Rotating a quarter circle -- how long has a point traveled. Question: see below quarter circle $AOB$. $P$ is the midpoint of $AO$. $OM$ is considered as the "ground surface". We keep rotating $AOB$ to the right, until $OB$ sits on the ground surface again. How long has $P$ travaled during this time?
This puzzle reminded me of this infamous SAT question: https://mindyourdecisions.com/blog/2015/07/05/everyone-got-this-sat-math-question-wrong-can-you-solve-it-sunday-puzzle/
But it looks even harder since it's not a full circle, rather, a oddly shaped quarter circle $AOB$...
[EDIT] as some hints suggested, the most difficult part is when "The circular arc rolls on the ground". How exactly do I calculate that. Looks like it's part of the https://mathworld.wolfram.com/CurtateCycloid.html and it looks awfully complicated..


The difficult segment is the second where the quarter circle rolls $\frac\pi2$ arc on the ground. Assume unit radius, you may parametrize the path
of $P$ with
$$x=t+\frac12\cos t ,\>\>\>\>\>y=-\frac12\sin t$$
Then, the path length of the second segment is
$$ \int_0^{\pi/2}\sqrt{(x_t’)^2 + (y_t’)^2 }dt=\int_0^{\pi/2} \sqrt{\frac54-\sin t}dt=1.19
$$
where the integral is elliptic, evaluated numerically.
