# Two disks on frictionless surface, demonstration

Consider a system composed of two plane disks, which we will designate as $D$ and $d$. Their radii are $R$ and $r$ respectively $(R>r)$. The disk $d$ is fixed over the disk $D$ and a distance $b$ from the center, as in the figure. The disk $D$ can spin and move freely in a friction-free platform, while the other disk $d$ is fixed on a point of disk $D$, but can spin freely without friction. No external forces act on the system. The masses of the disks are $M$ and $m$ respectively.

So let's just say the problem asks me in one of its to calculate the angular momentum of the system with respect to the center of $D$, this is a non inertial system of reference, but this does not influence a lot if I do not do forces yet right? In other words, the angular momentum will just be?...

$\mathbf{L_{D}}=\frac{1}{2}\left(\frac{1}{2}MR^{2}\right)\mathbf{\dot{\theta}}\hat{k}+\mathbf{b}\times m\left(\mathbf{v_{d}-v_{D}}\right)+\frac{1}{2}\left(\frac{1}{2}mr^{2}\right)\dot{\mathbf{\varphi}}\hat{k}$

Where $v_{d}$ and $v_{D}$ are the velocities of the center of mass of the small disk and big disk respectively. Bold are vectors. And if from there I wanted the total momentum do I just add $\mathbf{L}=\mathbf{R_{D}}\times(M\mathbf{v_{D}})$? Where $R_{D}$ stands for the vector that goes from the origin to the center of the disk $D$

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Have you succeded in proving that $\ddot \phi = 0$? –  pppqqq Jan 6 at 15:30
Hi Salvador2395 - your edit helps a lot, but still, it seems that you're mostly asking for someone to check your work. Could you expand on the actual conceptual problem you're having? In other words, what is actually preventing you from continuing with the problem? You've proposed a next step you could take; what happens if you do that? –  David Z Jan 9 at 20:02
Can you please tell about which axis/axes the discs are rotating ? –  Rijul Gupta Jan 15 at 11:09
Yes, the disk $D$ is rotating with respect to its center of mass and $d$ with respect to its center of mass. –  LagraMen0123 Jan 15 at 15:07