JOHN
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JOHN asked in Science & MathematicsMathematics · 5 years ago

A uniform cable AB of line density σ hangs freely under gravity with its two ends A, B fixed at points on a horizontal line. The?

A uniform cable AB of line density σ hangs freely under gravity with its two ends A, B fixed at points on a horizontal line. The tension at the lowest point is T₀. Prove that in equilibrium the cable assumes the shape y = ccosh(x/c), where c = T₀/ σ (the x-axis being below AB and || to it and the y-axis passes through the lowest point.

[This problem can be solved by minimising the potential energy using the methods of the calculus of variations. This is NOT the preferred solution, which is one using entirely elementary methods].

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