Physical System for $x'(t)=\sin (x(t))$

Question

I'm asked to find a system that is modeled (or at least approximated) by $x'(t)=\sin (x(t))$

Differentiate yields $x''(t)=\cos (x(t)) \sin (x(t))$ and integrating (care of Wolfram Alpha ) yields $x(t)=2 \cot ^{-1}(e^{-c-t})$ , but I am at a loss to find an actual physical example that follows this relationship

Answer 1

Imagine a bowl whose shape is given in cylindrical coordinates by $z=c-\frac{1}{2g}\sin^2(r)$ for $r\in[0,\frac{\pi}{2}]$ say. Then take a ball and place it at rest anywhere in this bowl. The energy of the ball is $E = \frac{1}{2}mr^{\prime2}+mgz$, so $\frac{1}{2}r^{\prime2} = \frac{E}{m}-g(c-\frac{1}{2g}\sin^2(r)) = \frac{E}{m}-gc+\frac{1}{2}\sin^2(r)$. So if we now choose $c=\frac{E}{mg}$, we end up with $\frac{1}{2}r^{\prime2} = \frac{1}{2}\sin^2(r)$ and $r^\prime = \sin(r)$.

For this to be proper, you'd have to worry about taking the square root in the end and the point $r=0$ and dimensions working out and stuff, but you get the idea.

Answer 2

An inverted pendulum in a very viscous medium.