momentum momentum

momentum Sentence Examples

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

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

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

• The 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.

• But 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.

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

• It 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.

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

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

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

• If 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.

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

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

• Since 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.

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

• There 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.

• There 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.

• Now 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.

• (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.

• At 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.

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

• Long 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.

• Consider, 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) .

• But 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.

• But 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.

• The 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.

• Substituting 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 .

• In 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.

• A 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.

• Taking 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.

• When 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.

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

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

• well 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.

• Since 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.

• the 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.

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

• Denoting 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.

• The 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.

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

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

• Sometimes, 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.

• These 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.

• The 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.

• Now consider the momentum leaving at A.

• The 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.

• Now consider the momentum leaving at A.

• The 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.

• In 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.

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

• Thus 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 =..

• Conversely, 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.

• In 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.

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

• On 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.

• For 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.

• Now momentum is transferred in two ways, viz.

• On 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.

• will 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'.

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

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

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

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

• bdo 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.

• Taking 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.

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

• Modern 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.

• If 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.

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

• The 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.

• As 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.

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

• He 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.

• If 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.

• But 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.

• The 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.

• (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.

• (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.

• From 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.

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

• Faster-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.

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

• This, 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.

• For 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.

• When 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.

• As 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.

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

• If the velocity changes, this is attributed to the action of force; and if we agree to measure the force (X) by the rate of change of momentum which it produces, we have the equation ~(mu)=X.

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

• The 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.

• In problems of impact we have to deal with cases of practically instantaneous impulse, where a very great and rapidly varying force produces an appreciable change of momentum in an exceedingly minute interval of time.

• We infer that on our reckoning the force of gravity on a mass m is to be measured by mg, the momentum produced per second when this force acts alone.

• If 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.

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

• Newtons 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.

• Since 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.

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

• Thus 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)

• Secondly, 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.

• The 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.

• 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.

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

• 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 ~j concentrated at G and moving with this point.

• 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 Cr.

• At 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.

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

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

• If extraneous forces act, it is seen in like manner that the resultant linear momentum of the system is in any given time modified by the geometric additiofi of the total impulse of the extraneous forces.

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

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

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

• If there are no extraneous forces, or if the extraneous forces have zero moment about any axis through G, the vector which represents the resultant angular momentum relative to G is constant in every respect.

• For example, if we have two particles connected by a string, the invariable plane passes through the string, and if w be the angular velocity in this plane, the angular momentum relative to G is mibiri ri +m1o~~rz - r2 (miri2 +mirl2)c~,,

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

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

• that of Ox) is equal to the total extraneous force in that direction, and that the rate of change of the -angular momentum about any fixed axis is equal to the moment of the extraneous forces about that axis.

• If we integrate with respect to 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.

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

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

• The 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.

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

• It 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.

• The 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.

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

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

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

• The 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.

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

• where I is the resultant angular momentum about 0.

• For 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.

• The 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.

• Again, 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.

• If 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 ~.

• Equating 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)

• In 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.

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

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

• The 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.

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

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

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

• The 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..

• (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.)

• Towards 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.

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

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

• INVARIABLE 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.

• We 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.

• But 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).

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

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

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

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

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

• Her momentum carried them through to the other side.

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

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

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

• angular momentum of the infalling gas is very low.

• angular momentum of the beam remains constant.

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

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

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

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

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

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

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

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

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

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

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

• With momentum rebuilt, they recruited a new bassist.

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

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

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

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

• continuity equation, momentum equation, energy equation ).

• continuum orbitals for each angular momentum can be included.

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

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

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

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

• gathered momentum sharply in the past year.

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

• impart momentum to the plasma ions.

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

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

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

• maintain the momentum.

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

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

• momentum of the particle.

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

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

• Our strategy is gaining positive momentum across the business.

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

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

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

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

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

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

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

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

• momentum tensor shown on the screen.

• momentum flux.

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

• momentum equation, energy equation ).

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

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

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

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

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

• overconfident, momentum investor.

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

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

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

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

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

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

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

• Dramatic momentum of tour in sparks parks amp recreation.

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

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

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

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

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

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

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

• sustain the momentum behind this initiative.

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

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

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

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

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

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

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

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

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

• A 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.

• Sometimes, 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.

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

• It 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.

• Taking 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.

• (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.

• The 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.

• bdo 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.

• Considered 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).

• Any 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.

• Taking 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.

• Denoting 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.

• The 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.

• Conversely, 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.

• Thus 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 =..

• These 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.

• In 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.

• well 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.

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

• Consider, 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) .

• will 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'.

• For 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.

• the 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.

• The 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.

• As 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.

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

• On 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.

• In 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.

• When 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.

• Since 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.

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

• Now 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.

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

• In 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.

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

• If 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.

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

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

• Substituting 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 .

• that the momentum issuing from A (fig.

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

• On 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.

• Since 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.

• But 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.

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

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

• Now momentum is transferred in two ways, viz.

• At 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.

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

• The 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.

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

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

• The 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.

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

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

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

• Modern 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.

• Long 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.

• If 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.

• He 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.

• If 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.

• According 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.

• But 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.

• The 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.

• (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.

• (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.

• From 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.

• He 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.

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

• The 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.

• Faster-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.

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

• This, 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.

• For 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.

• When 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.

• As 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.

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

• If the velocity changes, this is attributed to the action of force; and if we agree to measure the force (X) by the rate of change of momentum which it produces, we have the equation ~(mu)=X.

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

• The 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.

• In problems of impact we have to deal with cases of practically instantaneous impulse, where a very great and rapidly varying force produces an appreciable change of momentum in an exceedingly minute interval of time.

• We infer that on our reckoning the force of gravity on a mass m is to be measured by mg, the momentum produced per second when this force acts alone.

• If 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.

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

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

• Newtons 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.

• Since 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.

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

• Thus 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)

• Secondly, 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.

• The 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.

• 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.

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

• 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 ~j concentrated at G and moving with this point.

• 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 Cr.

• At 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.

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

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

• If extraneous forces act, it is seen in like manner that the resultant linear momentum of the system is in any given time modified by the geometric additiofi of the total impulse of the extraneous forces.

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

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

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

• If there are no extraneous forces, or if the extraneous forces have zero moment about any axis through G, the vector which represents the resultant angular momentum relative to G is constant in every respect.

• For example, if we have two particles connected by a string, the invariable plane passes through the string, and if w be the angular velocity in this plane, the angular momentum relative to G is mibiri ri +m1o~~rz - r2 (miri2 +mirl2)c~,,

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

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

• that of Ox) is equal to the total extraneous force in that direction, and that the rate of change of the -angular momentum about any fixed axis is equal to the moment of the extraneous forces about that axis.

• If we integrate with respect to 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.

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

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

• The 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.

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

• It 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.

• The 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.

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

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

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

• The 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.

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

• where I is the resultant angular momentum about 0.

• For 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.

• The 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.

• Again, 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.

• If 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 ~.

• Equating 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)

• In 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.

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

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

• The 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.

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

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

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

• The 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..

• (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.)

• Towards 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.

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

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

• INVARIABLE 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.

• We 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.

• But 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).

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

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

• Dramatic momentum of tour in sparks parks amp recreation.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

• So, 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.

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

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

• While 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.

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

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

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

• Kristen 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.

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

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

• Organic 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.

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

• This 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.

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

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

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

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

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

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

• The 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).

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

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

• The 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.

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

• The 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.

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

• Try 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.

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

• This 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.

• With 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.

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

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

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

• There'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.

• Even 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.

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

• Movements 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.

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

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

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

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

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

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

• One 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.

• If 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.

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

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

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

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

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

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

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

• Using 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.

• Be 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.

• Using momentum is not strength training.

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

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

• You 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.

• This 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.

• Look 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.

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

• Appearances 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.

• Fox 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.

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

• One office procedure gaining momentum is Thermage.

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

• We 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.