as unit.
s= Is sin 4', (22) the equation (21) would assume the same form as § 12 (5). The motion along the arc would then be accurately simpleharmonic, and the period 2rJ (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 ',k 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 Sv and vbe, where Se is the angle between the directions of the tangents at P,P'. Since Se = Ss/p, where Ss =PP'=vat and p is the radius of principal curvature at P, the component accelerations along the tangent and principal normal are dv/dt and vde/dt, respectively, or vdv/ds and v2/p. 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
my Ts = C, (23)
where 'C is the tangential component of the force. Integrating with respect to s we find
jmvlz — I.mve =f so Cds; (24)PP', we have SV =  SSs or
S= ay. (27)
In particular, by taking PP' parallel to each of the (rectangular) coordinate axes in succession, we find
Xax' Yay ZOz'
The equation (24) or (26) now gives
1mv12 +Vi = smvoz+Vo; (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
Imvz+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 0, where r denotes distance from 0 we have V = f Krdr= 2 Krz+const., whence
2mv2+zKrz=const. (31) It has been seen that the orbit is in this case an ellipse;also that if we put t= K/m the velocity at any point P is v= Jt1. OD, where OD is the semidiameter 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 w about Oz, which is fixed. In the interval at the projections of the line joining the origin to any point (x, y, z) on the directions of the coordinate axes at time t are changed from x, y, z to (x+Sx) cos coat— (y+Sy) sin coat, (x+Sx) sin wat+ (y+ay) cos coat, z respectively. Hence the component velocities parallel to the instantaneous positions of the coordinate axes at time t are
u=x—wy, v=Y+wz, w=i. (32) In the same way we find that the component accelerations are it—am, v+wu, ea.
(33) Hence if w be constant the equations of motion take the forms
m(z—2wywet)=X, m(y+2ec —wzy)=Y, mz=Z. (34) These become identical with the equations of motion relative
to
fixed axes provided we introduce a fictitious force mw2r acting outwards from the axis of z, where r=J (x2+y2), and a second fictitious force 2mwv 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 composee, 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
of x can
The value also be obtained as an infinite series by expanding the integrand in (18) by the binomial theorem, and integrating term by term. Thus
air 7 air
•I 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 I.O co
1z Iz z
r=2r NI g• I+2isin2za+22i42sin4as+...#. (19) If a be small, an approximation (usually sufficient) is
r=2rJ (l/g) . (I + .'zaz).
In the extreme case of a = r, the, equation (17) is immediately integrable; thus the time from the lowest position is
t = J (l/g) . log tan (!r + 10. (20) This becomes infinite for 0= r, 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
dzs
ate=  g sin IA (21)
where ' is the inclination of the tangent to the horizontal. If sin >G were accurately and not merely approximately proportional to the arc s, say
Ns,
' Fin. 67.
i
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(3z+yy+22)=Xx+Y51+Z2, (25) whence, on integration with respect to t,
2m(xzIyz+iz) = f (Xz+Yj'+Zi)dt+const. (26) =f (Xdx+Ydy+Zdz)+const.
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 or drawn in any direction from P, and S be the force due to the field, resolved in the direction
(28)
about a vertical axis halfway between the points of attachment of
the upper string. The equations of small motion are then of the
type (35)
z—2wy—w2x= —p2x,Y +2wx—w2Y= — q2y.
This is satisfied by (36)
z=A cos (at+s), y=B sin (ot+e),
provided (37)
(az } wz—p2)A+2vwB=o,
2awA+ (02 }w2 — g2) B = o.
Eliminating the ratio A:B we have (38)
(02+w2 p2) (02+w2 —q2) 402w2 =o.
It is easily proved that the roots of this quadratic in Q2 are always real, and that they are moreover both positive unless w2 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, s. 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=o, y=o is unstable. This occurs only when the period (22r/w) of revolution of the arm lies between the two periods (2w/p, 2a/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 0 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 St has zero moment about 0, the same will be true of the additional momentum generated. Hence the moment of the momentum (considered as a localized vector) about 0 will be constant. In symbols, if v be the velocity and p the perpendicular from 0 to the tangent to the path,
pv=h, (1)
where h is a constant. If Ss be an element of the path, Os is twice the area enclosed by Ss and the radii drawn to its extremities from O. Hence if SA be this area, we have SA=2 Os= 4 hilt, or
Hence equal areas are swept over by the radius vector in equal times.
If P be the acceleration towards 0, we have
dv dr
vds =  Pds'
since dr/ds is the cosine of the angle between the directions of r and Ss. We will suppose that P is a function of r only; then integrating (3) we find
a v2 =  fPdr+const., (4) which is recognized as the equation of energy. Combining this
with (1) we have
h2
pz =C 2 JPdr, (5) 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 =12/r2, and
z
p~ =C+2T.
Now in a conic whose focus is at 0 we have 1 2 I
P=r a'
where 1 is half the latusrectum, 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
t= h2/µ, a==u/C. (8)
Since
h2 =µ (2 I) ,
v2 = 1\r
it appears that the orbit is an ellipse, parabola or hyperbola, according as v2 is less than, equal to, or greater than 2µ/r. Now it appears from (6) that 2µ/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 crab is swept over in the time
zrab 2ra
T= eh =
since h=µll1=µlbal 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 rP; th=p
us
z
In the case of an ellipse described about the centre as pole we have
a b2=a2+b2—r2; (12)
hence P=µr, if µ=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/v µ) we infer that the orbit will be elliptic whatever the initial circumstances. Also the period is 22rab/h=21r/ J i, as previously found.
Again, in the equiangular spiral we have p =r sin a, and therefore P = z/r3, if u =h2/sin2 a. 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 =s/ z. sin a. Similarly, in the case of a circle with the pole on the circumference we have p2=r?/2a, P=µ/r5, if u=8h2a2; but this orbit is not a general one for the law of the inverse fifth power.
In astronomical and other investigations relating to central forces it is often convenient to use polar coordinates with
Again, the velocities parallel and perpendicular to OP change in the time St from u, v to uvSB, v+uSB, ultimately. The component accelerations at P in these directions are therefore
du do der de 2 dt vdt = r
dt2 dt
dv do 1 d f zde (14) )
dc,+udc=y at rdtJ'
respectively.
In the case of a central force, with 0 as pole, the transverse, acceleration vanishes, so that
r2dO/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
2
ater(d)_P,
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 do2 +It hP z• (17)
If, for example, the law be that of the inverse square, we have P=µu2, and the solution is of the form
u=h2{1+ecos(o—a)}, (18)
where e, a are arbitrary constants. This is recognized as the polar equation of a conic referred to the focus, the half latusrectum being h2/z.
dA
dt 2
(2)
(6) (7)
(9)
(to)
(3) the direction of 0 increasing),
the centre of force as pole. va$w Let P, Q be the positions of a moving point at times t, t + St, and write OP=r, OQ=r+Sr, LPOQ=SO, 0 being any fixed origin. If u, v be the component velocities at P along and perpendicular to OP (in we have
u =lim St dt' v =lim. r SO= rdt.
(13)
(16)
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 r, and since it cannot vary discontinuously the apsidal angle in a nearly circular orbit must be constant. Equating the expression (30) to r/m, we find that f(a)=C/a', 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, the inclination of the direction of motion to some fixed direction, the polar coordinates of V may be taken to be v, Ii; hence the velocities of V along and perpendicular to OV will be dv/dt and vdtti/dt. These expressions therefore give the tangential and normal accelerations of P; cf. § 13 (12).
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 semidiameter 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 0 as base, we have first a linear momentum whose components referred to rectangular axes through 0 are
The law of the inverse cube P=µ u8 is interesting by way of contrast. The orbits may be divided into two classes according as le. 'al, i.e. according as the transverse velocity (hu) is greater or less than the velocity s/ µ.0 appropriate to a circular orbit at the same distance. In the former case the equation (17) takes the form
d 1'1+m2u=o, (19)
(21)
u=Ae'"e{Be. o (22) If A, B have the same sign, this is equivalent to
au = cosh mO, (23) if the origin of 0 be suitably adjusted; hence r has a maximum value a, 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 mO, (24) this has an asymptote parallel to 0 =o, 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=µ, we have d2u/d92=o, and
u=AB+B; (25) the orbit is therefore a " reciprocal spiral," except in the special case of A=o, 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 (16821716), and the various curves obtained are known as Cotes'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 apseline. If the force. is always the same at the same distance any apseline 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 apselines 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),
jig =2 r Pdr ; (26)
this determines the form of the critical orbit, as it is called. If
P=p/r", its polar equation is (27)
r"cos me=a'',
where m=l (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 de/dt between (15) and (16) we obtain
cl2r h2
d 2—Ta=P=—f(r),
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,
d2xh2 ( 3x) _ —f (a) —x f,(a). dt2 a' ~a
Hence if h and a be connected by the relation h2=a3f(a) proper to a circular orbit, we have
d 2 + { f'(a)+af(a) } x=o. (28)
If the coefficient of x be positive the variations of x are simpleharmonic, and x can remain permanently small; the circular orbit is then said to be stable. The condition for this may be written
da{a2f(a)}>o,
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 halfperiod of x is r/~ { f'(a)+3alf(a) }, and since the angular velocity in the orbit is h/a2, approximately, the apsidal angle is, ultimately,
SS a
r~ af'( )+3f(a)
or, in the case of f(a) =k/r", r/ / (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
z(mx), Z(my), m(m2) ; (1) its representative vector is the same whatever point 0 be chosen. Secondly, we have an angular momentum whose components are
z{m(ya—23,)1, E{m(xy—y)1, (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 § it, if we imagine the two configurations of the system there referred to to be those corresponding to the instants t, t+bt. Thus
E (m• sP) =E(m).Gst Analytically we have
Z(mx) — Z(mx) =(m). • (4) with two similar formulae.
the solution of which is
au=sin m (0—a). (20) The orbit has therefore two asymptotes, inclined at an angle rim. In the latter case the differential equation is of the form
du 2 =m2u,
so that
(29)
(30)
(3)
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 m'D of momentum is equivalent to a single localized vector Z(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
E{m(yz—zy)I=2(m). (ydta—iddi
+z{m(n3"—Pi) 1. (5)
since ~(m%)=o, (m)=o, 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
m(v+dv) concentrated at G and moving with this point. If in (5) we make 0 coincide with' the instantaneous position of G, we have z=o, and the theorem follows.
Finally, the rates of change of the
components of the angular momen
tum 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
the momenta of the system are equivalent to a linear momentum represented by a localized vector E(m).(u+Su) 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 St, since the perpendicular distance of G from the tangent line at G' is of the order (St)2. Analytically we have from (5),
dtz{m(yi—zy)}=z(m). (ydt2—zdt~) +dtz{m(s3'—3'n))• (6)
If we put x, y, s=o, the theorem is proved as regards axes parallel to Ox.
Next consider the kinetic energy of the system. If from a
fixed point 0 we draw vectors OV1, OV2 . . . to represent the velocities of the several particles m,, m2, . . . , and if we construct the vector
(7)
this will represent the velocity of the masscentre, by (3). We
find, exactly as in the proof of Lagrange's First Theorem (§ r1),
that (8)
ZE (m . OV2) _ ,E (m) . OK2+ aE (m . KV2) ;
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 (2) \ 2 1j
iztm(z2+y2+e)}=;z(m) z+kd1/ / +(!)2
{ E{m(i2+i2+32)I. (9)
There is also an analogue to Lagrange's Second Theorem, viz.
(m.KV2)=1EE(mpmq.VpVg2)~ (Io) Z(m)
which expresses the internal kinetic energy in terms of the relative velocities of the several pairs of particles. This formula is due to Mobius.
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 St 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 masscentre 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 masscentre of a system free from extraneous force will describe a straight line with constant velocity. Again, the masscentre pf a chain of
Z(m.OV) 0 = z(m)
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 bt 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 bt 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 bt 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 w be the angular velocity in this plane, the angular momentum relative to G is
m1COlrl . r1 +m2wrz . r2 = (mlrl2 f m2r22)w,
where r1, r2 are the distances of ml, m2 from their masscentre G. Hence if the extraneous forces (e.g. gravity) have zero moment about G, w will be constant. Again, the tension R of the string is given by
__ 'n12 R=mlw2rl m,+m2w2a,
where a=rl+r2. Also by (Io) the internal kinetic energy is . m,m2 w2a2
1mlFmi
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 Rp9 (which we will reckon positive when attractive) between any two particles m,, mg is a function only of the distance rPg between them. In this case the work done by the internal forces will be represented by
E f Rpgdrpq,
when the summation includes every pair of particles, and each integral is to be taken between the proper limits. If we write V =EfRpgdrpq, (II)
when rP9 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 rigidbody 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 coordinates of a masselement m, the expressions mx, my, mz 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 (mi, my, m2) 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
(mx) =X, Z(m)) = Y, E(m2) =Z,
(m(yzzy)}=L, {m(zxxz)} = M, {m(xyyx)} = N, (I) where (X, Y, Z) and (L, M, N) are the force—and couple—constituents of the system of extraneous forces, referred to 0 as base, and the summations extend over all the masselements of the system. These equations may be written
d (mx) = X, dt E(mJ) = Y, dtE(mt) = Z,
dE(m(yzzy)}=L, atE{m(zxxz)}=M, dtE{m(xyyx)}=N,
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. Twodimensionali 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 coordinate, viz. the angle B through which some plane passing through the axis and fixed in the body has turned from a standard position in space. Then d9/dt, =w say, is the angular velocity of the body. The angular momentum of a particle m at a distance r from the axis is maw. r, and the total angular momentum is E(mr2) . w, or Iw, if I denote the moment of inertia (§ rr) about the axis. Hence if N be the moment of the extraneous forces about the axis, we have
dt(Iw) = N. (I)
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= o, w is constant.
}
(2)
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, Iw=Rb, mit=mg—R,
whence
mb2
bl~ = mb2 g. (2)
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 (§ ii) about the axis, we have
d(Mk2de) _ —Nigh sin B, (3)
where 0 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 [§ r3 (i5)] of length 1, provided 1=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 =1, the point P is called the centre of oscillation; the bob of a simple pendulum of length OP suspended from 0 will keep step with the motion of P, if properly started. If K be the radius of gyration about a parallel axis through G, we have k2=K2+h2 by § Ii (i6), and therefore l=h+K2/h, whence
GO . GP =K2. (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 v 1, or v (GO}OP) ; this is least, subject to the condition (4), when GO=GP=K. 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 knifeedges 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 knifeedges would in general occupy the positions of conjugate centres of suspension and oscillation ; and the distances between them would be the length 1 of the equivalent simple pendulum. For if h,+K2/hi = h2+K2/h2, then unless h1=h2, we must have K2=hih2, lhi+h2. Exact equality of the two observed periods (TI, T2, say) cannot of course be secured in practice, and a modification is necessary. If we write li = hi + K2/hi, 12 = h2 + K2/h2, we find, on elimination of K,
11+12 11 12
3 hi { hZ h 2 hi—h2— I'
whence
44 7r2_ (r12 1_7.22) (rig — r22)
g hi + h2 + hi — h2
The distance hi+h2, which occurs in the first term on the right hand can be measured directly. For the second term we require the values of hi, h2 separately, but if ri, r2 are nearly equal whilst hi, h2 are distinctly unequal this term will be relatively small, so that an approximate knowledge of hi, 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 masscentre 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 A= —KB, (6) and the period is therefore 7 =21rsl(I/K). If by the attachment of another body of known moment of inertia I', the period is altered from r tor, we have 7'=21r,/t(I+I')/Kl. We are thus enabled to determine both I and K, viz.
I/I'T2), K=4,r2r2I/(r'2—r2). (7)
The couple may be due to the earth's magnetism, or to the torsionof 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 0 the threads make angles ae/l, be/1 with the vertical, and the moment of the tensions about the vertical through G is accordingly —KB, 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 mw2r towards this axis, and its components are accordingly —w2mx, —w2my, 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 0 and a couple (L M N) given by
X= —w2E(mx) = —w2E(m) , Y = — w2E(my) = —w2Z(m) y, Z = o, L=w2Z(myz), M=—w2X(mzx), N =o, (8)
where x, y refer to the masscentre G. The reactions do not therefore reduce to a single force at 0 unless E (myz) = o, (mzx) = o, i.e. unless the axis of rotation be a principal axis of inertia (§ II) at O. In order that the force may vanish we must also have z, y=o, i.e. the masscentre' 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 0, 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.
Proceeding to the general motion of a rigid body in two dimensions we may take as the three coordinates of the body the rectangular Cartesian coordinates x, y of the masscentre G and the angle 6 through which the body has turned from some
standard position. The components of linear momentum are then Mx, Mq, and the angular
momentum relative to G as base is I9, 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
Mx = X, My = Y, Ie = N. (9) If the extraneous forces have zero moment about G the angular
velocity 0 is constant. Thus a
circular disk projected under Mg
gravity in a vertical plane spins FIG. 74.
with constant angular velocity, whilst its centre describes a parabola.
We may apply the equations (9) to the case of a solid of revolution rolling with its axis horizontal on a plane of inclination a. If the axis of x be taken parallel to the slope of the plane, with x increasing downwards, we have
MX= Mg sin a—F, o=Mg cos a—R, MK2B=Fa (io)
where K is the radius of gyration about the axis of symmetry, a is the constant distance of G from the plane, and R, F are the normal and tangential components of the reaction of the plane, as shown in fig. 74. We have also the kinematical relation z=ao. Hence
a2  K2
x=x21 gsina,R=Mg cos a, F=K2+as Mg sin a. (ii)
The acceleration of G is therefore less than in the case of frictionless sliding in the ratio a2/(K2+a2). For a homogeneous sphe:e this ratio is , for a uniform circular cylinder or disk 3, for a circular hoop or a thin cylindrical shell 4. 
The equation of energy for a rigid body has already been stated (in effect) as a corollary from fundamental assumptions.
(5)
R
It may also be deduced from the principles of linear and angular momentum as embodied in the equations (9). We have
M(zx+YY)+IBA+XX+Vy+N6, (12)
whence, integrating with respect to t,
1M (x2+512)+2I92= f (Xdx+Ydy+NdO)+const.
The lefthand side is 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 (§ 15); and the righthand member represents the integral work done by the extraneous forces in the successive infinitesimal displacements into which the motion may be resolved.
The formula (13) may be easily verified in the case of the compound pendulum, or of the solid rolling down an incline. As another example, suppose we have a circular cylinder whose masscentre is at an excentric point, rolling on a horizontal plane. This includes the case of a compound pendulum in which the knifeedge is replaced by a cylindrical pin. If a be the radius of the cylinder, h the distance of G from its axis (0), K the radius of gyration about a longitudinal axis through G, and 0 the inclination of OG to the vertical, the kinetic energy is 1MK202+ 2M . CG2. 82, by § 3, since the
body is turning about the line of contact (C) as instantaneous axis, and the potential energy is—Mgh cos O. The equation of energy is therefore
End of Article: ELASTICITY 

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