. Applied calculus; principles and applications . = »o J 0 CURVE OF A CORD 241 144. Curve of a Cord under Uniform Horizontal Load —Parabola. — When a flexible cord supports a load which isuniformly distributed over the horizontal projection of thecord, as, for example, the cable of a suspension bridge,which supports a load distributed uniformly per foot ofroadway, the curve assumedby the cord is a is evident geometri-cally, and the curve can beshown analytically to be aparabola. Consider a portion OP ofthe cord AOB, 0 being thelowest point of the equihbrating forcesac
. Applied calculus; principles and applications . = »o J 0 CURVE OF A CORD 241 144. Curve of a Cord under Uniform Horizontal Load —Parabola. — When a flexible cord supports a load which isuniformly distributed over the horizontal projection of thecord, as, for example, the cable of a suspension bridge,which supports a load distributed uniformly per foot ofroadway, the curve assumedby the cord is a is evident geometri-cally, and the curve can beshown analytically to be aparabola. Consider a portion OP ofthe cord AOB, 0 being thelowest point of the equihbrating forcesacting on OP are the ten-sions H and T at the ends, and the weight of the load, acting at the center of OM,Since three forces to be in equilibrium must meet in a point,the tangent to the cord at P passes through the middle ofOM. Now it is a property of the parabola that the sub-tangent is bisected at the vertex, in which case OC = ^NP = CM. Hence the curve assumed by the cord is aparabola. Otherwise the analytic conditions of equiUbrium give. XMp* = WX:^- Hy 0; y- 2H^ (1) the equation of a parabola. The same result is gotten bytaking the sides of ^he triangle PDT to represent the threeforces T, H, and wx, and then, dydx wx . _ A wx dx _ wo^~h ^~ Jo ~H~~ 2H (1) * Algebraic sum of moments of forces about point P. 242 INTEGRAL CALCULUS T coscj) = H and T sin 0 = wx give by division , ■ wx dy , „ tan ^ — ~Tj = -f^ as before. Also, T = = Vi/2 _|_ {y)xY gives the tension at any point of the cord. The curve is thus shown to be a parabola with its vertexat 0 and its axis vertical. If the supports at A and B are atthe same elevation and I is the span, the sag d at 0 is y, for X ^^ 2 l) .-. H = f., and Ti = -^. (3) 8 a cos 01 dy ^ , wl 4:d ... where <^i is the angle of inclination of PT at the length of the cord for a given span and sag is gotten by ^dyV\h , , dy wx , ^„ y) dx, where ~f- = -tt and s = OB. CM Let V? = -, or — = a, to simplify: then,Ha w s= f\i
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