RIGIDITY OF ROPES. In estimating the powers of machines, it is frequently necessary to take into consideration the effects arising from the rigidity or etiffness of the ropes which pass over tke pulleys or the axles of the wheels ; and, in order to understand how this con dition affects the relation between the moving power and the resistance, let it be observed that when a stiff rope is bent over the upper part of a wheel or pulley in a vertical plane, for example, the weights or powers applied at its extremities may not be sufficient to draw the descending portions into the poeitions of two vertical lines. Now, if one of the parte of the rope should take such a direction that a vertical line drawn through the weight attached to that part, cuts the hori zontal diameter of the wheel or pulley at a point between the centre and one extremity of the diameter, and if, at the same time, the other part should take such a direction that a vertical line drawn through*. the attached weight cuts the horizontal diameter at a point beyond the extremity of the latter, the distances of these vertical lines from the extremities of the diameter being represented by .r and reepec lively, the eorrespouiliug weights by w and w', and the radius of the wheel by st, the conditions of equilibrium, instead of being w - w', will be w (u .r)= But, if w be the weight which by descending raises tip the other, the value of .r is generally so small that it may be disregarded; so that we have, in the case of equilibrium, w u = w' (it + 2), or (w w') it w'.e', w' or again, w = ' that is, iu order to put the system in a state of equilibrium, the excess ' of w above w' should be equal to w' The formula given by Coulomb to express the force necessary for ws." overcoming the rigidity of a rope, or the equivalent of , ie r'" + f, ; r being the somieliameter of the rope, a the force arising from the warping or twisting of the rope, and b that which depends on the tension arising from the weight sv ; the values of in, a, and b may be determined by experiments made with cords of different diameters ; and thus .e' may be found. M. Coulomb ascertained that for slender
string, m=l. and that for stiff cordage the value of m varied (rein 1.5 to 2 ; also, from some experiments made with ropes consisting of 30 %Meads and 21 inches In circumference, he found that the weights requisite to overcome the rigidity, when the ropes paseed over a pulley 4 inches diameter, and were strained by weighta equal to 2.1 lbs,, 125 lba., and 423 lbs., were 5 Ilse Use., and 23 Ilia respectively.
Cul atzinately ropes of equal dimensions dill'er much in rigidity, so that little dependence can be placed on the results of general formulize lo estimating it value. White ropes when wet are more stiff than those which are dry, and the rigidity of rupee i3 greatly increased by tsrrins them. In general. the weights necessary to overcome the resistance of tarred ropes is proportional to the number of the threads of which they are compooed. General 3lorin has latterly revised the observations of Coulomb and of Navler on the subject of the rigidity of cordege, and has published the results of his inquiry in his Aide Ilscirsoire de Mdeanique Pratique; and in his' Lecona de Meesnique l'ratique.' Them essays have been translated and printed in the Engineers, Architect's, and Contractor's Pocket-Book; and from them it appears that Coulomb's rules, tables, and ferunthe were pre pared on a rather confined view of the case. Morin's own tables and Ionnulie are given in the translation thus quoted.
RING. [Axertrs.]