Rotating a quarter circle -- how long has a point traveled.

Question

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..

Answer 1

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.

Answer 2

Hint: Break up the path $P$ traveled into pieces:

1. The quarter-circle pivots about $B$
2. The circular arc rolls on the ground
3. The quarter-circle pivots about $A$
4. The quarter-circle pivots about $O$

The lengths of (1), (3), (4) are easy, (2) requires some calculation.

Answer 3

Hint: Try breaking the rotation up into 90-degree phases.