I try to solve a variant of the falling ladder problem, this time without a wall and the bottom of the ladder does not slide. There is a mass $m$ and the angle of the ladder with the vertical is $\phi$.
The rotating moment is caused by the gravitational force:
$$M_r = F_z l \sin \phi = m g l \sin \phi $$
Which equals the rotational inertial moment:
$$M_\phi = ml^2 \frac{d^2 \phi}{dt^2}$$
This results in the following differential equation:
$$\frac{d^2 \phi}{dt^2} = \frac{g}{l} \sin \phi$$
Any idea how to solve this?
This is the simple pendulum equation. Unfortunately, there is no closed form solution (except for the trivial $\varphi=0$ unstable ladder that never slides).