this might be a trivial problem but me and a friend have failed to solve it. This is not homework, it actually relates to a book of mine but that is of no interest. Imagine the following;
Two masses $M$ and $m$, where $M > m$, rotating around their center of mass $P$ at the angular velocity $\omega$. We further assume both $M$ and $m$ to be point masses.
$R$ is the distance from $M$ to $P$ as defined by $R = \frac{mr}{M+m}$, where $r$ is the distance between $M$ and $m$. Assume we know the angular velocity $\omega$.
Now, suppose a small amount of mass $\Delta M$ "drops" from $M$ at a constant rate. I.e. $M$ becomes smaller as time advances. What happens to the system ?
There are a few statements we can make. We know the momentum carried by the leaking mass is equal to $p = \Delta M v_{tangent}$. Where $v_{tangent} = \omega R$. We also know the total (angular) momentum has to be constant as well.
It then seems "What happens to the system" is a question easily answered by applying the conservation of angular momentum. Whatever momentum $\Delta M$ carries away has to subsequently be subtracted from the rotating system. Which intern means the Inertia, angular velocity or both of the rotating system have to decrease. Indeed it odd to be both since the Inertia changes by virtue of the system having lost mass.
But here is the problem, the Center of mass has also changed. It has shifted towards $m$. Thus $R$ has increased which again implies the system has to slow down.
This is all nice and well pondering wise. It should be obvious that the system cannot exactly speed up by losing mass in this way. But describing this mathematically seems a bit beyond me at the moment. What i struggle with the most is the apparent simultaneous change in angular velocity and inertia.
The Inertia changes because we lose $\Delta M$, and the Angular velocity changes because $R$ increases. And i dont really know how to determine how much each changes in a physically correct way. Of course i could say "We know the Angular momentum changes by $X$, so we say half of this change is done by a change in inertia and the other half by $\omega$" but i doubt that is correct.
This is how far i got with "reasoning". I say the system has to slow down, radical thought i know. But how this change is translated, i do not know. But i would like to D: So yeah, any help would be greatly appreciated ! Thanks for reading !
