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60. The preceding proposition is manifestly true in respect to a system of weights, these being pressures whose directions are always parallel, wherever their points of application may be moved. Now the centre of pressure of a system of weights is its centre of gravity (Art. 19). Thus then it follows, that if the weights composing such a system be separately moved in any directions whatever, and through any distances whatever, then the difference between the aggregate work done upwards in making this change of relative position and that done downwards is equal to the work necessary to raise the,sum of all the weights through a height equal to that through which their centre of gravity is raised or depressed. Moreover that if such a system of weights be supported in equilibrium by the resistance of any fixed point or points, and be put in motion, then (since the work of the resistance of each such point is nothing) the aggregate

This proposition has numerous applications. If, for instance, it be required to determine the aggregate expenditure of work in raising the different elements of a structure, its stone, cement, &c., to the different positions they occupy in it, we make this calculation by determining the work requisite to raise the whole weight of material at once to the height of the centre of gravity of the structure. If these materials have been carried up by labourers, an 1 we are desirous to include the whole of their labour in the calculation, we ascertain the probable amount of each load, and conceive the weight of a labourer to be added to each load, and then all these at once to be raised to the height of the centre of gravity.

Again, if it be required to determine the expenditure of work made in raising the material excavated from a well, or in pumping the water out of it, we know that (neglecting the effect of friction, and the weight and rigidity of the cord) this expenditure of work is the same as though the whole material hal been raised at one lift from the centre of gravity of the shaft to the surfac›. Let us take another application of this principle which offers so many practical results. The material of a railway excavation of considerable length is to b removed so as to form an embankinent across a valley at some distance, and it is required to determine the expenditure of work made in this transfer of th material. Here each load of material is. made to traverse a different distanc a resistance from the friction, &c., of the road being continually opposed to i motion. These resistances on the different loads constitute a system of paralel pressures, each of whose points of application is separately transferred fro one given point to another given point, the directions of transfer being als parallel. Now by the preceding proposition, the expenditure of work in a" these separate transfers is the same as it would have been had a pressure equil to the sum of all these pressures been at once transferred from the centre o resistance of the excavation to the centre of resistance of the embankment. Now the resistances of the parts of the mass moved are the frictions of its elements upon the road, and these frictions are proportional to the weights of the elements; their centre of resistance coincides therefore with the centre of gra vity of the mass, and it follows that the expenditure of work is the same as though all the material had been moved at once from the centre of gravity of the excavation to that of the embankment. To allow for the weight of the carriages, as many times the weight of a carriage must be added to the weight of the material as there are journeys made.

work of those weights which are made to descend, is equal to that of those which are made to ascend.

61. If a plane be taken perpendicular to the directions of any number of parallel pressures and there be two different positions of the points of application of certain of these pressures in which they are at different distances from the plane, whilst the points of application of the rest of these pressures remain at the same distance from that plane, und if in both positions the system be in equilibrium, then the centre of pressure of the first mentioned pressures will be at the same distance from the plane in both positions.

For since in both positions the system is in equilibrium, therefore in both positions P,+P,+P,+ . . =0, ::. (Y,—y,)P,+(Y‚—y2)P2+(Y,−y1)P,+. , −y1) P2 + ... + P(Y —y)=0: Now let P be any one of the pressures whose points of application is at the same distance from the given plane in both positions,

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where II represents the distance of the centre of pressure of P, P, .. P, from the given plane in the first position, and h its distance in the second position. Its distance in the first position is therefore the same as in the second. Therefore, &c.

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From this proposition, it follows that if a system of weights be supported by the resistances of one or more fixed points, and if there be any two positions whatever of the weights in both of which they are in equilibrium with the resistances of those points, then the height of the common centre of gravity of the weights is the same in both positions. And that if there be a series of positions in all of which the weights are in equilibrium about such a resisting point or points, then the centre of gravity remains continually at the same height as the system passes through this series of posi

tions.

If all these positions of equilibrium be infinitely near to

one another, then it is only during an infinitely small motion of the points of application that the centre of gravity ceases to ascend or descend; and, conversely, if for an infinitely small motion of the points of application the centre of gravity ceases to ascend or descend, then in two or more positions of the points of application of the system, infinitely near to one another, it is in equilibrium.

WORK OF PRESSURES APPLIED IN DIFFERENT DIRECTIONS TO A BODY MOVEABLE ABOUT A FIXED AXIS.

62. The work of a pressure applied to a body moveable about a fixed axis is the same at whatever point in its proper direction that pressure may be applied.

For let AB represent the direction of a pressure applie l

to a body moveable about a fixed axis O; the work done by this pressure will be the same whether it be ap plied at A or B. For conceive the body to revolve about O, through an exceedingly small angle AOC, or BOD, so that the points A and D may describe circular ares AC and BD. Draw Cm, Dn, and OE, perpendiculars to AB, then if P represent the pressure applied to AB, P. Am will represent the work done by P when applied at A (Art. 52.), and P. Bn will represent the work done by P when applied at B; therefore the work done by P at A is the same as that done by P at B, if Am is equal to Bn.

Now AC and BD being exceedingly small, they may be conceived to be straight lines. Since BD and BE are respectively perpendicular to OB and OE, therefore /DBE = BOE; and because AC and AE are perpendicular to OA and OE, therefore CAE = / AOE. Now Am =

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It is a well-known principle of Geometry, that if two lines be inclined at any angle, and any two others be drawn perpendicular to these, then the inclination of the last two to one another shall equal that of the first two.

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63. If any number of pressures oe in equilibrium about a fixed axis, then the whole work of those which tend to move the system in one direction about that axis is equal to the whole work of those which tend to move it in the opposite direction about the same axis. For let P be any one of such a system of pressures, and O a fixed axis, and OM perpendicular to the direction of P, then whatever may be the point of application of P, the work of that pressure is the same as though it were applied at M. Suppose the whole

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system to be moved through an exceeding small angle about the point O, and let OM be represented by p, then will p represent the space described by the point M, which will be actually in the direction of the force P, therefore the work of PP. p.. Now let P, P., P., &c. represent those pressures which act in the direction of the motion, and P',, P', &c. those which act in the opposite direction, and let P P P &c. be the perpendiculars on the first, and p'1, P'29 P', &c. be the perpendiculars on the second; therefore by the principle of the equality of moments P,p,+P2p2+P2P2 +&c. = P'1p', + P'2p'2+ P'‚p's +&c.; therefore multiplying both sides by, P1p + P2p2 + P‚p‚§ = P'‚p'‚ ̧◊ + P22p2‚a + Pp+&c.; but Pp, P'p', &c. are the works of the forces P., P',, &c.; therefore the aggregate work of those which tend to move the system in one direction is equal to the aggregate of those which tend to move it in the opposite direction.

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64. THE ACCUMULATION OF WORK IN A MOVING BODY.

In every moving body there is accumulated, by the action of the forces whence its motion has resulted, a certain amount of power which it reproduces upon any resistance. opposed to its motion, and which is measured by the work done by it upon that obstacle. Not to multiply terms, we shall speak of this accumulated power of working, thus measured by the work it is capable of producing, as ACCUMULATED WORK. It is in this sense that in a ball fired from

*Note (o) Ed. App.

a cannon there is understood to be accumulated the work it reproduces upon the obstacles which it encounters in its flight; that in the water which flows through the channel of a mill is accumulated the work which it yields up to the wheel; and that in the carriage which is allowed rapidly to descend a hill is accumulated the work which carries it a considerable distance up the next hill. It is when the pressure under which any work is done, exceeds the resistance opposed to it, that the work is thus accumulated in a moving body; and it will subsequently be shown (Art. 69.) that in every case the work accumulated is precisely equal to the work done upon the body beyond that necessary to overcome the resistances opposed to its motion, a principle which might almost indeed be assumed as in itself evident.

65. The amount of work thus accumulated in a body moving with a given velocity, is evidently the same, whatever may have been the circumstances under which its velocity has been acquired. Whether the velocity of a ball has been communicated by projection from a steam gun, or explosion from a cannon, or by being allowed to fall freely from a sufficient height, it matters not to the result; provided the same velocity be communicated to it in all three cases, and it be of the same weight, the work accumulated in it, estimated by the effect it is capable of producing, is evidently the same.

In like manner, the whole amount of work which it is capable of yielding to overcome any resistance is the same, whatever may be the nature of that resistance.

66. TO ESTIMATE THE NUMBER OF UNITS OF WORK ACCUMU

LATED IN A BODY MOVING WITH A GIVEN VELOCITY.

Let w be the weight of the body in pounds, and v its velocity in feet.

Now suppose the body to be projected with the velocity v in a direction opposite to gravity, it will ascend to the height h from which it must have fallen, to acquire that same velocity v (Art. 47.); there must then at the instant of projection have been accumulated in it an amount of work sufficient to raise it to this height h; but the number of units of work

This remark applies more particularly to the under-shot wheel, which is carried round by the rush of the water.

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