catenary

[logistic_guy]

Calculate the arc-length of the catenary $\displaystyle \gamma (t)=(t, \cosh t)$ starting at the point $\displaystyle (0, 1)$. This curve has the shape of a heavy chain suspended at its ends – see Exercise $\displaystyle 2.2.4$.

 

[khansaheb]

Calculate the arc-length of the catenary $\displaystyle \gamma (t)=(t, \cosh t)$ starting at the point $\displaystyle (0, 1)$. This curve has the shape of a heavy chain suspended at its ends – see Exercise $\displaystyle 2.2.4$.

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[logistic_guy]

The arc length is:

$\displaystyle s = \int_{0}^{t} \|\dot{\gamma}(u)\| \ du = \int_{0}^{t} \cosh u \ du = \sinh t$