ABSTRACT
To analyze the shape of a chain hung from two pegs, we model the chain as a line of length γ and constant mass per unit length ρ. The two ends of the line hang from two fixed points (a, A) and (b, B) in the x, y-plane with gravity in the negative y direction [and with ( b − a ) 2 + ( B − A ) 2 < γ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749821/5a9bd1b0-aa97-4dfe-adb3-8f69a65a6308/content/eq2226.tif"/> and b − a < γ]. Let the hanging shape of the chain be described by the plane curve y(x) with the coordinate axes adjusted so that y(x) > 0 for all x in [a, b] (see Fig. 10.1). From mechanics of deformable bodies, we expect the chain to hang in such a way that its potential energy is a minimum. A differential chain element of lenght ds at (x, y) has mass ρ ds and potential energy (relative to y = 0) ρgy ds. The total potential energy of the chain hanging in the shape y(x) is given by () J [ y ] = ∫ a b ρ g y 1 + ( y ′ ) 2 d x . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749821/5a9bd1b0-aa97-4dfe-adb3-8f69a65a6308/content/eq2227.tif"/> https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749821/5a9bd1b0-aa97-4dfe-adb3-8f69a65a6308/content/fig10_1_OB.tif"/>
