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Say I have this disk with radius $R$, mass $M$ that could rotate with angular velocity $\omega_0$ around it's CoM freely (no friction, etc). Then we have a smaller disk with radius $r$, mass $m$ that could also rotate around it's CoM freely with angular velocity $\omega_1$ relative to the larger disk. We then placed the smaller disk on the edge of the larger disk, so that the distance between their CoMs is $x$ (which in this case, $x$ = $R$). Both disks are rotating in the clockwise direction. So I tried to draw the system and it looks like this. the sketch

How do we calculate the angular velocity of the smaller disk relative to us (say we are observing the motion from above)?

If we had a different system e.g. we placed the smaller disk on top of the larger disk, their CoMs aligning with each other ($x$ is zero), calculating the angular velocity of the smaller disk from our frame of reference would be pretty straight-forward the sum of $\omega_0$ and $\omega_1$. But how do we calculate the angular velocity from our frame if the system is like the picture shown above? What I think is that the equation would obviously depend with the distance between the CoMs $x$ because I tried to trace the path created by a point of mass on the smaller disk and found that the larger the value $x$ is, the path tends to make this curling pattern (is this what I should be expecting?). And if the path i tried to trace was indeed correct, then is the angular velocity of the smaller disk from our frame of reference not constant with respect to time? How can I express it in a function?

I tried to look up the web to see similar topics that discuss about this particular system and couldn't really find one, so if you happen to know, please feel free to leave a link.

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If the small disk did not rotate relative to the large disk then its rotation rate (angular velocity) relative to us would be the same as that of the large disk $\omega_0$. This answer does not depend on the distance $x$ between the centres of the two disks.

If the small disk rotates at rate $\omega_1$ relative to the large disk then its rotation rate relative to us is $\omega_0+\omega_1$. Again this does not depend on the distance $x$ between the centres of the disks.

The linear velocity of any point on the small disk can be calculated in the same way as any other relative velocity. It is the vector sum of its linear velocity relative to the centre $O_1$ of the small disk plus the linear velocity of $O_1$ relative to the centre $O$ of the large disk (which is stationary relative to us).

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  • $\begingroup$ Hmm I'm starting to understand, but could the angular velocity of the smaller disk vary with respect to time? If we had a system where R = r, larger disk makes a rotation once every 4 seconds, so does the smaller disk (which is placed on the edge of the larger disk) surely ω0+ω1 applies because the path created by a point of mass on the smaller disk is larger, hence the sum. But if we had a different system such that the smaller disk rotates once every 2 seconds, wouldn't there be times when the point of mass is rotating "against" the larger disk hence this curling pattern that I'm seeing? $\endgroup$ – Felix Dec 29 '19 at 9:55
  • $\begingroup$ @Felix I understood from your question that both disks rotate with fixed angular velocities, either because there are motors controlling each, or because they are set in motion and rotate freely, there being no friction between them nor between them and the axles on which they rotate. If this is correct then neither angular velocity varies with time. $\endgroup$ – sammy gerbil Dec 29 '19 at 10:21
  • $\begingroup$ The paths traced out by points on the smaller disk will be circles when viewed from the larger disk, which is rotating, but when viewed by a stationary observer these paths are circles upon circles. $\endgroup$ – sammy gerbil Dec 29 '19 at 10:30
  • $\begingroup$ Yes, both disks rotate freely and frictional forces are negligible. So if I understood your response correctly, even though the path traced by a point on the smaller disk isn't a perfect circle rather in the form of "circles upon circles", it's angular velocity wouldn't vary with time? $\endgroup$ – Felix Dec 29 '19 at 11:08
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    $\begingroup$ The reason these two angular velocities are different is that for the rotation of the small disk the 2 points which you are tracking are both fixed in the small disk, whereas for the motion of one point on the small disk about one point on the large disk, the two points are fixed in different disks which are moving relative to each other. $\endgroup$ – sammy gerbil Dec 29 '19 at 16:12

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