Question

Q. Calculate the arc length of catenary γ(t) = (t, cosh t) starting at the point (0, 1).

Answer 1

Answer on Question #70830– Math – Geometry

Question

Calculate the arc length of catenary $\gamma(t) = (t, \cosh t)$ starting at the point $(0, 1)$.

Solution

The arc Length is given by

$$L = \int_{a}^{\beta} \sqrt{\left(\frac{dy}{dt}\right)^2 + \left(\frac{dx}{dt}\right)^2} dt,$$

where

$$x(t) = t, \quad y(t) = \cosh t$$ $$\frac{dx}{dt} = 1$$ $$\frac{dy}{dt} = (\cosh t)' = \sinh t$$ $$\left(\frac{dy}{dt}\right)^2 + \left(\frac{dx}{dt}\right)^2 = (\sinh t)^2 + 1 = \cosh^2 t.$$

Then

$$L = \int_{0}^{t} \sqrt{\cosh^2 t} dt = \int_{0}^{t} \cosh t dt = \sinh t \Big|_{0}^{t} = \sinh t - \sinh 0 = \sinh t$$

Answer:

The arc length of catenary is $L = \sinh t$

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