TABLE III. Friction of Gudgeons or Axle-ends, in Motion, upon their Bearings. P THE RIGIDITY OF CORDS. 142. It is evident that, by reason of that resistance to B ; deflexion which constitutes the rigidity of a cord, a certain force or pressure must be called into action whenever it is made to change its rectilineal direction, so as to adapt itself to the form of any curved surface over which it is made to pass; as, for instance, over the circumference of a pulley or wheel. Suppose such a cord to sustain tensions represented by P, and P,, of which P, is on the point of preponderating, and let the friction of the axis of the pulley be, for the present, neglected. It is manifest that, in order to supply the force necessary to overcome the rigidity of the cord and to produce its deflection at B, the tension P, must exceed P, whereas, if there were no rigidity, P, would equal P,; so that the effect of the rigidity in increasing the tension P, is the same as though it had, by a certain quantity, increase I the tension P. Now, from a very numerous series of experiments made by Coulomb upon this subject, it appears that the quantity by which the tension P, may thus be considered to be increased by the rigidity, is partly constant and partly dependent on the amount of P,; so as to be represented by an algebraical formula of two terms, one of which is a constant quantity, and the other the product of a constant quantity by P. Thus if D represent the constant part of this formula, and E the constant factor of P, then is the effect of the rigidity of the cord the same as though the tension P, were increased by the quantity D+E.P,. When the cord, instead of being bent, under different pressures, upon circular arcs of equal radii, was bent upon circular arcs of different radii, then this quantity D+E.P,; by which the tension P, may be considered to be increased by the rigidity, was found to vary inversely as the radii of the arcs; so that, on the whole, it may be represented by the formula where R represents the radius of the circular arc over which the rope is bent. Thus it appears that the yielding tension P, may be considered to have been increased by the rigidity of the rope, when in the state bordering upon motion, so as to become This formula applies only to the bending of the same cord under different tensions upon different circular arcs for dif ferent cords, the constants D and E vary (within certain limits to be specified) as the squares of the diameters or of the circumferences of the cords, in respect to new cords, wet or dry; in respect to old cords they vary nearly as the power of the diameters or circumferences. Tables have been furnished by Coulomb of the values of the constants D and E. These tables, reduced to English measures, are given on the next page. * The rigidity of the cord exerts its influence to increase resistance only at that point where the cord winds upon the pulley; at the point where it leaves the pulley its elasticity favours rather, and does not perceptibly affect, the conditions of the equilibrium. In all calculations of machines, in which the moving power is applied by the intervention of a rope passing over a pulley, one-half the diameter of rope is to be added to the radius of the pulley, or to the perpendicular on the direction of the rope from the point whence the moments are measured, the pressure applied to the rope producing the same effect as though it were all exerted along the axis of the rope. TABLE V. RIGIDITY OF ROPES. Table of the values of the constants D and E, according to the experiments of Coulomb (reduced to English measures). The radius R of the pulley is to be taken in feet. No. 1. New dry cords. Rigidity proportional to the square of the No. 2. New ropes dipped in water. Rigidity proportional to the square of the circumference. Circumference of the Rope in Inches. 1-96 2 4 1-052217 0230303 No. 3. Dry half-worn ropes. Rigidity proportional to the square root of the cube of the circumference. No. 4. Wetted half-worn cords. Rigidity proportional to the square root of the cube of the circumference. |