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or PP, 1+8 tan. ❤+ - tan. '+

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Now this relation of P, and P, obtains however small s be taken, or however great n be taken. Let n be taken infinitely great, so that the points A, B, C, &c. may be infinitely numerous and infinitely near to each other. supposed case thus passes into the actual case of a continuous surface, the fractions

above equation becomes

1 2 3

n' n' n'

The

&c. vanish, and the

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But the quantity within the brackets is the well known expansion (by the exponential theorem) of the functionetan. e,

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Since the length of cord S,, which passes over the point A, is the same with that S, which passes over the point Z, it follows that the modulus (Art. 152.) of such a cylindrical surface considered as a machine, and supposed to be fixed and to have a rope pulled and made to slip over it, is

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It is remarkable that these expressions are wholly inde pendent of the form and dimensions of the surface sustaining the tension of the rope, and that they depend exclusively upon the inclination or AdZ of the normals to the points A and Z, where the cord leaves the surface, and upon the co-efficient of friction (tan. ), of the material of which the rope is composed and the material of which the surface is composed. It matters not, for instance, so far as the fric

tion of the rope or band is concerned, whether it passes over a large pulley or drum, or a small one, provided the angle subtended by the arc which it embraces is the same, and the materials of the pulley and rope the same.

In the case in which a cord is made to pass m times round such a surface, 8=2mπ;

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And this is true whatever be the form of the surface, so that the pressure necessary to cause a cord to slip when wound completely round such a cylindrical surface a given number of times is the same (and is always represented by this quantity), whatever may be the form or dimension of the surface, provided that its material be the same. It matters not whether it be square, or circular, or elliptical.

1

187. If P, P", P,"", &c. represent the pressures which must be applied to one extremity of a rope to cause it to slip when wound once, twice, three times, &c. round any such surface, the same tension P, being in each case supposed to be applied to the other extremity of it, we have

P=P, tan., P," Petan. 6, P.,""P, tan. o, &c.=&c.

So that the pressures P', P.", P."", &c. are in a geometrical progression, whose common ratio is 2 tan., which ratio is always greater than unity. Thus it appears by the experiments of M. Morin (p. 135.), that the co-efficient of friction between hempen rope and oak free from unguent is 33, when the rope is wetted. In this case tan. ¢=33 and 2 tan. =2×3.14159 × 33=2.07345. The common ratio of the progression is therefore in this case 207345, or it is the number whose hyperbolic logarithm is 2.07345. This number is 795; so that each additional coil increases the friction nearly eight times. Had the rope been dry, this proportion would have been much greater. If an additional half coil had been supposed continually to be put upon the rope instead of a whole coil, the friction would have been found in the same way to increase in geometrical progression, but the common ratio would in this case have been tan instead of 2 tan. ò̟ ̧ In the above example the. value of this ratio would for each half coil have been

2.82.

The enormous increase of friction which results from

each additional turn of the cord upon a capstan or drum, may from these results be understood.

Fig. 1.

Fig. 2.

Fig. 8.

188. We may, from what has been stated above, readily explain the reason why a knot connecting the two extremities of a cord effectually resists the action of any force tending to separate them. If a wetted cord be wound round a cylinder of oak as in fig. 1., and its extremities be acted upon by two forces P and R, it has been shown that P will not overcome R, unless it be equal to some Now if the string to which R is attached be brought underneath the other string so as to be pressed by it against the surface of the cylinder, as at m, jig. 2.; then, provided the friction produced by this pressure be not less than one eighth of P, the string will not move even although the force R cease to act. And if both extremities of the string be thus made to pass between the coil and the cylinder, as in fig. 3., a still less pressure upon each will be requisite. Now, by diminishing the radius of the cylinder, this pressure can be increased to any extent, since, by a known property of funicular curves, it varies inversely as the radius. We may, therefore, so far diminish the radius of a cylinder, as that no force, however great, shall be able to pull away a rope coiled upon it, as represented in fig. 3., even although one extremity were loose, and acted upon by no force.

where about eight times that force.

Fig. 4.

*

Let us suppose the rope to be doubled as in fig. 4., and coiled as before. Then it is apparent, from what has been said, that the cylinder may be made so small, that no forces P and P applied to the extremities of either of the double cords will be sufficient to pull them from

it, in whatever directions these are applied.

This property will be proved in that portion of the work which treats of

the THEORY OF CONSTRUCTION.

Now let the cylinder be removed. The cord then being drawn tight, instead of being coiled round the cylinder, will be coiled round portions of itself, at the points m and n; and instead of being pressed at those points upon the cylinder, by a force acting on one portion of its circumference, it will be pressed by a greater force acting all round its circumference. All that has been proved before, with regard to the impossibility of pulling either of the cords away from the coil when the cylinder is inserted, will therefore now obtain in a greater degree; whence it follows that no forces P and P' acting to pull the extremities of the cords asunder, may be sufficient to separate the knot.

THE FRICTION BREAK.

189. There are certain machines whose motion tends, at certain stages, to a destructive acceleration; as, for instance, a crane, which, having raised a heavy weight in one position of its beam, allows it to descend by the action of gravity in another; or a railway train, which, on a certain portion of its line of transit, descends a gradient, having an inclination greater than the limiting angle of resistance. In each of these cases, the work done by gravity on the descending weight exceeds the work expended on the ordinary resistance due to the friction of the machine; and if some other resistance were not, under these circumstances, opposed to its motion, this excess (of the work done by gravity upon it over that expended upon the friction of its rubbing surfaces) would be accumulated in it (Art. 130.) under the form of vis viva, and be accompanied by a rapid acceleration and a destructive velocity of its moving parts. The extraordinary resistance required to take up its excess of work, and to prevent this accumulation, is sometimes supplied in the crane by the work of the laborer, who, to let the weight down gradually, exerts upon the revolving crank a pressure in a direction opposite to that which he used in raising it. It is more commonly supplied in the crane, and always in the railway train, without any work at all of the laborer, by simple pressure of his hand or foot on the lever of the fric tion break, which useful instrument is represented in the accompanying figure under the form in which it is com

monly applied to the crane, a form of it which may serve to illustrate the principle of its application under every

B

other. BC represents a wheel fixed commonly upon that axis of the machine to which the crank is attached, and which axis is carried round by it with greater velocity than any other. The periphery of this wheel, which is usually of cast iron, is em

braced by a strong band* ABCE of wrought iron, fixed firmly by its extremity A to the frame of the machine, and by its extremity E to the short arm AE of a bent lever PAE, which turns upon a fixed axis or fulcrum, at A, and whose arm PA, being prolonged, carries a counterpoise D just sufficient to overbalance the weight of the arm AP, and to relieve the point E of all tension, and loosen the strap from the periphery of the wheel, when no force P is applied to the extremity of the arm AP, or when the break is out of action.

It is evident that a pressure P applied to the extremity of the lever will produce a pressure upon the point E, and a tension upon the band in the direction ABCE, and that being fixed at its extremity A, the band will thus be tightened upon the wheel, producing by its friction a certain resistance upon the circumference of the wheel.

Moreover, it is evident that this resistance of friction upon the circumference of the wheel is precisely equal to the tension upon the extremity A of the band, being, indeed, wholly borne by that tension; and that it is the same whether the wheel move, as in this case it does, under the band at rest, or whether the band move (under the same tensions upon its extremities, but in the opposite direction) over the wheel at rest. Let R and Q represent the tensions upon the extremities A and E of the band; then if we sup pose the wheel to be at rest, and the band to be drawn over it in the direction ECB by the tension R, and to represent the angle subtended at the centre of the wheel by that part of its circumference which the band embraces, we have (equation 205)

*Blocks of wood are interposed between the band, the periphery of the break wheel. This case will be discussed in the Appendix.

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