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CLASS18 - 1 CHAPTER 18 THE CATENARY 18.1 Introduction If a...

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1 CHAPTER 18 THE CATENARY 18.1 Introduction If a flexible chain or rope is loosely hung between two fixed points, it hangs in a curve that looks a little like a parabola, but in fact is not quite a parabola; it is a curve called a catenary , which is a word derived from the Latin catena , a chain. 18.2 The Intrinsic Equation to the Catenary FIGURE XVIII.1 We consider the equilibrium of the portion AP of the chain, A being the lowest point of the chain. See figure XVIII.1 It is in equilibrium under the action of three forces: The horizontal tension T 0 at A; the tension T at P, which makes an angle ψ with the horizontal; and the weight of the portion AP. If the mass per unit length of the chain is µ and the length of the portion AP is s , the weight is µ sg . It may be noted than these three forces act through a single point. Clearly, T T 0 = cos ψ 18.2.1 and µ ψ sg T = sin , 18.2.2 from which ( ) µ sg T T 2 0 2 2 + = 18.2.3 and tan . ψ µ = gs T 0 18.2.4 T T 0 A P ψ mg
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2 Introduce a constant a having the dimensions of length defined by a T g = 0 µ . 18.2.5 Then equations 18.2.3 and 4 become T g s a = + µ 2 2 18.2.6 and s a = tan . ψ 18.2.7 Equation 18.2.7 is the intrinsic equation (i.e. the s , ψ equation) of the catenary. 18.3 Equation of the Catenary in Rectangular Coordinates, and Other Simple Relations The slope at some point is , tan ' a s dx dy y = ψ = = from which ds dx a d y dx = 2 2 . But, from the usual pythagorean relation between intrinsic and rectangular coordinates ds y dx = + ( ' ) / 1 2 1 2 , this becomes ( ' ) ' . / 1 2 1 2 + = y a dy dx 18.3.1 On integration, with the condition that y ' = 0 where x = 0, this becomes y x a ' sinh( / ), = 18.3.2 and, on further integration, ( ) . / cosh C a x a y + = 18.3.3 If we fix the origin of coordinates so that the lowest point of the catenary is at a height a above the x
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