In considering the forces that produce derailment the total mass of the vehicle or locomotive may be supposed to be concentrated at its centre of gravity.
The centre of gravity in Hebrew religion was shifted from ceremonial observance and local sacra to righteous conduct.
Its nearness to Washington, the material and manufacturing resources concentrated in it, and the moral importance attached to its possession by both sides, caused it to be regarded as the centre of gravity of the military operations in the east to which the greatest leaders and the finest armies were devoted from 1861 to 1865.
But the triumph of the navy in 480 and the great expansion of commerce and industry had definitely shifted the political centre of gravity from the yeoman class of moderate democrats to the more radical party usually stigmatized as the " sailor rabble."
Though this prince continued to develop the city, giving it a municipality in 1866 1 and new harbour works in 1871-1878, he developed Cairo still more; and the centre of gravity definitely shifted to the inland capital.
Assuming as an axiom that the centre of gravity of any number of interdependent bodies cannot rise higher than the point from which it fell, he arrived, by anticipating in the particular case the general principle of the conservation of vis viva, at correct although not strictly demonstrated conclusions.
The fifth method consists in observing the displacement in the direction of the sun, or of one of the nearer planets, due to the motion of the earth round the common centre of gravity of the earth and moon.
If a magnetized needle were supported so that it could move freely'about its centre of gravity it would not generally settle with its axis in a horizontal position, but would come to rest with its north-seeking pole either higher or lower than its centre.
For the practical observation of this phenomenon it is usual to employ a needle which can turn freely in the plane of the magnetic meridian upon a horizontal axis passing through the centre of gravity of the needle.
For this reason a thin bar suspended at its centre of gravity between a pair of magnetic poles will, if paramagnetic, set itself along the line joining the poles, where the field is strongest, and if diamagnetic, transversely to the line.
He had not attempted to include in his calculations the orbital variations of the disturbing bodies; but Lagrange, by the happy artifice of transferring the origin of coordinates from the centre of the sun to the centre of gravity of the sun and planets, obtained a simplification of the formulae, by which the same analysis was rendered equally applicable to each of the planets severally.
Lord Roberts's plan was first to concentrate to his left, taking every measure to induce the Boers to believe that the original scheme of invasion by the centre would now be resumed, and in this purpose he succeeded so well that his field army with the necessary transport for a cross-country march was assembled between the Orange and the Modder without serious mishap. Cronje at the new centre of gravity was not reinforced, all available Boers drawing down towards Colesberg.
- A body immersed in a fluid is buoyed up by a force equal to the weight of the liquid displaced, acting vertically upward through the centre of gravity of the displaced liquid.
For if the body is removed, and replaced by the fluid as at first, this fluid is in equilibrium under its own weight and the thrust of the surrounding fluid, which must be equal and opposite, and the surrounding fluid acts in the same manner when the body replaces the displaced fluid again; so that the resultant thrust of the fluid acts vertically upward through the centre of gravity of the fluid displaced, and is equal to the weight.
In the steady motion under no force of such a body in medium, the centre of gravity describes a helix, while the axis escribes a cone round the direction of motion of the centre of ravity, and the couple causing precession is due to the dislacement of the medium.
Each impinging molecule exerts an impulsive pressure equal to mu on the boundary before the component of velocity of its centre of gravity normal to the boundary is reduced to zero.
The centroid is sometimes called the centre of volume, centre of area, or centre of arc. The proof of the existence of the centroid of a figure is the same as the proof of the existence of the centre of gravity of a body.
In the case, therefore, of any solid whose cross-section at distance x from one end is a quadratic function of x, the position of the crosssection through the centroid is to be found by determining the position of the centre of gravity of particles of masses proportional to So, S2, and 4S 1, placed at the extremities and the middle of a line drawn from one end of the solid to the other.
Algol, in fact, travels at the rate of 26.3 miles a second round the centre of gravity of the system which it forms with an invisible companion, while the two together approach the sun with an unvarying speed of 2.3 miles per second.
This, and the movements of the 4th Army, which had set its face towards Haicheng and no longer seemed to be part of a threat on Liao-Yang, led to the idea being entertained at Kuropatkin's headquarters that the centre of gravity was shifting to the south.
If it revolves about a vertical axis d its centre of gravity must always lie in that axis; if it rolls the centre of gravity must always lie over the e abutment.
having two components revolving about their common centre of gravity - and the first to have its orbit calculated.
But the importance of these modifications is something more than the doubt which they have thrown on WH's theories: they have really shifted the centre of gravity of the textual problem.
Herschel, who discovered that many of them formed systems of two stars revolving round their common centre of gravity.
long have been resorted to, provided with a steel eye at the top and a ring near the centre of gravity to enable them to be worked (fig.
(c) Wren and Huygens further proved that the law of equal action and reaction, already experimentally established by the former, is deducible from the conservation of the velocity of the common centre of gravity, which is the same as the common velocity of the bodies, that is, deducible from the fact that their common centre of gravity does not change its state of motion or rest by the actions of the bodies between themselves; and they further extended the law to bodies, qua elastic.
The socialistic idea, with which the " Democratie Chretienne " had identified itself both in France and Belgium, regards numbers as the centre of gravity of the whole state organism.
On the whole, however, what may be called the speculative centre of gravity of Great Britain's export business in cotton goods is not in Manchester but abroad.
Pascal solved the hitherto refractory problem of the general quadrature of the cycloid, and proposed and solved a variety of others relating to the centre of gravity of the curve and its segments, and to the volume and centre of gravity of solids of revolution generated in various ways by means of it.
of the German problem; with his accustomed calculated bluntness he had more than once openly asserted that this problem could only be settled by Austria ceasing to influence the German courts and transferring her centre of gravity towards Budapest; with equal bluntness he told the committee on the budget, on the 30th of September 1862, that the problem could not be solved by parliamentary decrees, but only by blood and iron.
In Aeolis, of course, the centre of gravity moved to the Attalid capital, Pergamum.
(ga 2 /2b); or if the height of the centre of gravity of a column like a gravestone above the base on which it rests is y, and x is the horizontal distance of this centre from the edge over which it has turned, then the acceleration or suddenness of motion which caused its overthrow is measured, as pointed out by C. D.
Antioch became the capital and court-city of the western Seleucid empire under Antiochus I., its counterpart in the east being Seleucia-on-Tigris; but its paramount importance dates from the battle of Ancyra (240 B.C.), which shifted the Seleucid centre of gravity from Asia Minor, and led indirectly to the rise of Pergamum.
17) be a section of the polar axis; it is then easy to adjust the weight P attached to its lower end so that the centre of gravity X of the whole moving parts of the instrument shall be in the vertical (V V) of a line passing through the apex of the hollowed flange p q at q, which flange forms part of the polar axis.
The difficulties of relief friction could probably be best overcome by a large hollow cylinder concentric with the polar axis fixed near the centre of gravity of the whole instrument and floated in mercury, on the plan adopted in the Mount Wilson 60-in.
The magnet is in equilibrium under the influence of the couple VM due to the vertical component V, and the couple due to the fact that the centre of gravity is slightly on one side of the knife-edge.
The chief difficulty with this form of instrument is that it is very sensitive to changes of temperature, for such changes not only alter M but also in general cause the centre of gravity of the system to be displaced with reference to the knife-edge.
In the Eschenhagen form of vertical force balance two deflecting magnets are used to partly neutralize the vertical component, so that the centre of gravity is almost exactly over the support.
It contains the theory of the centre of gravity as ordinarily understood.
If the system be subject to gravity, the corresponding part of the virtual work can be calculated from the displacement of the centre of gravity.
.l (~) =(Wi+W2+ ...)&s, where ~ is the depth of the centre of gravity (see 8 (14) and 11 (6)).
An important conclusion is that in any displacement of a system of bodies in equilibrium, such that the virtual work of all forces except gravity may be ignored, the depth of the centre of gravity is stationary.
The question as to stability of equilibrium belongs essentially to kinetics; but we may state by anticipation that in cases where gravity is the only force which does work, the equilibrium of a body or system of bodies is stable only if the depth of the centre of gravity be a maximum.
For equilibrium, the altitude of the centre of gravity G must be stationary; hence G must lie in the same vertical line with the point of contact J of the two curves.
It is a classical problem in the calculus of variations to deduce the equation (9) from the condition that the depth of the centre of gravity of a, chain of given length hanging I I between fixed points must be catenary; it determines the scale of the curve, all cate } stationary (~ 9).
The vertical through the centre of gravity of the arc AP (see fig.
This contains the theory of the centre of gravity (~ 4, 9).
where 0 is the angle which the plane containing the axis and the centre of gravity G makes with the vertical, and h is the distance of G from the axis.
The resultant of a parallel projection of any system of forces is the projection of their resultant; and the centre of gravity of a parallel projection of a solid is the projection of the centre of gravity of the first solid.
Of this kind is the weight of any piece in the mechanism whose centre of gravity alternately rises and falls; for during the rise of the centre of gravity that weight acts as a resistance, and energy is employed in lifting it to an amount expressed by the product of the weight into the vertical height of its rise; and during the fall of the centre of gravity the weight acts as an effort, and exerts in assisting to perform the work of the machine an amount of energy exactly equal to that which had previously been employed in lifting it.
Centrifugal Force of a Rotating BodyThe centrifugal force exerted by a rotating body on its axis of rotation is the same in magnitude as if the mass of the body were concentrated at its centre of gravity, and acts in a plane passing through the axis of rotation and the centre of gravity of the body.
If the axis of rotation traverses the centre of gravity of the body, the centrifugal force exerted on that axis is nothing.
In balancing the mechanism of a steam engine it is often sufficiently accurate to consider the motion of the pistons as simple harmonic, and the effect on the framework of the acceleration of the connecting rod may be approximately allowed for by distributing the weight of the rod between the crank pin and the piston inversely as the centre of gravity of the rod divides the distance between the centre of the cross head pin and the centre of the crank pin.
* Centrifugal Whirling of Shafts.When a system of revolv ing masses is balanced so that the conditions of the preceding sectiol are fulfilled, the centre of gravity of the system lies on the axis o revolution.
To take a simple case, suppose a shaft supported on two bearings to carry a disk of weight W at its centre, I and let the centre of gravity of the disk be at a distance e from the axis of rotation, this small distance being due to imperfections of material or faulty construction.
through its centre of gravity.
133) let the required deviation be a rotation of the bod3 BB about an axis 0, not traversing the centre of gravity G, di being, as before, the deviation of angular velocity to be produced in the interval dl.
134) be any link moving in any manner in a plane, and let G be its centre of gravity.
Then its motion may be analysed into (I) a translation of its centre of gravity; and (2) a rotation about an axis through its centre of gravity perpendicular to its plane of motion.
Let a be the acceleration of be the angular acceleration about the axis through the Centre of gravity; then the K I1~~ the centre of gravity and let A
force required to produce the - translation of the centre of - gravity is F =Wa/g, and the - couple required to produce the - angular acceleration about the Centre of gravity is M = IA/g, FiG.
W and I being respectively the weight and the moment of inertia of the link about the axis through the Centre of gravity.
Og is then the acceleration of the centre of gravity and the force F can therefore be immediately calculated.
Hence the magnitude F and the position of F relatively to the centre of gravity of the link, necessary to give rise to the couple M, are known, and this force is therefore the resultant force required.
I35, so that the centre of gravity G is not in the line joining them.
Through G, the centre of gravity of the rod, draw Gg parallel to the line of stroke, thus dividing the image at g in the proportion that the connecting rod is divided by G.
Hence Og represents the acceleration of the centre of gravity and, the weight of the connecting J.
The motion of the common centre of gravity of the two bodies is unchanged by the collision.
The loss of energy consists of a certain proportion of that part of the actual energy of the bodies which is due to their motion relatively to their common centre of gravity.
To say that it displaced the centre of gravity in politics and commerce, substituting the ocean for the Mediterranean, dethroning Italy from her seat of central importance in traffic, depressing the eastern and elevating the western powers of Europe, opening a path for Anglo-Saxon expansiveness, forcing philosophers and statesmen to regard the Occidental nations as a single group in counterpoise to other groups of nations, the European community as one unit correlated to other units of humanity upon this planet, is truth enough to vindicate the vast significance of these discoveries.
If, in an unsharp image, a patch of light corresponds to an object point, the " centre of gravity " of the patch may be regarded as the image point, this being the point where the plane receiving the image, e.g.
of Salamis and Plataea definitely shattered the offensive powe of the empire; that the centre of gravity in the worlds history had shifted from Susa and Babylon to the Aegean Sea; and that the Persians were conscious that in spite of all their courage they were henceforward in the presence of an enemy, superior in arms as well as in intellect, whom they could not hope to subdue by their own strength.
Corps occupied the plain from opposite Novak to Kikuricani, with its centre of gravity on the Prilep road.'
The plane of the ecliptic is that plane in or near which the centre of gravity of the earth and moon.
Owing to the action of the planets, especially Venus and Jupiter, on the earth, the centre of gravity of the earth and moon deviates by a yet minuter amount, generally one or two tenths of a second, from the plane of the ecliptic proper.
The social fabric was built up not on the towns, but on the great landlords; and when the centre of gravity began to move, first of all in Italy, to the towns, and crowded populations began to be massed together in them, the parochial systems broke down under the weight of the new conditions, and the people were in a state of spiritual and moral no less than physical destitution.
thick of a monolithic dam, subject to the pressure of water against its vertical side to the full depth ab= d in feet, the horizontal _ eL 2 pressure of water against the section of the dam, inI creasing uniformly with the depth, is properly represented by the isosceles right-angled triangle abe, in which be is the maximum water-pressure due to the Cent full depth d, while the area 2 abe = d is t h e total hor12 d3 6 If x be the width of the base, and p the density of the masonry, the weight of the masonry in terms of a cubic foot of water will be acting at its centre of gravity g, situated at 3x from the outer toe, and the moment of resistance to overturning on the outer toe, p x 2 d (2) In countries where good clay or retentive earth cannot be obtained, numerous alternative expedients have been adopted with more or less success.
The theory of the scale-beam is stated by Weisbach in his Mechanics of Machinery and Engineering, as follows - In fig I D is the fulcrum of the balance, S the centre of gravity of the beam alone without the scales, chains or weights; A and B the points of suspension of the chains.
In order to increase the sensitiveness of a balance, the line AB joining the points of suspension and the centre of gravity of the balance must be brought nearer to each other.
In order to ensure a high degree of sensitiveness, balances are sometimes constructed so that Z is slightly below the line joining X and Y, and is only slightly above H, the centre of gravity of the beam with the scale - pans and chains attached.
7 let Z be the fulcrum knife-edge, X the knife-edge on which the load R is hung, and H the centre of gravity of the weights to the right of Z, viz.
gravity of the long body CD, and be the centre of gravity of the three vertical forces acting downwards at the points x i, and considered as weights collected at those points; then if be above the line z i y i it can be shown that this arrangement of the knife-edges of CD favours the" acceleration " principle, and is suited to act with and assist an " accelerating " steelyard, and similarly if the point h2 be above the line z 2 y 2 in the case of the short body EF.
If the centre of gravity of a pulley is on the axis of rotation, and the whole mass is distributed so that the axis of inertia coincides with the axis of rotation, there can be no unbalanced force or unbalanced couple as the pulley revolves.
They knew well that the centre of gravity in their own belief lay elsewhere than in the elaborately detailed scheme of relations within the Godhead or in the Theanthropic person.
These were: First, the law of the conservation of the centre of gravity.
which act upon each other, their motions are so related that the centre of gravity of the entire system moves in a straight line with a constant velocity.
Taking as the radius vector of each body the line from the body to the common centre of gravity of all, the sum of the products formed by multiplying each area described, by the mass of the body, remains a constant.
Among the new truths detected by him was the valuable mechanical principle that if any number of bodies be so connected that, by their motion, their centre of gravity can neither ascend nor descend, then those bodies are in equilibrium.
At the Marchese's request he wrote, in 1588, a treatise on the centre of gravity in solids, which obtained for him, together with the title of "the Archimedes of his time," the honourable though not lucrative post of mathematical lecturer at the Pisan university.
Suppose that from the centre of gravity of the solar system (instead of which we may, if we choose, take the centre of the sun), lines or radii vectores be drawn to every body of the solar system.
The second of these contains a remarkable discussion of the relation between the centre of the moon's figure and its centre of gravity.
If the perpendiculars from the vertices to the opposite faces of a tetrahedron be concurrent, then a sphere passes through the four feet of the perpendiculars, and consequently through the centre of gravity of each of the four faces, and through the mid-points of the segments of the perpendiculars between the vertices and their common point of intersection.
I, 1907, p. 328) that this is the result of physical causes and that the existence of the Pacific Ocean shows that the centre of gravity of the earth does not coincide with the centre of figure.
If it is an engine, particular attention must be directed to the type, weight, arrangement of wheels and height of centre of gravity above rail level.
As the vehicle sweeps round the curve the centre of gravity tends to be thrown outwards,.
From consideration of the rigid triangular frame described above, it is clear that the " overturning " force acts horizontally from the centre of gra'Vity, and that the length of its lever arm is, at any instant, the vertical distance from the centre of gravity to the level of the outer rail.
- The girders are longer than the span and the part overhanging the abutment is counter-weighted so that the centre of gravity is over the abutment when the bridge is rolled forward (fig.
If the deviation be a translationthat is, an equal change of motion of all the particles of the bodythe centre of percussion is obviously the centre of gravity itself; and, according to the second law of motion, if dv be the deviation of velocity to be produced in the interval dt, and W the weight of the body, then W dv r-.~ (82)
I27.~ Alternative construction for finding the position of F relatively to the centre of gravity of the link.Let B and K be any two points in the link which for greater generality are taken in fig.
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