Contraction of a rotating system Consider a system of two masses that rotates with constant angular velocity. When a force contracts the system the velocitie of the two masses increase.
I understand this in terms of conservation of angular momentum but I would like to understand how does the force that cause the contraction accelerates the two masses.

Using polar coordinates, the force is central, therefore radial. This means that $v_r$ of the two masses increase, while $v_{\theta}$ should remain constant. During the contraction the motion is a spiral, so the velocity is not perpendicular to the force, hence the magnitude of the velocity vector changes. But at the end, when the system is compressed the two masses follow a circular motion which is faster than the one at the beginning. This means that $v_{\theta}$ has somehow increased, but how?
The increase of the magnitude of the velocity does not imply the increase of the component perpendicular to the radial direction. This increase seems impossible to me since the force itself is radial.
How can $v_{\theta}$ increase during the motion?
 A: Let's look at the hodograph of a constant radius & constant velocity motion.

Left: trajectory of one of the masses. Right: hodograph, i.e. locus of the velocity vectors.
Now, let's look closer at how the velocity changes during a small time interval $\mathrm dt$.

A force is needed to rotate it (difference between the brown and red arrows). If you exert a larger force, you see that:


*

*the velocity increases in norm (the magenta arrow is longer)

*the velocity rotates faster (the angle red-magenta is bigger than red-brown)


The key to understanding the phenomenon is realizing the radial & orthoradial directions are not fixed: the radial direction at time $t$ will soon be the orthoradial direction at some time $t'$. Thus, when you say "the radial force changes $v_r$", in fact you should say "the radial force changes both $v_r$ and $v_θ$".
For a more formal explanation, note the acceleration along $\hat r$ and $\hat θ$ is not the derivative of the amplitude of velocity along $\hat r$ and $\hat θ$. Indeed, $\vec v=v_r \hat r+v_θ \hat θ$, so $\vec a=(\dot{v_r}-v_θ\dot θ)\hat r+(v_r\dot θ+\dot{v_θ})\hat θ$, that is $a_r=\dot{v_r}-v_θ\dot θ$ and $a_θ=v_r\dot θ+\dot{v_θ}$. Hence $a_θ=0$ does not imply $v_θ=\text{const.}$, rather $\dot{v_θ}=-v_r\dot θ$: because of rotation ($\dot θ≠0$), radial velocity ($v_r$) is "converted" into variation of orthoradial velocity ($\dot{v_θ}$).
