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Unformatted text preview: y 2 = Cy 2-1. Thus dy p Cy 2-1 = ± dx. (9) The integral of the left hand side of this equation can be found from a table of indeﬁnite integrals. We obtain ln( p Cy 2-1 + √ C y ) √ C = ± ( x-x ) , (10) where x is a constant of integration. We may change the origin of the x axis such that x = 0. Further solving the equation for y , we get y = e √ Cx + e-√ Cx 2 √ C . (11) Introducing a length parameter L = 1 / √ C , we have y = L cosh x L , (12) a neat result! It is plotted in Figure 1 (next page). The hyperbolic cosine, cosh θ ≡ e θ + e-θ 2 , is actually the cosine of an imaginary angle : cosh θ = cos( iθ ), with i ≡ √-1. 2 M 2 M 1 1 2 x L 1 2 3 4 y L Figure 1: The soap ﬁlm is formed by any segment of the above curve (say x 1 < x < x 2 ) rotated about the x axis. 3...
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This note was uploaded on 01/22/2012 for the course PHYS 3202 taught by Professor Tan during the Spring '11 term at Georgia Institute of Technology.
- Spring '11