A Classical Mechanics challenge for fun

[zwierz]

I composed a problem and propose it here. I know the solution so it just for fun of the participants.

There is a cylindrical bobbin of radius $r$; the bobbin rotates about its central axis with angular velocity $\omega=const>0$. An inextensible weightless string is coiled around the bobbin. The string is very thin and very long. On a free end of the string there is a point mass $m$. Let $l=l(t)$ stand for the length of the free part of the string. There is no gravity.
Initially the value $l_0=l(0)>0$ and the velocity $v=\frac{d}{dt} l\mid_{t=0}$ are given. Moreover it is known that $-\omega r<v<0$. Find minimal value of the function $l(t)$ that is $\min_{t\ge 0}l(t)$ -- ?.
We consider motion such that the string remains strained for all time

 

[JoePhysics]

Sounds like an interesting problem. I've seen similar ones in graduate qualifiers in the classical mechanics section.

 

[Greg Bernhardt]

After some discussion we are allowing this as a stand alone but unofficial challenge. Good luck!

 

[zwierz]

Spoiler

the Lagrangian is
$$L\Big(l,\frac{d l}{dt}\Big)=\frac{m}{2}\Big(l^2\omega^2+\frac{l^2}{r^2}\Big(\frac{d l}{dt}\Big)^2 \Big);$$
the answer is $$l_0\sqrt{1-\Big(\frac{v}{\omega r}\Big)^2}$$