(42)
if p denotes the perpendicular drawn from O in the direction (λ, μ, ν) to a tangent plane of the ellipsoid
(43)
This is called theellipsoid of gyrationat O; it was introduced into the theory by J. MacCullagh. The ellipsoids (41) and (43) are reciprocal polars with respect to a sphere having O as centre.
If A = B = C, the momental ellipsoid becomes a sphere; all axes through O are then principal axes, and the moment of inertia is the same for each. The mass-system is then said to possess kinetic symmetry about O.
If all the masses lie in a plane (z = 0) we have, in the notation of (25), c2= 0, and therefore A = Mb2, B = Ma2, C = M(a2+ b2), so that the equation of the momental ellipsoid takes the formb2x2+ a2y2+ (a2+ b2) z2= ε4.(44)The section of this by the plane z = 0 is similar tox2+y2= 1,a2b2(45)which may be called themomental ellipseat O. It possesses the property that the radius of gyration about any diameter is half the distance between the two tangents which are parallel to that diameter. In the case of a uniform triangular plate it may be shown that the momental ellipse at G is concentric, similar and similarly situatedto the ellipse which touches the sides of the triangle at their middle points.Fig. 59.The graphical methods of determining the moment of inertia of a plane system of particles with respect to any line in its plane may be briefly noticed. It appears from § 5 (fig. 31) that the linear moment of each particle about the line may be found by means of a funicular polygon. If we replace the mass of each particle by its moment, as thus found, we can in like manner obtain the quadratic moment of the system with respect to the line. For if the line in question be the axis of y, the first process gives us the values of mx, and the second the value of Σ(mx·x) or Σ(mx2). The construction of a second funicular may be dispensed with by the employment of a planimeter, as follows. In fig. 59 p is the line with respect to which moments are to be taken, and the masses of the respective particles are indicated by the corresponding segments of a line in the force-diagram, drawn parallel to p. The funicular ZABCD ... corresponding to any pole O is constructed for a system of forces acting parallel to p through the positions of the particles and proportional to the respective masses; and its successive sides are produced to meet p in the points H, K, L, M, ... As explained in § 5, the moment of the first particle is represented on a certain scale by HK, that of the second by KL, and so on. The quadratic moment of the first particle will then be represented by twice the area AHK, that of the second by twice the area BKL, and so on. The quadratic moment of the whole system is therefore represented by twice the area AHEDCBA. Since a quadratic moment is essentially positive, the various areas are to taken positive in all cases. If k be the radius of gyration about p we findk2= 2 × area AHEDCBA × ON ÷ αβ,Fig. 60.where αβ is the line in the force-diagram which represents the sum of the masses, and ON is the distance of the pole O from this line. If some of the particles lie on one side of p and some on the other, the quadratic moment of each set may be found, and the results added. This is illustrated in fig. 60, where the total quadratic moment is represented by the sum of the shaded areas. It is seen that for a given direction of p this moment is least when p passes through the intersection X of the first and last sides of the funicular;i.e.when p goes through the mass-centre of the given system; cf. equation (15).
If all the masses lie in a plane (z = 0) we have, in the notation of (25), c2= 0, and therefore A = Mb2, B = Ma2, C = M(a2+ b2), so that the equation of the momental ellipsoid takes the form
b2x2+ a2y2+ (a2+ b2) z2= ε4.
(44)
The section of this by the plane z = 0 is similar to
(45)
which may be called themomental ellipseat O. It possesses the property that the radius of gyration about any diameter is half the distance between the two tangents which are parallel to that diameter. In the case of a uniform triangular plate it may be shown that the momental ellipse at G is concentric, similar and similarly situatedto the ellipse which touches the sides of the triangle at their middle points.
The graphical methods of determining the moment of inertia of a plane system of particles with respect to any line in its plane may be briefly noticed. It appears from § 5 (fig. 31) that the linear moment of each particle about the line may be found by means of a funicular polygon. If we replace the mass of each particle by its moment, as thus found, we can in like manner obtain the quadratic moment of the system with respect to the line. For if the line in question be the axis of y, the first process gives us the values of mx, and the second the value of Σ(mx·x) or Σ(mx2). The construction of a second funicular may be dispensed with by the employment of a planimeter, as follows. In fig. 59 p is the line with respect to which moments are to be taken, and the masses of the respective particles are indicated by the corresponding segments of a line in the force-diagram, drawn parallel to p. The funicular ZABCD ... corresponding to any pole O is constructed for a system of forces acting parallel to p through the positions of the particles and proportional to the respective masses; and its successive sides are produced to meet p in the points H, K, L, M, ... As explained in § 5, the moment of the first particle is represented on a certain scale by HK, that of the second by KL, and so on. The quadratic moment of the first particle will then be represented by twice the area AHK, that of the second by twice the area BKL, and so on. The quadratic moment of the whole system is therefore represented by twice the area AHEDCBA. Since a quadratic moment is essentially positive, the various areas are to taken positive in all cases. If k be the radius of gyration about p we find
k2= 2 × area AHEDCBA × ON ÷ αβ,
where αβ is the line in the force-diagram which represents the sum of the masses, and ON is the distance of the pole O from this line. If some of the particles lie on one side of p and some on the other, the quadratic moment of each set may be found, and the results added. This is illustrated in fig. 60, where the total quadratic moment is represented by the sum of the shaded areas. It is seen that for a given direction of p this moment is least when p passes through the intersection X of the first and last sides of the funicular;i.e.when p goes through the mass-centre of the given system; cf. equation (15).
Part II.—Kinetics
§ 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 + δt let the point be at Q, and let OQ = x + δx. Themean velocityof the point in the interval δt is δx/δt. The limiting value of this when δt is infinitely small, viz. dx/dt, is adopted as the definition of thevelocityat the instant t. Again, let u be the velocity at time t, u + δu that at time t + δt. The mean rate of increase of velocity, or themean acceleration, in the interval δt is then δu/δt. The limiting value of this when δt is infinitely small, viz., du/dt, is adopted as the definition of theaccelerationat 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 ẋ and the acceleration by u̇ or áº. 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 thecurve of positionsorspace-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 thecurve of velocitiesorvelocity-time curve. Its gradient represents the acceleration, and the area (∫u dt) 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 suitablemassorinertia-coefficientm. The productmuof the mass into the velocity is called themomentumor (in Newton’s phrase) thequantity 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
(1)
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 thedyne. Units of this kind are calledabsoluteon 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
mu′ − mu =∫t′tX dt.
(2)
The time-integral on the right hand is called theimpulseof the force on the interval t′ − t. The statement that the increase ofmomentum 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 = δt, and taking δt 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 u̇ or Ạis, according to (1), constant, and we have
(3)
say, the general solution of which is
x =1â„2αt2+ At + B.
(4)
The “arbitrary constants†A, B enable us to represent the circumstances of any particular case; thus if the velocity ẋ 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 about1â„2%. 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 μ be the acceleration at unit distance, the equation of motion becomes
(5)
the solution of which may be written in either of the forms
x = A cos σt + B sin σt, x = a cos (σt + ε),
(6)
where σ= √μ, and the two constants A, B or a, ε 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 2π/σ. The type of motion represented by (6) is of fundamental importance in the theory of vibrations (§ 23); it is called asimple-harmonicor (shortly) asimplevibration. If we imagine a point Q to describe a circle of radius a with the angular velocity σ, its orthogonal projection P on a fixed diameter AA′ will execute a vibration of this character. The angle σt + ε (or AOQ) is called thephase; the arbitrary elements a, ε are called theamplitudeandepoch(or initial phase), respectively. In the case of very rapid vibrations it is usual to specify, not theperiod(2π/σ), but its reciprocal thefrequency,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
(7)
provided the inertia of the spring itself be neglected. This becomes identical with (5) if we put μ = K/M; and the period is therefore 2π√(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 2π√(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
(8)
The motion is therefore simple-harmonic, of period τ = 2π√(l/g). This indicates an experimental method of determining g with considerable accuracy, using the formula g = 4π2l/τ2.
In the case of a repulsive force varying as the distance from the origin, the equation of motion is of the typed2x= μx,dt2(9)the solution of which isx = Aent+ Be−nt,(10)where n = √μ. 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 μ = 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 the case of a repulsive force varying as the distance from the origin, the equation of motion is of the type
(9)
the solution of which is
x = Aent+ Be−nt,
(10)
where n = √μ. 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 μ = 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
(11)
where σ1is prescribed. A particular solution is
(12)
This represents aforced oscillationwhose period 2π/σ1, coincides with that of the disturbing force; and the phase agrees with that of the force, or is opposed to it, according as σ12< or > μ;i.e.according as the imposed period is greater or less than the natural period 2π/√μ. The solution fails when the two periods agree exactly; the formula (12) is then replaced by
(13)
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 onlybe 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 ξ = a cos σt in a horizontal line, the equation of small motion of the bob ismẠ= −mgx − ξ,lorẠ+gx= gξ.ll(14)Fig.63.This is the same as if the point of suspension were fixed, and a horizontal disturbing force mgξ/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 μ be the acceleration at unit distance, we haveudu= −μdxx2(15)whenceu2=2μ+ C.x(16)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, μ/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 √(2ga). If we take as rough values a = 21 × 106feet, 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 = − 2μ/c, and thereforedx= u = −√ {2μ (c − x)},dtcx(17)the minus sign indicating motion towards the origin. If we put x = c cos21â„2φ, we findt =c3/2(φ + sin φ),√(8μ)(18)no additive constant being necessary if t be reckoned from the instant of starting, when φ = 0. The time t of reaching the origin (φ = Ï€) ist1=Ï€ c3/2.√(8μ)(19)This may be compared with the period of revolution in a circular orbit of radius c about the same centre of force, viz. 2Ï€c3/2/ √μ (§ 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.
A simple mechanical illustration is afforded by the pendulum. If the point of suspension have an imposed simple vibration ξ = a cos σt in a horizontal line, the equation of small motion of the bob is
or
(14)
This is the same as if the point of suspension were fixed, and a horizontal disturbing force mgξ/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 μ be the acceleration at unit distance, we have
(15)
whence
(16)
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, μ/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 √(2ga). If we take as rough values a = 21 × 106feet, 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 = − 2μ/c, and therefore
(17)
the minus sign indicating motion towards the origin. If we put x = c cos21â„2φ, we find
(18)
no additive constant being necessary if t be reckoned from the instant of starting, when φ = 0. The time t of reaching the origin (φ = π) is
(19)
This may be compared with the period of revolution in a circular orbit of radius c about the same centre of force, viz. 2πc3/2/ √μ (§ 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
(20)
which is equivalent to (1), with respect to x between the limits x0, x1, we obtain
1â„2mu12−1â„2mu02=∫x1x0X dx.
(21)
We recognize the right-hand member as theworkdone by the force X on the particle as the latter moves from the position x0to 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 product1â„2mu2is called thekinetic energyof 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 P0to rest in any assigned position P, will depend only on the position of P; it is called thestaticalorpotential energyof the particle with respect to the field, in the position P. Denoting this by V, we have δV − Xδx = 0, whence
(22)
The equation (21) may now be written
1â„2mu12+ V1=1â„2mu02+ 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 =∫x0Kx dx =1â„2Kx2, whence1â„2Mu2+1â„2Kx2= 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 mv2had been called by G. W. Leibnitz the “vis vivaâ€; the name “energy†was substituted by T. Young; finally the name “actual energy†was appropriated to the expression1â„2mv2by 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 formdu= g − ku2;dt(24)this shows that the velocity u will send asymptotically to a certain limit V (called theterminal velocity) such that kV2= g. The solution isu = V tanhgt,   x =V2log coshgt,VgV(25)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 havedu= −g − ku2,dt(26)the positive direction being now upwards. This leads totan−1u= tan−1u0−gt,   x =V2logV2+ u02,VVV2gV2+ u2(27)where u0is the velocity of projection. The particle comes to rest whent =Vtan−1u0,   x =V2log(1 +u02).gV2gV2(28)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 becomesẠ= −μx − kẋ.(29)If k2< 4μ the solution isx = a e−t/Ï„cos (σt + ε),(30)whereÏ„ = 2/k,   σ = √(μ −1â„4k2),(31)and the constants a, ε are arbitrary. This may be described as a simple harmonic oscillation whose amplitude diminishes asymptotically to zero according to the law e−t/Ï„. The constant Ï„ is called themodulus of decayof the oscillations; if it is large compared with 2Ï€/σ the effect of friction on the period is of the second order of small quantities and may in general be ignored. We have seen thata 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> 4μ, the solution of (29) is, in real form,x = a1e−t/Ï„1+ a2e−t/Ï„2,(32)where1/Ï„1, 1/Ï„2=1â„2k ± √(1â„4k2− μ).(33)The body now passes once (at most) through its equilibrium position, and the vibration is therefore styledaperiodic.To find the forced oscillation due to a periodic force we haveẠ+ kẋ + μx = Æ’ cos (σ1t + ε).(34)The solution isx =Æ’cos (σ1t + ε − ε1),R(35)providedR = { (μ − σ12)2+ k2σ12}1/2,   tan ε1=kσ1.μ − σ12(36)Hence the phase of the vibration lags behind that of the force by the amount ε1, which lies between 0 and1â„2Ï€ or between1â„2Ï€ and Ï€, according as σ12≶ μ. If the friction be comparatively slight the amplitude is greatest when the imposed period coincides with the free period, being then equal to Æ’/kσ1, 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/Ï„the influence of the initial conditions gradually disappears.
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
(24)
this shows that the velocity u will send asymptotically to a certain limit V (called theterminal velocity) such that kV2= g. The solution is
(25)
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
(26)
the positive direction being now upwards. This leads to
(27)
where u0is the velocity of projection. The particle comes to rest when
(28)
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
Ạ= −μx − kẋ.
(29)
If k2< 4μ the solution is
x = a e−t/τcos (σt + ε),
(30)
where
Ï„ = 2/k,   σ = √(μ −1â„4k2),
(31)
and the constants a, ε are arbitrary. This may be described as a simple harmonic oscillation whose amplitude diminishes asymptotically to zero according to the law e−t/τ. The constant τ is called themodulus of decayof the oscillations; if it is large compared with 2π/σ the effect of friction on the period is of the second order of small quantities and may in general be ignored. We have seen thata 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> 4μ, the solution of (29) is, in real form,
x = a1e−t/τ1+ a2e−t/τ2,
(32)
where
1/Ï„1, 1/Ï„2=1â„2k ± √(1â„4k2− μ).
(33)
The body now passes once (at most) through its equilibrium position, and the vibration is therefore styledaperiodic.
To find the forced oscillation due to a periodic force we have
Ạ+ kẋ + μx = ƒ cos (σ1t + ε).
(34)
The solution is
(35)
provided
(36)
Hence the phase of the vibration lags behind that of the force by the amount ε1, which lies between 0 and1â„2Ï€ or between1â„2Ï€ and Ï€, according as σ12≶ μ. If the friction be comparatively slight the amplitude is greatest when the imposed period coincides with the free period, being then equal to Æ’/kσ1, 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/Ï„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 ξ changes the velocity of a mass m from u to u′ we have
mu′ − mu = ξ.
(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, m2moving 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
m1u1′2+ 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
(42)
The further theoretical discussion of the subject belongs toElasticity.
This is perhaps the most suitable place for a few remarks on the theory of “dimensions.†(See alsoUnits, 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 thedyne, and the unit of work or energy theerg. 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 √(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 μ of equation (15). The dimensions of μ/x2are those of an acceleration; hence the dimensions of μ are L3T−2. Assuming that the time in question varies as axμy, whose dimensions are Lx+3yT−2y, we must have x + 3y = 0, −2y = 1, so that the time of falling will vary as a3/2/√μ, 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 equationsd2x= −μ,d2x′= −μ′,dt2x2dt′2x′2(43)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 thatx:x′=μ:μ′,t2t′2x2x′2(44)and that there is a suitable correspondence between the initial conditions. The relation (44) is equivalent tot : t′ =x3/2:x′3/2,μ1/2μ′1/2(45)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.
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 √(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 μ of equation (15). The dimensions of μ/x2are those of an acceleration; hence the dimensions of μ are L3T−2. Assuming that the time in question varies as axμy, whose dimensions are Lx+3yT−2y, we must have x + 3y = 0, −2y = 1, so that the time of falling will vary as a3/2/√μ, 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
(43)
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
(44)
and that there is a suitable correspondence between the initial conditions. The relation (44) is equivalent to
(45)
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 + δt respectively. A vectorOU>drawn parallel to PQ, of length proportional to PQ/δt on any convenient scale, will represent themean velocityin the interval δt,i.e.a point moving with a constant velocity having the magnitude and direction indicated by this vector would experience the same resultant displacementPQ>in the same time. As δt is indefinitely diminished, the vectorOU>will tend to a definite limitOV>; this is adopted as the definitionof thevelocityof the moving point at the instant t. ObviouslyOV>is parallel to the tangent to the path at P, and its magnitude is ds/dt, where s is the arc. If we projectOV>on the co-ordinate axes (rectangular or oblique) in the usual manner, the projections u, v, w are called thecomponent velocitiesparallel to the axes. If x, y, z be the co-ordinates of P it is easily proved that