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momentum

Using her momentum, he swung her over his head.

309119Her body continued its momentum down the hill and she fell, twisting so that she wouldn't fall on the kid.

13985Hence the angular momentum of the part between A and B remains constant, or as much enters at B as leaves at A.

4839The velocity of propagation of a torsional disturbance along a wire of circular section may be found by the transfer of momentum method, remembering that we must now replace linear momentum by angular momentum.

4132But the tension at P is T, parallel to the tangent, and T sin 4 parallel to PM, and through this - T sin is the momentum passing out at P per second.

4135Read on to see how that momentum has built over time, and continues to build.

4032It was a long and wearing fight, in which the British lost 485 killed and wounded, and what was more serious, Lord Methuen (himself wounded) found that his force had exhausted its forward momentum, and that he would have to collect supplies and reinforcements on the Modder before fighting his next battle.

3631Consider a submarine boat or airship moving freely with the direction of the resultant momentum horizontal, and the axis at a slight inclination 0.

3326The pressure on CD is equal to the A C momentum which it receives per second.

3026Momentum (quantity of motion) is the product of mass and velocity.

2318If the velocity of a particle at A relative to the undisturbed parts is u from left to right, the velocity of the matter moving out at A is U - u, and the momentum carried out by the moving matter is p(U - u) 2.

2113Equating this to the momentum entering at B and subtracting P' from each X+W+p(U - u)2 =poU 2.

1914In fluid movements, he shook out a noose, swung it a few times to get momentum and then threw it at the limb.

1915Since the conditions in the region PQ remain always the same, the momentum perpendicular to AB entering the region at Q is equal to the momentum perpendicular to AB leaving the region at P. But, since the motion at Q is along AB, there is no momentum there perpendicular to AB.

1937She lit with one foot underneath her body, and the momentum of her fall threw her forward - over the ledge.

1834There is also the " external " applied pressure X, and the total momentum flowing out per second is X-I-P4-W-1-p(U - u)2.

1512There is also the " external " applied pressure X, and the total momentum flowing out per second is X-I-P4-W-1-p(U - u)2.

1512Now the transfer of momentum across a surface occurs in two ways, firstly by the carriage of moving matter through the surface, and secondly by the force acting between the matter on one side of the surface and the matter on the other side.

1513(2) y The rate of generation of momentum in the interior of S by the component of force, X per unit mass, is fffpXdxdydz, f pXdxdydz, (3) and by the pressure at the surface S is -f.

1211At B there is only the latter kind, and since the transfer of matter is powoU, where po is the undisturbed density and wo is the undisturbed cross-section, since its velocity is U the passage of momentum per second is powoUo 2.

1212Therefore the momentum entering through a square centimetre at B per second is equal to the momentum leaving through a square centimetre at A.

119Long previously Lord Kelvin himself came nearer this view, in offering the opinion that magnetism consisted, in some way, in the angular momentum of the material molecules, of which the energy of irregular translations constitutes.

119Consider, for example, a submarine boat under water; the inertia is different for axial and broadside motion, and may be represented by (1) c 1 =W+W'a, c2=W+W'/3' where a, R are numerical factors depending on the external shape; and if the C.G is moving with velocity V at an angle 4) with the axis, so that the axial and broadside component of velocity is u = V cos 0, v =V sin 4), the total momentum F of the medium, represented by the vector OF at an angle 0 with the axis, will have components, expressed in sec. Ib, F cos 0 =c 1 - = (W +W'a) V cos 43, F sin 0 = c 2.11 = (W +W'/3) V sin 4) .

1111But the matter to the right of A is also receiving momentum from the matter to the left of it at the rate indicated by the force across A.

1112But the matter to the right of A is also receiving momentum from the matter to the left of it at the rate indicated by the force across A.

1112The transfer of angular momentum through A is of two kinds - first, that due to the passage of rotating matter, and, secondly, that due to the couple with which matter to the right of A acts upon matter to the left of A.

108Substituting in the momentum equation, we obtain Pv 1 7V + y 2 I V / +PoU 2 I - v) V) = PoU2, whence U 2 = Po (I }-y21 U J .

1012In later memoirs Reynolds followed up this subject by proceeding to establish definitions of the velocity and the momentum and the energy at an element of volume of the molecular medium, with the precision necessary in order that the dynamical equations of the medium in bulk, based in the usual manner on these quantities alone, without directly considering thermal stresses, shall be strictly valid - a discussion in which the relation of ordinary molar mechanics to the more complete molecular theory is involved.

1013A violent gust strikes the plate, which is driven back and carried by its own momentum far past the position in which a steady wind of the same force would place it; by the time the motion has reached the pen it has been greatly exaggerated by the springiness of the connexion, and not only is the plate itself driven too far back, but also its position is wrongly recorded by the pen; the combined errors act the same way, and more than double the real maximum pressure may be indicated on the chart.

1015Taking the fixed direction parallel to the axis of x, the time-rate of increase of momentum, due to the fluid which crosses the surface, is - f'fpuq cos OdS = - f f (lpu 2 -+mpuv+npuw)dS, (1) which by Green's transformation is (d(uiu 2) dy dz dxdydz.

99When A is held still, and B rotated, centrifugal action sets up vortex currents in the water in the pockets; thus a continuous circulation is caused between B and A, and the consequent changes of momentum give rise to oblique reactions.

911Sulpicius Galba and others, and along with it the development of prose composition, went on with increased momentum till the age of Cicero.

912Since no angular momentum goes out on the whole Z nwra 4 d0/dx -?- 2 pwra 4 Ud0/dt = o.

89well as of the body from the vector OF to O'F' requires an impulse couple, tending to increase the angle F00', of magnitude, in sec. foot-pounds F.00'.sin FOO'=FVt sin (0-0), (4) equivalent to an incessant couple N=FV sin (0-0) = (F sin 0 cos 0-F cos 0 sin ¢)V = (c 2 -c i) (V /g) sin 0 cos 4) =W'(13-a)uv/g (5) This N is the couple in foot-pounds changing the momentum of the medium, the momentum of the body alone remaining the same; the medium reacts on the body with the same couple N in the opposite direction, tending when c 2 -c 1 is positive to set the body broadside to the advance.

810Since the condition of the medium between A and B remains constant, even though the matter is continually changing, the momentum possessed by the matter between A and B is constant.

810the moment of inertia of the body about the axis, denoted by But if is the moment of inertia of the body about a mean diameter, and w the angular velocity about it generated by an impluse couple M, and M' is the couple required to set the surrounding medium in motion, supposed of effective radius of gyration k', If the shot is spinning about its axis with angular velocity p, and is precessing steadily at a rate about a line parallel to the resultant momentum F at an angle 0, the velocity of the vector of angular momentum, as in the case of a top, is C i pµ sin 0- C2µ 2 sin 0 cos 0; (4) and equating this to the impressed couple (multiplied by g), that is, to gN = (c 1 -c 2)c2u 2 tan 0, (5) and dividing out sin 0, which equated to zero would imply perfect centring, we obtain C21 2 cos 0- (c 2 -c 1)c2u 2 sec 0 =o.

811But it has velocity U, and therefore momentum poU 2 is carried in.

75Denoting the effective inertia of the liquid parallel to Ox by aW' the momentum aW'U = 4)0W' (24) _ U i -AO' 25) in this way the air drag was calculated by Green for an ellipsoida pendulum.

76The time rate of increase of momentum of the fluid inside S is )dxdydz; (5) and (5) is the sum of (I), (2), (3), (4), so that /if (dpu+dpu2+dpuv +dpuw_ +d p j d xdyd z = o, (b)` dt dx dy dz dx / leading to the differential equation of motion dpu dpu 2 dpuv dpuv _ X_ (7) dt + dx + dy + dz with two similar equations.

78Lord Kelvin was thereby induced to identify magnetic force with rotation, involving, therefore, angular momentum in the aether.

78The total momentum moving in at B is therefore P+poU 2.

79Sometimes, however, a sharp incline occurring on an otherwise easy line is not reckoned as the ruling gradient, trains heavier than could be drawn up it by a single engine being helped by an assistant or " bank " engine; sometimes also " momentum " or " velocity " grades, steeper than the ruling gradient, are permitted for short distances in cases where a train can approach at full speed and thus surmount them by the aid of its momentum.

711These equations are proved by taking a line fixed in space, whose direction cosines are 1, then dt=mR-nQ,' d'-t = nP =lQ-mP. (5) If P denotes the resultant linear impulse or momentum in this direction P =lxl+mx2+nx3, ' dP dt xl+, d y t x2' x3 +1 dtl dt 2 +n dt3, =1 ('+m (dt2-x3P+x1R) ' +n ('-x1Q-{-x2P) ' '= IX +mY+nZ, / (7) for all values of 1, Next, taking a fixed origin and axes parallel to Ox, Oy, Oz through 0, and denoting by x, y, z the coordinates of 0, and by G the component angular momentum about 1"2 in the direction (1, G =1(yi-x2z+x3y) m 2-+xlz) n(y(y 3x 1 x3x y + x 2 x) (8) Differentiating with respect to t, and afterwards moving the fixed.

66The equations of motion can be established in a similar way by considering the rate of increase of momentum in a fixed direction of the fluid inside the surface, and equating it to the momentum generated by the force acting throughout the space 5, and by the pressure acting over the surface S.

69Now consider the momentum leaving at A.

69The equations of motion can be established in a similar way by considering the rate of increase of momentum in a fixed direction of the fluid inside the surface, and equating it to the momentum generated by the force acting throughout the space 5, and by the pressure acting over the surface S.

69Now consider the momentum leaving at A.

69The partial differential coefficient of T with respect to a component of velocity, linear or angular, will be the component of momentum, linear or angular, which corresponds.

610In addition there is a pressure between the layers of the medium, and if this pressure in the undisturbed parts of the medium is P, momentum P per second is being transferred from right to left across each square centimetre.

57In the momentum equation (4) we may now omit X and it becomes 0.+P(U - u) 2 =poU2.

57Thus if T is expressed as a quadratic function of U, V, W, P, Q, R, the components of momentum corresponding are dT dT dT (I) = dU + x2=dV, x3 =dW, dT dT dT Yi dp' dQ' y3=dR; but when it is expressed as a quadratic function of xi, 'x2, x3, yi, Y2, Y3, U = d, V= dx, ' w= ax dT Q_ dT dT dy 1 dy2 dy The second system of expression was chosen by Clebsch and adopted by Halphen in his Fonctions elliptiques; and thence the dynamical equations follow X = dt x2 dy +x3 d Y = ..., Z ..., (3) = dt1 -y2?y - '2dx3+x3 ' M =..

58Conversely, if the kinetic energy T is expressed as a quadratic function of x, x x3, y1, y2, y3, the components of momentum, the partial differential coefficient with respect to a momentum component will give the component of velocity to correspond.

45In the absence of a medium the inertia of the body to transtion is the same in all directions, and is measured by the (3) But the change of the resultant momentum F of the medium as.

45The material between A and B, though continually changing, is always in the same condition, and therefore the momentum within AB is constant.

45On the other hand, if the effects arose from balanced stresses set up inside the globe by the radiation, the effects on the vanes and on the case would be of the nature of action and reaction, so that the establishment of motion of the vanes in one direction would involve impulsion of the case in the opposite direction; but when the motion became steady there would no longer be any torque either on the vanes or on the case, and the latter would therefore come back to its previous position of equilibrium; finally, when the light was turned off, the decay of the motion of the vanes would involve impulsion of the case in the direction of their motion until the moment of the restoring torque arising from the suspension of the case had absorbed the angular momentum in the system.

46For the body alone the resultant of the components of momentum W V -cos andW V sin 0 is W V -sec. lb, acting along 00', and so is unaltered.

47Now momentum is transferred in two ways, viz.

48On the whole the air S within ABCD neither gains nor g D loses momentum, so that on the whole it receives as much through AB as it gives up to CD.

35will have moved from 0 to 0', where 00' = Vt; and at 0' the momentum is the same in magnitude as before, but its vector is displaced from OF to O'F'.

36If P is the undisturbed pressure and P+w the pressure at AB, the momentum entering through AB per second isJ01(P+w-+pu2)dt.

36If p is the density at A, and w the cross-section, then the momentum carried past A is pc,(U - u) 2.

36So that no angular momentum enters at B, and therefore on the whole none leaves at A.

36If P is the undisturbed pressure and P+w the pressure at AB, the momentum entering through AB per second isJ01(P+w-+pu2)dt.

36bdo 7rpb 2 (u, b 2 a2 Uibb +¢z), and the difference X-X 1 is the component momentum of the liquid in the interspace; with similar expressions for Y and Y1.

312Taking two planes x = =b, and considering the increase of momentum in the liquid between them, due to the entry and exit of liquid momentum, the increase across dy in the direction Oy, due to elements at P and P' at opposite ends of the diameter PP', is pdy (U - Ua 2 r2 cos 20 +mr i sin 0) (Ua 2 r 2 sin 2 0+mr 1 cos 0) + pdy (- U+Ua 2 r 2 cos 2 0 +mr1 sin 0) (Ua 2 r 2 sin 2 0 -mr 1 cos 0) =2pdymUr '(cos 0 -a 2 r 2 cos 30), (8) and with b tan r =b sec this is 2pmUdo(i -a 2 b2 cos 30 cos 0), (9) and integrating between the limits 0 = 27r, the resultant, as before, is 27rpmU.

319Hence the angular momentum conveyed per second outwards is 2prra 4 Ud0/dt.

25Modern theory accepts the deduction, but ascribes the momentum to the revolving ions in the molecules of matter traversed by the light; for the magneto-optic effect is present only in material media.

25If At seconds is the time during which the resistance of the air, R it), causes the velocity of the shot to fall Av(f/s), so that the velocity drops from v+2Av to v-2Av in passing through the mean velocity v, then (3) Rot = loss of momentum in second-pounds, =w(v-+ZAv)/g - w(v - 2 Av)/g = wAv/g so that with the value of R in (I), (4) At =wAv/nd2pg.

25It was time to pursue his duties relentlessly to continue to build momentum.

00The effect of an external circulation of vortex motion on the motion of a cylinder has been investigated in § 29; a similar procedure will show the influence of circulation through a hole in a solid, taking as the simplest illustration a ring-shaped figure, with uniplanar motion, and denoting by the resultant axial linear momentum of the circulation.

00As the ring is moved from 0 to 0' in time t, with velocity Q, and angular velocity R, the components of liquid momentum change from aM'U +E and SM'V along Ox and Oy to aM'U'+ and /3M'V' along O'x' and O'y', (I) the axis of the ring changing from Ox to O'x'; and U = Q cos 0, V = Q sin 0, U' =Q cos (o - Rt), V' =Q sin (0 - Rt), (2) so that the increase of the components of momentum, X 1, Y 1, and N1, linear and angular, are X 1 = (aM'U'+ 0 cos Rt - aM'U - - 1 3M'V' sin Rt =(a - (3)M'Q sin_(0 - Rt) sin Rt - ver Rt (3) Y 1 = (aM'U'+) sin Rt-[-13M'V' cos Rt - (3M'V = (a - (3) M'Q cos (0 - Rt) sin Rt +t sin RT, N1=[ - (aM'U'+E) sin (0 - Rt)+ 1 3M'V' cos (o - Rt)]OO' = [- (a - 1 3) M'Q cos (o - Rt) sin (o - Rt) - sin (o - Rt) ]Qt.

00We shall find the velocity of propagation, just as in previous cases, from the consideration of transfer of momentum.

00He supplied the momentum which was necessary to counteract the caution of Burghley and Elizabeth; but it was probably fortunate that his headstrong counsels were generally overruled by the circumspection of his sovereign.

00If the hounds jump at the brook, even though they fail to clear it, the rider may take it for granted that at that place the leap is within the capacity of any ordinary hunter in his stride; hence if, when going at three parts speed, a horse's feet come just right to take off, the mere momentum of his body would take him over a place 15 ft.

00But instead of returning to the concrete fact of the equivalence of momentum, by which each body moving makes the other move oppositely, he denied that bodies do reciprocally act on one another, and even that bodies as mutually resisting substances press one another apart in collision.

00The true order of discovery, however, was as follows: (a) Sir Christo p her Wren made many experiments before the Royal Society, which were afterwards repeated in a corrected form by Sir Isaac Newton in the Principia, experimentally proving that bodies of ascertained comparative weights, when suspended and impelled against one another, forced one another back by impressing on one another opposite changes of velocity inversely as their weights and therefore masses; that is, by impressing on one another equal and opposite changes of momentum.

00(b) Wallis showed that such bodies reduce one another to a joint mass with a common velocity equal to their joint momentum divided by their joint weights or masses.

00(d) Hence, first inductively and then deductively, the third law was originally discovered only as a law of collision or impact between bodies of ascertained weights and therefore masses, impressing on one another equal and opposite changes of momentum, and always reducing one another to a joint mass with a common velocity to begin with, apart from the subsequent effects of elasticity.

00From slow beginnings these factors kept gaining momentum until they compassed the overthrow of the mighty order of the Jesuits, and culminated in the revolutionary spoliation of the Church.

00What progress reform made during his pontificate was due to its acquired momentum, rather than to the zeal of the pope.

00Faster-moving particles diffusing from A to B carry their momentum with them, and tend to accelerate B; an equal number of slower particles diffusing from B to A act as a drag on A.

00The mechanism is the same, but in one case we have diffusion of momentum, in the other case diffusion of heat.

00This, at first sight, paradoxical result is explained by the fact that the mean free path of each molecule increases in the same proportion as the density is diminished, so that as the number of molecules crossing each square centimetre decreases, the distance to which each carries its momentum increases, and the total transfer of momentum is unaffected by variation of density.

00For the more equal distribution of the water over the surface of the beds from the conductor and feeders, small masses, such as stones or solid portions of earth or turf fastened with pins, are placed in them, in order to retard the momentum which the water may have acquired.

00When the angular momentum is too great for the usual spheroidal form to persist, this gives place to an ellipsoid with three unequal axes; this is succeeded by a pear-shaped form.

00As the simple star radiates heat and contracts, it retains its angular momentum; when this is too great for the spheroidal form to persist, the star may ultimatel y separate into two components, which are driven farther and farther apart by their mutual tides.

00The product mu of the mass into the velocity is called the momentum or (in Newtons phrase) the quantity of motion.

00If 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 ~(mu)=X.

00The unit of force implied in (I) is that which produces unit momentum in unit time.

00The statement that the increase of momentum is equal to the impulse is (it may be remarked) equivalent to Newtons own formulation of his Second Law.

00In 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.

00We 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.

00If two masses m1, nil moving in the same straight line impinge, with the result that the velocities are changed from u1, u2, to ui, ui, then, since the impulses on the two bodies must be equal and opposite, the total momentum is unchanged, i.e.

00The momentum of a particle is the vector obtained by multiplying the velocity by the mass in.

00Newtons 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.

00Since the impulse of the force in any element of time & has zero moment about 0, the same will be true of the additional momentum generated.

00Hence th,l moment of the momentum (considered as a localized vector) about 0 will be constant.

00Thus taking nny point 0 as base, we have first a linear momentum whose components referred to rectangular axes through 0 are ~(m~), Z(m~), ~(mb); - (I)

00Secondly, we have an angular momentum whose components are ~{m(y~z3)}, ~lm(z~xb)1, ~{m(xi?yi~)}, (2) these being the sums of the moments of the momenta of the several particles about the respective axes.

00The linear momentum is the same as if the whole mass wer(concentrated at the centre of mass G, and endowed with thi velocity of this point.

00Again, 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.

00The aggregate of the components intl of momentum is equivalent to a single localized vector ~(~n).

00This expresses that the moment of momentum about any fixed axis (e.g.

00Ox) 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 ~j concentrated at G and moving with this point.

00referred 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 Cr.

00At the instant t+t5t the momenta of the system are equivalent to a linear momentum represented by a localized vector ~(m).(U+U) in a line through G tangential to the path of G, together with a certain angular momentum.

00The first of these may be called the Principle of Linear Momentum.

00If there are no extraneous forces, the resultant linear momentum is constant in every respect.

00If 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 additiofi of the total impulse of the extraneous forces.

00The second general result is the Principle of Angular Momentum.

00If there are no extraneous forces, the moment of momentum about any fixed axis is constant.

00If extraneous forces act, the total angular momentum about any fixed axis is in time t increased by the total extraneous impulse about that axis.

00If 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.

00For 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 mibiri ri +m1o~~rz - r2 (miri2 +mirl2)c~,,

00On this basis we can predicate the principles of linear and angular momentum, as in 15.

00and so express that the rate of change of the linear momentum in any fixed direction (e.g.

00that 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.

00If we integrate with respect to I 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.

00The angular momentum of a particle m at a distance r from the axis is mwr.

00r, and the total angular momentum is ~(mr2).

00The components of linear momentum are then M~, Mi, and the angular momentum relative to G as base is JO, where M is the mass and I the moment of inertia about G.

00It may also be deduced from the principles of linear and angular momentum as embodied in the equations (9).

00It is to be carefully noticed that the axis of resultant angular momentum about 0 does not in general coincide with the instantaneous axis of rotation.

00The axis of resultant angular momentum is therefore normal to the tangent plane at J, and does not coincide with OJ unless the latter be a principal axis.

00Again, if F be the resultant angular momentum, so that ~ii~,ijs (10)

00The components ~, il, ~ of linear momentum are a~ aT

00whilst those of the relative angular momentum are given by (l).

00The slightest frictional forcessuch as the resistance of the aireven if they act in lines through the centre of the rolling sphere, and so do not directly affect its angular momentum, will cause the centre gradually to descend in an ever-widening spiral path.

00Again, the vector which represents the angular momentum with respect to 0 will be constant in every respect.

00where I is the resultant angular momentum about 0.

00For we have seen that r is constant when there are no extraneous forces; and r is evidently not affected by an instantaneous impulse which leaves the angular momentum Cr, about the axis of symmetry, unaltered.

00The components of angular momentum about the axis of the flywheel and about the vertical will be Cn and A ~ respectively, where A is the moment of inertia about any axis through the masscentre (or through the fixed point 0) perpendicular to that of symmetry.

00Again, the components of angular momentum about OC, OA are Cn,A sin 0~, and therefore the angular momentum (u, say) about OZ is pA sini 0 ~+Cn cosU.

00If 0 is constant the points C, A will in time at come to positions C, A such that CC = sin 0 i~, AA = cos 0 ~, and the angular momentum about OB will become Cn sin 0 ~ A sin 0 ~.

00Equating this to the rate of increase of the angular momentum about OB, investigated as above, we find (C+Ma2+A~ cos e)~~=Mg ~- cot 0, (4)

00In the case of the top, the equation of energy and the condition of constant angular momentum (~l) about the vertical OZ are sufficient to determine the motion of the axis.

00The components of angular momentum about Ox, Qy, 01 will be denoted as usual by X, u, v.

00Hence the altered angular momentum about Ox will be X+OX+(is+Ou)(rOt)+ (v+Oe)qOt.

00The former is, in fact, equal to 2T, and the latter to ~2, where T is the kinetic energy an.d r the resultant angular momentum.

00By an obvious analogy, the expressions OTfO4r may be called the generalized components of momentum; they are usually denoted by Pr, thus = OT/aq,.

00The latter equation expresses that the angular momentum mP sing O~t about the vertical OZ is constant.

00If)~, u, 1 be the components of momentum, we have X = AO, aT

00The meaning of these quantities is easily recognized; thus X is the angular momentum about a horizontal axis normal to the plane of 0, u is the angular momentum about the vertical OZ, and s is the angular momentum about the axis of symmetry..

00(The product wv/g of the mass of a body by its velocity is called its momentum; so that the effort required is found by dividing the increase of momentum by the time in which it is produced.)

00Towards the beginning of the century the first Oireachtas was held in Dublin; it was the equivalent of the Welsh Eisteddfod, and became an annual event, and from this time forward the movement (which had now added to its aims a new clause - the support of Irish industries) began to go forward of its own momentum.

00In the language of theoretical mechanics, the moment of momentum of the entire system is a constant quantity.

00His theory of the inclined plane, combined with his satisfactory definition of "momentum," led him towards the third law of motion.

00INVARIABLE PLANE, in celestial mechanics (see Astronomy), that plane on which the sum of the moments of momentum of all the bodies which make up a system is a maximum.

00We shall thus have a projected areal velocity, the product of which by the mass of the planet is the moment of momentum of the latter.

00But if, instead of rotating around PP, it rotates around some other axis, RR, making a small angle, POR, with the axis of figure PP; then it has been known since the time of Euler that the axis of rotation RR, if referred to the spheroid regarded as fixed, will gradually rotate round the axis of figure PP in a period defined in the following way: - If we put C = the moment of momentum of the spheroid around the axis of figure, and A = the corresponding moment around an axis passing through the equator EQ, then, calling one day the period of rotation of the spheroid, the axis RR will make a revolution around PP in a number of days represented by the fraction C/(C - A).

00Using her momentum, he swung her over his head.

00She lit with one foot underneath her body, and the momentum of her fall threw her forward - over the ledge.

00It was time to pursue his duties relentlessly to continue to build momentum.

00Her body continued its momentum down the hill and she fell, twisting so that she wouldn't fall on the kid.

00In fluid movements, he shook out a noose, swung it a few times to get momentum and then threw it at the limb.

00Her momentum carried them through to the other side.

00The collective mind of the Board of Directors had become the antithesis of the momentum Tesla had gained in his lifetime.

00The promise of media reform which gathered momentum in the 1980s and 90s unfortunately did not actualize into performance.

00angular momentum can be made to feel very real to a child with the use of simple props.

00angular momentum of the infalling gas is very low.

00angular momentum of the beam remains constant.

00Abstract Jones matrices describe the polarization, or spin angular momentum, of a light beam as it passes through an optical system.

00As a result, to conserve angular momentum, the rate of rotation must increase.

00Finally, if the infalling gas has too much angular momentum it cannot form circumstellar disks and all goes into a circumbinary disk.

00We devise an equivalent of the Jones matrix formulation for light possessing orbital angular momentum.

00angular momentum DWBA (Distorted Wave Born Approximation) cross sections for a range of transferred angular momenta.

00Madabhushi, S.P.G., Schofield, A.N., Lesley, S. A new stored angular momentum (SAM) based earthquake actuator.

00Spin: The intrinsic angular momentum possessed by fundamental particles - giving the appearance of them ' spinning ' .

00As more pairs are broken, the nucleons contribute more single-particle spin to the total angular momentum.

00The destination of the gas depends on its specific angular momentum relative to the binary.

00The large angular momentum in this massive disk leads to turbulence and increased interaction of the constituents.

00They can be used very generally to study the angular momentum and spectroscopic factors associated with specific single particle states.

00With momentum rebuilt, they recruited a new bassist.

00In another quick breakaway, Varney fed the overlapping Wilson, whose momentum made him a good bet to get past Ben Moore.

00I started to attend Relaxation classes which are held in an office block in the City Center where I go for my Momentum classes.

00Cgi-bin prnh nyf to wpte's significant Las Vegas Review Journal: Shooting Stars a pink commemorative to generate momentum.

00conservation of angular momentum Figure 5: Angular momentum of a spinning disk.

00continuity equation, momentum equation, energy equation ).

00continuum orbitals for each angular momentum can be included.

00entrained flow must balance the total momentum of the expelled fluid.

00The disagreement gained momentum and eventually the bishop of Carlisle and the royal exchequer were called upon to resolve the matter.

00turning the flywheel: building momentum by building the brand.

00forward momentum to the Courtauld Institute since our independence was achieved.

00gathered momentum sharply in the past year.

00gathering momentum from now on, as these two powerful families became government policemen.

00impart momentum to the plasma ions.

00integrationist countries would seek to maintain the momentum toward a political union.

00l p = h. If the momentum of the electrons is zero, then the wavelength they have is infinite.

00lass="ex">l p = h. If the momentum of the electrons is zero, then the wavelength they have is infinite.

00maintain the momentum.

00modifyStored Angular Momentum (SAM) earthquake actuator was modified to accommodate a self-contained swinging platform.

00momentum of each photon can be changed in direction without any change of speed.

00momentum of the particle.

00momentum of electrons in the page is scarcely a concern to the reader.

00The global economic recovery has gathered less momentum than forecast in Budget 2002.

00Our strategy is gaining positive momentum across the business.

00They also have their sights on running the New York Marathon next year to maintain funding momentum.

00Mr Douglas has organized an intensive program of more than two events a month to sustain momentum in the first year.

00Happily, George judged things correctly and regained some forward momentum before stepping back too far.

00The upward swing enjoyed over the past three years is losing momentum.

00This angular momentum leads to a magnetic moment of fixed size for each particle.

00His determined focus was such that his march toward more major triumph had begun to assume unstoppable momentum.

00We devise an equivalent of the Jones matrix formulation for light possessing orbital angular momentum.

00He suggested that the challenge ahead is to maintain an upward momentum.

00momentum tensor shown on the screen.

00momentum flux.

00momentum stocks in a snap and choose only the best every day.

00momentum equation, energy equation ).

00The rise of the Gordons and the Campbells gained a gathering momentum from now on, as these two powerful families became government policemen.

00mosey along at the momentum of the other frames or move up the gears into full Metaphor process using Clean Language proper.

00spin multiplicity must not change, i.e. D S = 0. Orbital Angular momentum quantum number L may change by 0 or ±1.

00newtons law states that the rate of change of momentum is equal to the force acting.

00office block in the City Center where I go for my Momentum classes.

00overconfident, momentum investor.

00Using the equation l p = h. If the momentum of the electrons is zero, then the wavelength they have is infinite.

00You should be very controlled and not allow the momentum to dictate the pace.

00Momentum The Planck mission is in ESA's implementation phase.

00polarization of the light is carried by the direction of the angular momentum or spin of the photons.

00For Hammond all contemporary craft potters, however diverse in method and intention, contributed to the momentum of the discipline.

00As the measurement of position, gets more precise, the value of the momentum gets more indefinite.

00recombination loss mechanism, by relaxing the normal momentum conservation requirements.

00Dramatic momentum of tour in sparks parks amp recreation.

00repositioning of the company which I am pleased to report is taking place with increasing momentum.

00Political momentum to build a missile shield for the entire United States is now strong.

00sometime, arrangements toward the commencement of legal clinic are so slipshod that the momentum is either lost or reduced considerably.

00They gather momentum around the homes and reach a climax at the Manor, followed by a brief sortie to the Cross.

00Discover momentum stocks in a snap and choose only the best every day.

00The ongoing momentum gained by the visitors eventually accumulated into an opening goal, scored by prolific striker Duncan Stoddart.

00They are pounded by surf created from the mighty Atlantic Ocean, which has the momentum from 3000 uninterrupted miles.

00sustain the momentum behind this initiative.

00The upward swing enjoyed over the past three years is losing momentum.

00A scalar field with a potential V, will have the energy momentum tensor shown on the screen.

00Abstract The purpose of this paper is to present an engineering assessment of a proposal for a deployment brake for a momentum exchange tether.

00troubleeston reply never gained any momentum with both openers departing without troubling the scorers.

00virtually unknown outside the party, Davis needs to declare early to get momentum going.

00The gathering was widely seen as a missed opportunity to inject the necessary urgency, fresh momentum and political will into the process.

00Clearly a gross violation of the law of conservation of momentum is implied.

00Once the momentum is broken, the habit will soon wither.

00The momentum had now shifted, and try as she might, Grinham could not wrest it back.

00A violent gust strikes the plate, which is driven back and carried by its own momentum far past the position in which a steady wind of the same force would place it; by the time the motion has reached the pen it has been greatly exaggerated by the springiness of the connexion, and not only is the plate itself driven too far back, but also its position is wrongly recorded by the pen; the combined errors act the same way, and more than double the real maximum pressure may be indicated on the chart.

00Sometimes, however, a sharp incline occurring on an otherwise easy line is not reckoned as the ruling gradient, trains heavier than could be drawn up it by a single engine being helped by an assistant or " bank " engine; sometimes also " momentum " or " velocity " grades, steeper than the ruling gradient, are permitted for short distances in cases where a train can approach at full speed and thus surmount them by the aid of its momentum.

00The momentum of their commercial enterprise and stalwart patriotism proved irresistible.

00It was a long and wearing fight, in which the British lost 485 killed and wounded, and what was more serious, Lord Methuen (himself wounded) found that his force had exhausted its forward momentum, and that he would have to collect supplies and reinforcements on the Modder before fighting his next battle.

00Taking the fixed direction parallel to the axis of x, the time-rate of increase of momentum, due to the fluid which crosses the surface, is - f'fpuq cos OdS = - f f (lpu 2 -+mpuv+npuw)dS, (1) which by Green's transformation is (d(uiu 2) dy dz dxdydz.

00(2) y The rate of generation of momentum in the interior of S by the component of force, X per unit mass, is fffpXdxdydz, f pXdxdydz, (3) and by the pressure at the surface S is -f.

00The time rate of increase of momentum of the fluid inside S is )dxdydz; (5) and (5) is the sum of (I), (2), (3), (4), so that /if (dpu+dpu2+dpuv +dpuw_ +d p j d xdyd z = o, (b)` dt dx dy dz dx / leading to the differential equation of motion dpu dpu 2 dpuv dpuv _ X_ (7) dt + dx + dy + dz with two similar equations.

00bdo 7rpb 2 (u, b 2 a2 Uibb +¢z), and the difference X-X 1 is the component momentum of the liquid in the interspace; with similar expressions for Y and Y1.

00Considered by itself, with the cylinders held fixed, the vortex sets up a circumferential velocity m/r on a radius r, so that the angular momentum of a circular filament of annular cross section dA is pmdA, and of the whole vortex is pm7r(b2-a2).

00Any circular filament can be started from rest by the application of a circumferential impulse 7rpmdr at each end of a diameter; so that a mechanism attached to the cylinders, which can set up a uniform distributed impulse rpm across the two parts of a diameter in the liquid, will generate the vortex motion, and react on the cylinder with an impulse couple-pmira 2 and pm7rb 2, having resultant pm7r(b 2 -a 2), and this couple is infinite when b = oo, as the angular momentum of the vortex is infinite.

00Taking two planes x = =b, and considering the increase of momentum in the liquid between them, due to the entry and exit of liquid momentum, the increase across dy in the direction Oy, due to elements at P and P' at opposite ends of the diameter PP', is pdy (U - Ua 2 r2 cos 20 +mr i sin 0) (Ua 2 r 2 sin 2 0+mr 1 cos 0) + pdy (- U+Ua 2 r 2 cos 2 0 +mr1 sin 0) (Ua 2 r 2 sin 2 0 -mr 1 cos 0) =2pdymUr '(cos 0 -a 2 r 2 cos 30), (8) and with b tan r =b sec this is 2pmUdo(i -a 2 b2 cos 30 cos 0), (9) and integrating between the limits 0 = 27r, the resultant, as before, is 27rpmU.

00Denoting the effective inertia of the liquid parallel to Ox by aW' the momentum aW'U = 4)0W' (24) _ U i -AO' 25) in this way the air drag was calculated by Green for an ellipsoida pendulum.

00The partial differential coefficient of T with respect to a component of velocity, linear or angular, will be the component of momentum, linear or angular, which corresponds.

00Conversely, if the kinetic energy T is expressed as a quadratic function of x, x x3, y1, y2, y3, the components of momentum, the partial differential coefficient with respect to a momentum component will give the component of velocity to correspond.

00Thus if T is expressed as a quadratic function of U, V, W, P, Q, R, the components of momentum corresponding are dT dT dT (I) = dU + x2=dV, x3 =dW, dT dT dT Yi dp' dQ' y3=dR; but when it is expressed as a quadratic function of xi, 'x2, x3, yi, Y2, Y3, U = d, V= dx, ' w= ax dT Q_ dT dT dy 1 dy2 dy The second system of expression was chosen by Clebsch and adopted by Halphen in his Fonctions elliptiques; and thence the dynamical equations follow X = dt x2 dy +x3 d Y = ..., Z ..., (3) = dt1 -y2?y - '2dx3+x3 ' M =..

00These equations are proved by taking a line fixed in space, whose direction cosines are 1, then dt=mR-nQ,' d'-t = nP =lQ-mP. (5) If P denotes the resultant linear impulse or momentum in this direction P =lxl+mx2+nx3, ' dP dt xl+, d y t x2' x3 +1 dtl dt 2 +n dt3, =1 ('+m (dt2-x3P+x1R) ' +n ('-x1Q-{-x2P) ' '= IX +mY+nZ, / (7) for all values of 1, Next, taking a fixed origin and axes parallel to Ox, Oy, Oz through 0, and denoting by x, y, z the coordinates of 0, and by G the component angular momentum about 1"2 in the direction (1, G =1(yi-x2z+x3y) m 2-+xlz) n(y(y 3x 1 x3x y + x 2 x) (8) Differentiating with respect to t, and afterwards moving the fixed.

00In the absence of a medium the inertia of the body to transtion is the same in all directions, and is measured by the (3) But the change of the resultant momentum F of the medium as.

00well as of the body from the vector OF to O'F' requires an impulse couple, tending to increase the angle F00', of magnitude, in sec. foot-pounds F.00'.sin FOO'=FVt sin (0-0), (4) equivalent to an incessant couple N=FV sin (0-0) = (F sin 0 cos 0-F cos 0 sin ¢)V = (c 2 -c i) (V /g) sin 0 cos 4) =W'(13-a)uv/g (5) This N is the couple in foot-pounds changing the momentum of the medium, the momentum of the body alone remaining the same; the medium reacts on the body with the same couple N in the opposite direction, tending when c 2 -c 1 is positive to set the body broadside to the advance.

00Consider a submarine boat or airship moving freely with the direction of the resultant momentum horizontal, and the axis at a slight inclination 0.

00Consider, for example, a submarine boat under water; the inertia is different for axial and broadside motion, and may be represented by (1) c 1 =W+W'a, c2=W+W'/3' where a, R are numerical factors depending on the external shape; and if the C.G is moving with velocity V at an angle 4) with the axis, so that the axial and broadside component of velocity is u = V cos 0, v =V sin 4), the total momentum F of the medium, represented by the vector OF at an angle 0 with the axis, will have components, expressed in sec. Ib, F cos 0 =c 1 - = (W +W'a) V cos 43, F sin 0 = c 2.11 = (W +W'/3) V sin 4) .

00will have moved from 0 to 0', where 00' = Vt; and at 0' the momentum is the same in magnitude as before, but its vector is displaced from OF to O'F'.

00For the body alone the resultant of the components of momentum W V -cos andW V sin 0 is W V -sec. lb, acting along 00', and so is unaltered.

00the moment of inertia of the body about the axis, denoted by But if is the moment of inertia of the body about a mean diameter, and w the angular velocity about it generated by an impluse couple M, and M' is the couple required to set the surrounding medium in motion, supposed of effective radius of gyration k', If the shot is spinning about its axis with angular velocity p, and is precessing steadily at a rate about a line parallel to the resultant momentum F at an angle 0, the velocity of the vector of angular momentum, as in the case of a top, is C i pµ sin 0- C2µ 2 sin 0 cos 0; (4) and equating this to the impressed couple (multiplied by g), that is, to gN = (c 1 -c 2)c2u 2 tan 0, (5) and dividing out sin 0, which equated to zero would imply perfect centring, we obtain C21 2 cos 0- (c 2 -c 1)c2u 2 sec 0 =o.

00The effect of an external circulation of vortex motion on the motion of a cylinder has been investigated in § 29; a similar procedure will show the influence of circulation through a hole in a solid, taking as the simplest illustration a ring-shaped figure, with uniplanar motion, and denoting by the resultant axial linear momentum of the circulation.

00As the ring is moved from 0 to 0' in time t, with velocity Q, and angular velocity R, the components of liquid momentum change from aM'U +E and SM'V along Ox and Oy to aM'U'+ and /3M'V' along O'x' and O'y', (I) the axis of the ring changing from Ox to O'x'; and U = Q cos 0, V = Q sin 0, U' =Q cos (o - Rt), V' =Q sin (0 - Rt), (2) so that the increase of the components of momentum, X 1, Y 1, and N1, linear and angular, are X 1 = (aM'U'+ 0 cos Rt - aM'U - - 1 3M'V' sin Rt =(a - (3)M'Q sin_(0 - Rt) sin Rt - ver Rt (3) Y 1 = (aM'U'+) sin Rt-[-13M'V' cos Rt - (3M'V = (a - (3) M'Q cos (0 - Rt) sin Rt +t sin RT, N1=[ - (aM'U'+E) sin (0 - Rt)+ 1 3M'V' cos (o - Rt)]OO' = [- (a - 1 3) M'Q cos (o - Rt) sin (o - Rt) - sin (o - Rt) ]Qt.

00Sulpicius Galba and others, and along with it the development of prose composition, went on with increased momentum till the age of Cicero.

00On the other hand, if the effects arose from balanced stresses set up inside the globe by the radiation, the effects on the vanes and on the case would be of the nature of action and reaction, so that the establishment of motion of the vanes in one direction would involve impulsion of the case in the opposite direction; but when the motion became steady there would no longer be any torque either on the vanes or on the case, and the latter would therefore come back to its previous position of equilibrium; finally, when the light was turned off, the decay of the motion of the vanes would involve impulsion of the case in the direction of their motion until the moment of the restoring torque arising from the suspension of the case had absorbed the angular momentum in the system.

00In later memoirs Reynolds followed up this subject by proceeding to establish definitions of the velocity and the momentum and the energy at an element of volume of the molecular medium, with the precision necessary in order that the dynamical equations of the medium in bulk, based in the usual manner on these quantities alone, without directly considering thermal stresses, shall be strictly valid - a discussion in which the relation of ordinary molar mechanics to the more complete molecular theory is involved.

00When A is held still, and B rotated, centrifugal action sets up vortex currents in the water in the pockets; thus a continuous circulation is caused between B and A, and the consequent changes of momentum give rise to oblique reactions.

00Since the condition of the medium between A and B remains constant, even though the matter is continually changing, the momentum possessed by the matter between A and B is constant.

00Therefore the momentum entering through a square centimetre at B per second is equal to the momentum leaving through a square centimetre at A.

00Now the transfer of momentum across a surface occurs in two ways, firstly by the carriage of moving matter through the surface, and secondly by the force acting between the matter on one side of the surface and the matter on the other side.

00But it has velocity U, and therefore momentum poU 2 is carried in.

00In addition there is a pressure between the layers of the medium, and if this pressure in the undisturbed parts of the medium is P, momentum P per second is being transferred from right to left across each square centimetre.

00The total momentum moving in at B is therefore P+poU 2.

00If the velocity of a particle at A relative to the undisturbed parts is u from left to right, the velocity of the matter moving out at A is U - u, and the momentum carried out by the moving matter is p(U - u) 2.

00Equating this to the momentum entering at B and subtracting P' from each X+W+p(U - u)2 =poU 2.

00In the momentum equation (4) we may now omit X and it becomes 0.+P(U - u) 2 =poU2.

00Substituting in the momentum equation, we obtain Pv 1 7V + y 2 I V / +PoU 2 I - v) V) = PoU2, whence U 2 = Po (I }-y21 U J .

00that the momentum issuing from A (fig.

00The pressure on CD is equal to the A C momentum which it receives per second.

00On the whole the air S within ABCD neither gains nor g D loses momentum, so that on the whole it receives as much through AB as it gives up to CD.

00Since the conditions in the region PQ remain always the same, the momentum perpendicular to AB entering the region at Q is equal to the momentum perpendicular to AB leaving the region at P. But, since the motion at Q is along AB, there is no momentum there perpendicular to AB.

00But the tension at P is T, parallel to the tangent, and T sin 4 parallel to PM, and through this - T sin is the momentum passing out at P per second.

00We shall find the velocity of propagation, just as in previous cases, from the consideration of transfer of momentum.

00The material between A and B, though continually changing, is always in the same condition, and therefore the momentum within AB is constant.

00Now momentum is transferred in two ways, viz.

00At B there is only the latter kind, and since the transfer of matter is powoU, where po is the undisturbed density and wo is the undisturbed cross-section, since its velocity is U the passage of momentum per second is powoUo 2.

00If p is the density at A, and w the cross-section, then the momentum carried past A is pc,(U - u) 2.

00The velocity of propagation of a torsional disturbance along a wire of circular section may be found by the transfer of momentum method, remembering that we must now replace linear momentum by angular momentum.

00Hence the angular momentum of the part between A and B remains constant, or as much enters at B as leaves at A.

00So that no angular momentum enters at B, and therefore on the whole none leaves at A.

00The transfer of angular momentum through A is of two kinds - first, that due to the passage of rotating matter, and, secondly, that due to the couple with which matter to the right of A acts upon matter to the left of A.

00Hence the angular momentum conveyed per second outwards is 2prra 4 Ud0/dt.

00Since no angular momentum goes out on the whole Z nwra 4 d0/dx -?- 2 pwra 4 Ud0/dt = o.

00Lord Kelvin was thereby induced to identify magnetic force with rotation, involving, therefore, angular momentum in the aether.

00Modern theory accepts the deduction, but ascribes the momentum to the revolving ions in the molecules of matter traversed by the light; for the magneto-optic effect is present only in material media.

00Long previously Lord Kelvin himself came nearer this view, in offering the opinion that magnetism consisted, in some way, in the angular momentum of the material molecules, of which the energy of irregular translations constitutes.

00If At seconds is the time during which the resistance of the air, R it), causes the velocity of the shot to fall Av(f/s), so that the velocity drops from v+2Av to v-2Av in passing through the mean velocity v, then (3) Rot = loss of momentum in second-pounds, =w(v-+ZAv)/g - w(v - 2 Av)/g = wAv/g so that with the value of R in (I), (4) At =wAv/nd2pg.

00He supplied the momentum which was necessary to counteract the caution of Burghley and Elizabeth; but it was probably fortunate that his headstrong counsels were generally overruled by the circumspection of his sovereign.

00If the hounds jump at the brook, even though they fail to clear it, the rider may take it for granted that at that place the leap is within the capacity of any ordinary hunter in his stride; hence if, when going at three parts speed, a horse's feet come just right to take off, the mere momentum of his body would take him over a place 15 ft.

00According to the mechanics of Newton, when two bodies collide each body makes the other move equally and oppositely; but it has become a convenient habit to express this concrete fact in abstract language by calling it the conservation of momentum, by talking of one body communicating its motion to the other; as if bodies exchanged motion as men do money.

00But instead of returning to the concrete fact of the equivalence of momentum, by which each body moving makes the other move oppositely, he denied that bodies do reciprocally act on one another, and even that bodies as mutually resisting substances press one another apart in collision.

00The true order of discovery, however, was as follows: (a) Sir Christo p her Wren made many experiments before the Royal Society, which were afterwards repeated in a corrected form by Sir Isaac Newton in the Principia, experimentally proving that bodies of ascertained comparative weights, when suspended and impelled against one another, forced one another back by impressing on one another opposite changes of velocity inversely as their weights and therefore masses; that is, by impressing on one another equal and opposite changes of momentum.

00(b) Wallis showed that such bodies reduce one another to a joint mass with a common velocity equal to their joint momentum divided by their joint weights or masses.

00(d) Hence, first inductively and then deductively, the third law was originally discovered only as a law of collision or impact between bodies of ascertained weights and therefore masses, impressing on one another equal and opposite changes of momentum, and always reducing one another to a joint mass with a common velocity to begin with, apart from the subsequent effects of elasticity.

00From slow beginnings these factors kept gaining momentum until they compassed the overthrow of the mighty order of the Jesuits, and culminated in the revolutionary spoliation of the Church.

00He made experiments, simultaneously with Wallis and Wren, on the collision of hard spherical bodies, and his statement of the results (1669) included a clear enunciation of the conservation of linear momentum, as demonstrated for these cases of collision, and apparently correct in certain other cases, mass being estimated by weight.

00What progress reform made during his pontificate was due to its acquired momentum, rather than to the zeal of the pope.

00The effect of each of these additions will be lessened by the future improvements in processes of manufacture, and more particularly by the progressive replacemerit of that ephemeral source of energy, coal, by the secular sources, the winds, waves, tides, sunshine, the earth's heat and, greatest of all, its momentum.

00Faster-moving particles diffusing from A to B carry their momentum with them, and tend to accelerate B; an equal number of slower particles diffusing from B to A act as a drag on A.

00The mechanism is the same, but in one case we have diffusion of momentum, in the other case diffusion of heat.

00This, at first sight, paradoxical result is explained by the fact that the mean free path of each molecule increases in the same proportion as the density is diminished, so that as the number of molecules crossing each square centimetre decreases, the distance to which each carries its momentum increases, and the total transfer of momentum is unaffected by variation of density.

00For the more equal distribution of the water over the surface of the beds from the conductor and feeders, small masses, such as stones or solid portions of earth or turf fastened with pins, are placed in them, in order to retard the momentum which the water may have acquired.

00When the angular momentum is too great for the usual spheroidal form to persist, this gives place to an ellipsoid with three unequal axes; this is succeeded by a pear-shaped form.

00As the simple star radiates heat and contracts, it retains its angular momentum; when this is too great for the spheroidal form to persist, the star may ultimatel y separate into two components, which are driven farther and farther apart by their mutual tides.

00The product mu of the mass into the velocity is called the momentum or (in Newtons phrase) the quantity of motion.

00If 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 ~(mu)=X.

00The unit of force implied in (I) is that which produces unit momentum in unit time.

00The statement that the increase of momentum is equal to the impulse is (it may be remarked) equivalent to Newtons own formulation of his Second Law.

00In 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.

00We 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.

00If two masses m1, nil moving in the same straight line impinge, with the result that the velocities are changed from u1, u2, to ui, ui, then, since the impulses on the two bodies must be equal and opposite, the total momentum is unchanged, i.e.

00The unit of momentum is MLT~ the unit force generates unit momentum in.

00The momentum of a particle is the vector obtained by multiplying the velocity by the mass in.

00Newtons 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.

00Since the impulse of the force in any element of time & has zero moment about 0, the same will be true of the additional momentum generated.

00Hence th,l moment of the momentum (considered as a localized vector) about 0 will be constant.

00Thus taking nny point 0 as base, we have first a linear momentum whose components referred to rectangular axes through 0 are ~(m~), Z(m~), ~(mb); - (I)

00Secondly, we have an angular momentum whose components are ~{m(y~z3)}, ~lm(z~xb)1, ~{m(xi?yi~)}, (2) these being the sums of the moments of the momenta of the several particles about the respective axes.

00The linear momentum is the same as if the whole mass wer(concentrated at the centre of mass G, and endowed with thi velocity of this point.

00Again, 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.

00The aggregate of the components intl of momentum is equivalent to a single localized vector ~(~n).

00This expresses that the moment of momentum about any fixed axis (e.g.

00Ox) 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 ~j concentrated at G and moving with this point.

00referred 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 Cr.

00At the instant t+t5t the momenta of the system are equivalent to a linear momentum represented by a localized vector ~(m).(U+U) in a line through G tangential to the path of G, together with a certain angular momentum.

00The first of these may be called the Principle of Linear Momentum.

00If there are no extraneous forces, the resultant linear momentum is constant in every respect.

00If 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 additiofi of the total impulse of the extraneous forces.

00The second general result is the Principle of Angular Momentum.

00If there are no extraneous forces, the moment of momentum about any fixed axis is constant.

00If extraneous forces act, the total angular momentum about any fixed axis is in time t increased by the total extraneous impulse about that axis.

00If 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.

00For 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 mibiri ri +m1o~~rz - r2 (miri2 +mirl2)c~,,

00On this basis we can predicate the principles of linear and angular momentum, as in 15.

00and so express that the rate of change of the linear momentum in any fixed direction (e.g.

00that 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.

00If we integrate with respect to I 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.

00The angular momentum of a particle m at a distance r from the axis is mwr.

00r, and the total angular momentum is ~(mr2).

00The components of linear momentum are then M~, Mi, and the angular momentum relative to G as base is JO, where M is the mass and I the moment of inertia about G.

00It may also be deduced from the principles of linear and angular momentum as embodied in the equations (9).

00It is to be carefully noticed that the axis of resultant angular momentum about 0 does not in general coincide with the instantaneous axis of rotation.

00The axis of resultant angular momentum is therefore normal to the tangent plane at J, and does not coincide with OJ unless the latter be a principal axis.

00Again, if F be the resultant angular momentum, so that ~ii~,ijs (10)

00The components ~, il, ~ of linear momentum are a~ aT

00whilst those of the relative angular momentum are given by (l).

00The slightest frictional forcessuch as the resistance of the aireven if they act in lines through the centre of the rolling sphere, and so do not directly affect its angular momentum, will cause the centre gradually to descend in an ever-widening spiral path.

00Again, the vector which represents the angular momentum with respect to 0 will be constant in every respect.

00where I is the resultant angular momentum about 0.

00For we have seen that r is constant when there are no extraneous forces; and r is evidently not affected by an instantaneous impulse which leaves the angular momentum Cr, about the axis of symmetry, unaltered.

00The components of angular momentum about the axis of the flywheel and about the vertical will be Cn and A ~ respectively, where A is the moment of inertia about any axis through the masscentre (or through the fixed point 0) perpendicular to that of symmetry.

00Again, the components of angular momentum about OC, OA are Cn,A sin 0~, and therefore the angular momentum (u, say) about OZ is pA sini 0 ~+Cn cosU.

00If 0 is constant the points C, A will in time at come to positions C, A such that CC = sin 0 i~, AA = cos 0 ~, and the angular momentum about OB will become Cn sin 0 ~ A sin 0 ~.

00Equating this to the rate of increase of the angular momentum about OB, investigated as above, we find (C+Ma2+A~ cos e)~~=Mg ~- cot 0, (4)

00In the case of the top, the equation of energy and the condition of constant angular momentum (~l) about the vertical OZ are sufficient to determine the motion of the axis.

00The components of angular momentum about Ox, Qy, 01 will be denoted as usual by X, u, v.

00Hence the altered angular momentum about Ox will be X+OX+(is+Ou)(rOt)+ (v+Oe)qOt.

00The former is, in fact, equal to 2T, and the latter to ~2, where T is the kinetic energy an.d r the resultant angular momentum.

00By an obvious analogy, the expressions OTfO4r may be called the generalized components of momentum; they are usually denoted by Pr, thus = OT/aq,.

00The latter equation expresses that the angular momentum mP sing O~t about the vertical OZ is constant.

00If)~, u, 1 be the components of momentum, we have X = AO, aT

00The meaning of these quantities is easily recognized; thus X is the angular momentum about a horizontal axis normal to the plane of 0, u is the angular momentum about the vertical OZ, and s is the angular momentum about the axis of symmetry..

00(The product wv/g of the mass of a body by its velocity is called its momentum; so that the effort required is found by dividing the increase of momentum by the time in which it is produced.)

00Towards the beginning of the century the first Oireachtas was held in Dublin; it was the equivalent of the Welsh Eisteddfod, and became an annual event, and from this time forward the movement (which had now added to its aims a new clause - the support of Irish industries) began to go forward of its own momentum.

00In the language of theoretical mechanics, the moment of momentum of the entire system is a constant quantity.

00His theory of the inclined plane, combined with his satisfactory definition of "momentum," led him towards the third law of motion.

00INVARIABLE PLANE, in celestial mechanics (see Astronomy), that plane on which the sum of the moments of momentum of all the bodies which make up a system is a maximum.

00We shall thus have a projected areal velocity, the product of which by the mass of the planet is the moment of momentum of the latter.

00But if, instead of rotating around PP, it rotates around some other axis, RR, making a small angle, POR, with the axis of figure PP; then it has been known since the time of Euler that the axis of rotation RR, if referred to the spheroid regarded as fixed, will gradually rotate round the axis of figure PP in a period defined in the following way: - If we put C = the moment of momentum of the spheroid around the axis of figure, and A = the corresponding moment around an axis passing through the equator EQ, then, calling one day the period of rotation of the spheroid, the axis RR will make a revolution around PP in a number of days represented by the fraction C/(C - A).

00Momentum Arts will continue to test exciting ways of delivering projects and research new ways of pushing the frontiers of the arts forward.

00We shall also investigate whether this mixing enhances the direct CHCC Auger recombination loss mechanism, by relaxing the normal momentum conservation requirements.

00Dramatic momentum of tour in sparks parks amp recreation.

00I would now like to turn to the repositioning of the Company which I am pleased to report is taking place with increasing momentum.

00The recording has been gathering serious momentum with present results causing a few people at Sugar Shack Records to salivate rather uncontrollably !

00Political momentum to build a missile shield for the entire United States is now strong.

00Sometime, arrangements toward the commencement of legal clinic are so slipshod that the momentum is either lost or reduced considerably.

00They gather momentum around the homes and reach a climax at the Manor, followed by a brief sortie to the Cross.

00The ongoing momentum gained by the visitors eventually accumulated into an opening goal, scored by prolific striker Duncan Stoddart.

00They are pounded by surf created from the mighty Atlantic Ocean, which has the momentum from 3000 uninterrupted miles.

00It is important that we now sustain the momentum behind this initiative.

00Abstract The purpose of this paper is to present an engineering assessment of a proposal for a deployment brake for a momentum exchange tether.

00The Neston reply never gained any momentum with both openers departing without troubling the scorers.

00Virtually unknown outside the party, Davis needs to declare early to get momentum going.

00The gathering was widely seen as a missed opportunity to inject the necessary urgency, fresh momentum and political will into the process.

00Clearly a gross violation of the law of conservation of momentum is implied.

00Once the momentum is broken, the habit will soon wither.

00The momentum had now shifted, and try as she might, Grinham could not wrest it back.

00Large companies do large product launches, and that is appropriate because they have the money, the platform, prior testing and the momentum.

00This lending process has gained momentum as the Internet continues to grow.

00As the green build movement gathers momentum, prospective home buyers will start to add sustainability to the top of their home shopping list.

00Momentum traders purchase a large volume of shares of a particular stock with the intention of selling all shares before the market closes.

00Hoping to continue the momentum set by the first two seasons, ABC has announced the third season of the popular Dancing with the Stars.

00So, there was a huge interest from young actors in LA and that developed a momentum to where we started to get people like Ben Savage and Autumn Reeser.

00While Ray has met her fair share of criticisms from gourmet foodies, she never loses momentum.

00The American version of the British hit had a slow start, but was renewed for a second season and quickly gained momentum.

00While it is reported that her move to L.A. was to follow her boyfriend at the time, it really was fate, because it was there that Garner's career really began gaining momentum.

00Reports state that Seinfeld swerved the small car to the right in an attempt to slow its forward momentum.

00The Adam Sandler biography starts to gain momentum after Sandler's move to Los Angeles in 1988.

00Matt Damon's career keeps gaining momentum and it's showing no signs of slowing down.

00Kristen Stewart's career has been gathering momentum for several years, but with the success of the Twilight movie series and the fan hysteria surrounding it, it seems like her popularity is exploding.

00Thousands of fans joined the group, which started to gain momentum.

00These sailboats catapult individuals with the explosive momentum of flying cannonballs.

00Organic farms and the organic movement in general have been gaining momentum in the U.S. for over a decade amid reports of health issues related to widespread pesticide use.

00Finding the right fitness clothing can make a big difference in keeping the momentum in your fitness program.

00This includes the ability to "drift" repeatedly through the straight portions of the tracks, sliding from the left to right and back again to pick up even more momentum.

00This ability was perfect for escaping precarious situations and quickly gaining momentum for loop-the-loops and other aerial sections.

00At 18 to 24 months of age toddlers come to understand that there are words for everything and their language development gains momentum.

00Despite this, the dances continued to gain momentum in Europe.

00Keep the momentum going by implementing a few of these tips into your daily routine!

00This brand has revolutionized swimwear and has maintained momentum as one of the most in-demand brands around.

00Star Wars never seems to lose momentum, especially with little (and full grown!) boys.

00The method has recently gained momentum among experimental home chefs, though there has not been an easy set up to cook sous vide at home, given the expensive equipment costs (upwards of several thousand dollars).

00The middle is where you are building the momentum in order to get the ending.

00Coach has never lost its momentum for being popular and always in style.

00The challenge for an Aries who is pursuing romance will be to maintain enough momentum to feed his interest level while striving to partner equally with another person.

00This method of gaining momentum in repayment is similar to a snowball rolling down a hill.

00The debt snowball method gets its name because the approach helps you build momentum in repaying debt in a way that's similar to a snowball getting bigger as it rolls down a hill.

00Reebok Easy Tones help wearers move their feet forward, enabling forward momentum and drive to finish longer runs faster.

00Try WatchTrueBlood.net, which allows you to watch episodes in HDTV quality, and also provides fans with the opportunity to repost on Twitter or Facebook to keep the momentum going.

00While tribal scarification was once widespread throughout Africa, the practice has been losing momentum over the last few decades.

00This technique has begun to gain some momentum in the West where cigar ash and other irritants are used to create scars that are purple in color.

00With no signs of losing momentum, Nike continued to dominate the sports industry, logically choosing to expand its product line in the 1990s to include accessories like high precision timepieces.

00Take care not to let momentum swing you back, as that might cause an injury.

00The popular men's Momentum pant is organic cotton.

00Consider the Momentum Yoga Short, which is a loose-fitting, 3/4-thigh length short is made from a 4-way stretch cotton-lycra blend.

00There's no way to do this hesitantly, especially since the base depends on the momentum of the flyer's jump to push her up into the air, feet held solidly together.

00Even though they're lifting you, if you use your arm strength and your momentum to assist with the lifts, everyone will have an easier time of it.

00These alternatives paved the way for the Momentum plan, which has replaced the FlexPoints plan and strives to nudge members into making healthier choices.

00Movements should be slow and controlled, letting the muscle do the work rather than momentum, and the movements should extend through the full range of motion for each exercise.

00If you're disappointed to find that the FlexPoints plan has been discontinued, you may want to consider the Weight Watchers Momentum plan.

00In many cases, you'll find that you're coaxed into eating more within the guidelines of the Momentum program.

00The Weight Watchers Momentum Plan is a relatively new weight loss plan from Weight Watchers.

00The Weight Watchers Momentum Plan incorporates the best aspects of Weight Watcher's previous two plans, the Flex Plan and the Core Plan.

00While customers used to chose one or the other of those two plans, now all Weight Watcher's clients are put on the Momentum Plan.

00The Weight Watchers Points system is the mainstay of all Weight Watchers plans, and it is no different with the Momentum Plan.

00One common criticism of the Momentum Plan is that it doesn't provide enough protein and is therefore less satiating than plans that rely more heavily on protein.

00If you have multiple food sensitivities and allergies, it may be best to talk with a nutritionist, who can help you incorporate your own specific needs into the Momentum Plan.

00Keeping your eyes on the ceiling can help prevent you from using your forward momentum to "fling" your shoulders forward.

00Conversely, it's also important not to lose the momentum by dilly-dallying between the warm up and the actual exercise.

00If you swing your leg, you are involving your hip flexors and using momentum.

00These allow you to rest and cause you to rely on momentum instead of muscle control.

00Controlled motions allow you to avoid using momentum and instead rely on full muscle usage.

00If you rapidly work through your pushups, momentum is propelling you through your set.

00The weight of the hoop can help keep the hoop against your back or front by slowing down the momentum of the hoop.

00Using an unweighted hoop can be a great workout all on its own, since it takes additional energy and momentum to keep the hoop going for extended periods of time.

00Be sure to use good posture and form with muscular endurance training to avoid using momentum to lift and lower, which increases the risk of injury and decreases the effectiveness of the exercise.

00Using momentum is not strength training.

00To make sure that you are relying on your muscles rather than momentum, use careful, controlled movements.

00This will help prevent you from using your arms to generate momentum.

00You may find working your entire body in one session a more effective approach to exercise by taking advantage of the momentum you build while working out.

00This forces you to eliminate momentum and pay attention to both the positive and negative (lifting and lowering) aspects of the motion, creating muscular strength and endurance.

00Look for a class that meets at least twice a week, as once-weekly classes tend to lose momentum, as do the students enrolled in them.

00It currently boasts over 80 million users and is only gaining in its momentum.

00Appearances on The Tonight Show, Late Night with Conan O'Brien, and Jimmy Kimmel Live continued the band's momentum, and their song "Spin" was chosen for the popular Madden 07 video game.

00Fox Reality TV Shows have continued to gain momentum over the years, as television primetime has evolved from sitcoms and crime dramas, into America's obsession with…itself.

00The story has enough momentum at this point to continue on without him.

00One office procedure gaining momentum is Thermage.

00Keep up the momentum with quality skincare and your oily complexion should be managed and kept well under control.

00We encourage people to support each other in their weight loss efforts, and want to reward them with products that keep their weight loss momentum rolling.

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The word usage examples above have been gathered from various sources to reflect current and historial usage. They do not represent the opinions of YourDictionary.com.