ELASTICITY . This is perhaps the most suitable See also:

• PLACE (through Fr. from Lat. platea, street; Gr. IrAar6s, wide)

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 See also:

• PERIOD
• PERIOD (Gr. irepioSos, a going or way round, circuit, aepi, round, and &Sis, way, road)

§ 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 See also:

• DRAWN

Taking this as the plane xy, with the See also:

• AXIS (Lat. for " axle ")

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 See also:

• ELLIPSE (adapted from Gr. EXkei 'tc, a deficiency, iXXeirely, to fall behind)

It is sometimes convenient to resolve the accelerations in directions having a more See also:

• INTRINSIC (through Fr. intrinsique, from Lat. intrinsecus, inwardly; inter, within, secus, following, from root of sequi, to follow)

The tangential See also:

• RESOLUTION

The See also:

• DEVICE

(30) This applies to motion on a smooth curve, as well as to the See also:

• FREE

(34) These become identical with the equations of motion relative to fixed axes provided we introduce a fictitious force mw2r acting out-wards 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 See also:

• FRENCH
• FRENCH, DANIEL CHESTER (1850– )
• FRENCH, NICHOLAS (1604-1678)

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(xz-I-yz+iz) = f (Xz+Yj'+Zi)dt+const. (26) =f (Xdx+Ydy+Zdz)+const. If the axes be rectangular, this has the same See also:

• INTERPRETATION (from Lat. interpretari, to expound, explain, inter pres, an agent, go-between, interpreter; inter, between, and the root pret-, possibly connected with that seen either in Greek 4 p4'ew, to speak, or irpa-rrecv, to do)

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 -}-See also:

• W2(P — R1)

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 See also:

• AREA

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 See also:

• POLE (FAMILY)
• POLE, REGINALD (1500-1558)
• POLE, RICHARD DE LA (d. 1525)
• POLE, WILLIAM (1814—1900)

The component accelerations at P in these directions are therefore du do der de 2 dt -vd-t =- -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 ate-r(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 See also:

• DIFFERENTIAL

Moreover, the case n=2 is the only one in which the See also:

• CRITICAL (in alphabetical order of authors)

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. See also:

• KINETICS (from Gr. taveiv, to move)

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. See also:

• COTES, ROGER (1682-1716)

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 See also:

• VARIATIONS
• VARIATIONS, CALCULUS
• VARIATIONS, CALCULUS OF

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 d-u 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), See also:

• DTZ

. . , and if we construct the vector (7) this will represent the velocity of the mass-centre, by (3). We find, exactly as in the See also:

• PROOF (in M. Eng. preove, proeve, preve, &°c., from O. Fr . prueve, proeve, &c., mod. preuve, Late. Lat. proba, probate, to prove, to test the goodness of anything, probus, good)

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 See also:

• ASTRONOMY (from Gr. &arpov, a star, and v ,usiv, to classify or arrange)

For example, the mass-centre of a system free from extraneous force will describe a straight line with constant velocity. Again, the mass-centre pf a See also:

• CHAIN (through the O. Fr. citable, chcene, &c., from Lat. catena)

Also by (Io) the internal kinetic energy is . m,m2 w2a2 1ml-Fmi 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 See also:

• PLAIN (O. Fr. plain, from Lat. plenum)

One See also:

• PLAN
• PLAN (from Lat. planes, flat)

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 See also:

• PECULIAR

(I) This may be compared with the equation of rectilinear motion of a particle, viz. d/dt.(Mu) =X; it shows that I See also:

• MEASURES

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 See also:

• KATER, HENRY (1777-1835)

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 See also:

• EARTH
• EARTH (a word common to Teutonic languages, cf. Ger. Erde, Dutch aarde, Swed. and Dan. jord; outside Teutonic it appears only in the Gr. 'pq'e, on the ground; it has been connected by some etymologists with the Aryan root ar-, to plough, which is seen in
• EARTH, FIGURE OF
• EARTH, FIGURE OF THE

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 mass-centre G. The reactions do not there-fore 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 mass-centre' must See also:

• LIE, JONAS LAURITZ EDEMIL (1833—1908)
• LIE, MARIUS SOPHUS (1842–1899)

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 See also:

• UNIFORM

- 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 See also:

• LEFT