Some months ago a professor of mine showed us a 'proof' of why $g\approx 9.8 ~\text{m}/\text{s}^2$ (the gravitational acceleration at the surface of the Earth) is 'equal' to $\pi^2\approx9.86\dots$ Using a differential equation that I think is used to model the movement of a pendulum of something like that.
Does anyone know the DE I'm talking about? Or, has anyone heard such story?
Maybe this helps: link Looks like some time ago the meter was defined to be essentially the length of the "seconds pendulum", i. e. the pendulum whose period is two seconds
The oscillation time of a pendulum is given by $T = 2\pi\sqrt{\frac{L}{g}}$. With $T = 2$ and $L = 1$ this gives $g = \pi^2$