We all have done this!

As no one has given a solution till now, I am giving now.

If we consider the motion of the loop in the centre of the loop frame, the loop performs uniform rotation. With this in mind, we can use

$T = \rho v^2$

where $T$ is the tension, $\rho$ is the linear mass density, and $v$ is the speed of all elements of the loop.

Now, we know $F = T$

$\Large \Rightarrow \boxed{v=\sqrt{\dfrac{FL}{M}}}$

So that $\begin{aligned}\ell &= I\omega\\&=2\pi R\rho R^2 \dfrac{v}{R}\\&=2\pi R^2\sqrt{\dfrac{MF}{L}}\end{aligned}$ $\Large\Rightarrow \boxed{\ell = 2\pi R^2\sqrt{\dfrac{MF}{L}}}$

And, $\begin{aligned}KE_{tot} &= I\omega^2\\&=2\pi R \rho v^2\\&=2\pi\rho R \dfrac{F}{\rho}\end{aligned}$ $\Large \Rightarrow \boxed{KE=2\pi RF}$

$\large\color{#3D99F6}{\therefore\boxed{v+\ell+KE = \sqrt{\dfrac{FL}{M}}+ 2\pi R^2\sqrt{\dfrac{MF}{L}}+2\pi RF}}$

Does this really happens?? I have never seen it in reality!