Suppose a uniform exible cable is suspended between two points .L;H/ at equal heights

Chapter 3, Problem 69

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Suppose a uniform exible cable is suspended between two points .L;H/ at equal heights located symmetrically on either side of the x-axis (Fig. 1.4.12). Principles of physics can be used to show that the shape y Dy.x/of the hanging cable satises the differential equationad2y dx2 Ds1C#dy dx$2; where the constant a D T=' is the ratio of the cables tension T at its lowest point x D 0 (where y0.0/ D 0 ) and its (constant) linear density '. If we substitutev Ddy=dx, dv=dx D d2y=dx2 in this second-order differential equation, we get the rst-order equationadv dx Dp1Cv2: Solve this differential equation for y0.x/ D v.x/ D sinh.x=a/. Then integrate to get the shape function y.x/D acosh!x a"CC of the hanging cable. This curve is called a catenary, from the Latin word for chain.