Why is e seen in seen in gravity? (catenary curves)

Question

I have been learning about how catenary curves, described as a rope suspended between two poles supported by only its own weight can be represented as some transformation of the curve cosh(x). This function is derived from the number e, and I am wondering why exactly e is present here in this phenomenon?

Answer 1

Nothing special with that, when we write down the governing equation for equilibrium, under the load due to the self weight of the rope, we obtain a differential equation which has that shape as a solution.

Note that when the rope is not much sloped then the shape is very similar to a parabolic arch, which indeed if the shape of equilibrium for an uniform load (i.e. as for the cables of suspension bridges).

Answer 2

The equation for a catenary is $y=\frac a2\left(e^{\frac xa}+e^{-\frac xa}\right)$. If you want to use some other base for the exponentials, that just changes the value of $a$. For example, if you want to use $5$, note that $e^{\frac xa}=\left(e^{\log 5}\right)^\frac x{a \log 5}=5^\frac x{a \log 5}$ and you can absorb it into the value of $a$.