Physics 325 Spring 2011 Problem Session 9 Solutions

Physics 325 Spring 2011 Problem Session 9 Solutions - = 1 c...

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Physics 325 Spring 2011 Discussion 9 April 4, 2011 A uniform chain of mass per unit length μ is suspended on two points, one at x = - L , and the other at x = L . Assume that both support points are at the same height, that is y ( L ) = y ( - L ) . Due to gravity g , the chain forms a curve y ( x ) that minimizes the potential energy. Find this curve y ( x ) . You may leave one undetermined constant in the answer. Solutions : The potential energy is U = Z gy dm = Z gyμds = Z L - L μgy q 1 + y 0 2 dx = μg Z L - L f dx, where f = y q 1 + y 0 2 . In order to minimize U , f must satisfy the Euler-Lagrange equation. d dx ± ∂f ∂y 0 ² - ∂f ∂y = 0 0 = d dx ± y y 0 p 1 + y 0 2 ² - q 1 + y 0 2 0 = y 0 y 0 p 1 + y 0 2 + y y 00 p 1 + y 0 2 - y y 0 2 y 00 (1 + y 0 2 ) 3 / 2 - q 1 + y 0 2 0 = yy 00 p 1 + y 0 2 - yy 0 2 y 00 (1 + y 0 2 ) 3 / 2 + y 0 2 - (1 + y 0 2 ) p 1 + y 0 2 0 = yy 00 (1 + y 0 2 ) (1 + y 0 2 ) 3 / 2 - yy 00 y 0 2 (1 + y 0 2 ) 3 / 2 - 1 p 1 + y 0 2 0 = yy 00 (1 + y 0 2 ) 3 / 2 - 1 p 1 + y 0 2 Multiply both sides by - y 0 , and notice that d dx ± 1 p 1 + y 0 2 ² = - 1 2 1 (1 + y 0 2 ) 3 / 2 2 y 0 y 00 = - y 0 y 00 (1 + y 0 2 ) 3 / 2 . 1
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Thus, the Euler-Lagrange equation becomes 0 = - yy 0 y 00 (1 + y 0 2 ) 3 / 2 + y 0 p 1 + y 0 2 0 = y d dx ± 1 p 1 + y 0 2 ² + dy dx 1 p 1 + y 0 2 0 = d dx ± y p 1 + y 0 2 ² The last step in the above comes from product rule. y p 1 + y 0 2 = constant
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Unformatted text preview: = 1 c c 2 y 2 = 1 + y 2 y = dy ddx = p c 2 y 2-1 Z dy p c 2 y 2-1 = Z dx To compute the integral on the left hand side, substitute cy = cosh . Z 1 c sinh d p cosh 2 -1 = x + A c = x + A y ( x ) = 1 c cosh = 1 c cosh c ( x + A ) The constant A can be determined using the condition y ( L ) = y (-L ) . 1 c cosh c (-L + A ) = 1 c cosh c ( L + A ) c (-L + A ) = c ( L + A ) If we take the positive sign, we get c = 0 , which is not physical. If we take the negative sign, we get A = 0 . Thus, y ( x ) = 1 c cosh cx. The constant c can be determined implicitly from calculating the total mass of the chain. 2...
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Physics 325 Spring 2011 Problem Session 9 Solutions - = 1 c...

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