Chap. XI.

MECHANICS.

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Fig. 1250

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necessary that the surface on which it rests should move also, so that the friction is removedin a great degree by the curved surface rolling on with the rope.

"When two cords or ropes, suffering the same tension in every part of their length, arcparallel, there will be equilibrium of the forces, if each P and P' is one half of the weight, W ;

for the arch embraced by the rope being 180 degrees, the chord of which -

is equal to twice the radius, therefore each of the powers is only one halfthe resistance, and when one of these chords is fixed, then one power,

P, only being in action, it will be in equilibrio and equal to halfW. A system of pulleys with four ropes shows the weight to be 16times the power, upon the same principle; for supposing the first tocarry the weight \V, and the cord which passes round to have one endfixed at M, and the other attached to the block of the second pulley, andother cords to pass round the various other pulleys in a similar manner ;the tension of the first rope is equal to the powers. The first movablepulley sustains a weight equal to twice the power, but the weight whichit bears is the tension of the second rope ; hence the tension of the secondrope is twice that of the first, the tension of the third twice that of the second,and so on. If there be four ropes, the tension of the first is P, the second2 P, the third 2 x 2P, or 4 P; that of the fourth 2 x 2 x 2, or 8 P. Each

rope added to this system doubles its power, so that W = 2 n P, n _

expressing the number of distinct ropes there may be applied ; or,if n be the number of movable pulleys employed, there will be anequilibrium when F

P : \V = 1 : 2 n — 1.

To determine the ratio of the power to the resistance when thereis an equilibrium with any number of fixed or movable pulleys, andwhen the cords are parallel.

Spanish Bartons consist of two ropes and two movable pulleys ;and by such an arrangement there is a considerable gain of power.

The tension of the rope P 13 is equal to the power, and this ropebeing finally attached to the pulley, which sustains the weight, sup-ports the part of the weight equal to the power : the rope from Ato 13 balances the united tensions of both parts of the rope, extend-ing from B to the weight and power, and therefore its tension istwice the power; being brought under the pulley which sustainsthe weight W, and finally attached to the fixed point, it sustains apart of the weight equal to four times the power: thus thewhole weight must be equal to five times the power. The rope Ipasses by F over the wheel A, and it is attached to the wheel B.

Another system of two ropes is here shown, which does not pos-sess the power of the other : the tension of the rope extending fromthe power to the fixed point is equal to the power, and the tension ofthat extending from the pulley A to the weight is obviously equalto twice the power : thus the weight is four timesthe power. II is the point at which is suspendedthe rope that passes under the wheel, G C, andover that at A. B is the fixed wheel over whichthe rope attached to A and E pasres: as thewheel A is caused to descend by the power P,so that of GC, to which the weight W is attached,rises.

In a system of four ropes the weight is sixteentimes the power, for the tension of the ropeE C is evidently equal to the power, because itsustains it; D being a movable pulley, mustsustain a weight equal to twice the power : butthe weight which it sustains is the tension of thesecond rope; hence the tension of the secondrope is twice that of the first; in like mannerthe tension of the third rope is twice that ofthe second, and so on, the weight being equalto twice the tension of the last rope. The fourropes, then, of this example have the tensionof the first equal to P, of the second equal to2 P, that of the third 4 P, that of the fourth8 P, therefore the weight W will be 16 P: each rope so added doubling its effect, and the

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Fig. 1257.

Fig. 1259.

Fig. 1258.