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I'm trying to verify a line of thought.

Consider, we have two cases. In the first case, we have a rotating disc about a point, and we have two points marked on the disc at distance $r_1$ and $r_2$ from the center. In the second case, we have two independent objects revolving about the point, at distances $r_1$ and $r_2$, following the inverse square law.

In the first case, the velocity increases as we move away from the center of the sphere. Hence, if we have $r_1 \lt r_2$, then we essentially have $v_1 \lt v_2$.

In the second case, we have a completely opposite behavior. Because of the inverse square law, we have velocity decrease, as the distance increases. So, here we have $v_1 \gt v_2$.

Both are examples of circular motion, however, the two systems seem to act very differently. Is this effect purely because, in the first case, by considering an extended body we have fixed the value of $T$ or $\omega$ for the entire body, and every point on it. Hence, as distance increases, so does the velocity.

In the second case, however, the value of $T$ is no longer set and depends upon the inverse square force. If we equate the gravitational force to the centripetal force needed to maintain circular orbit, we find that velocity decreases as radius increases.

Is this the reason, why the two cases differ so much, even though they both represent uniform circular motion ? A better explanation would be extremely welcome.

Also, in Kepler motion, angular momentum is conserved. However, if angular momentum is conserved, velocity should be inversely proportional to distance. Is angular momentum conserved along different points of the same orbit, or is it also conserved between two separate orbits ? Like, I know I can find the velocity at different points of an elliptic orbit due to conservation of angular momentum, but I can't compare two separate orbits that way, can I ?

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Both are examples of circular motion, however, the two systems seem to act very differently. Is this effect purely because, in the first case, by considering an extended body we have fixed the value of $T$ or $\omega$ for the entire body, and every point on it. Hence, as distance increases, so does the velocity.

In the second case, however, the value of $T$ is no longer set and depends upon the inverse square force. If we equate the gravitational force to the centripetal force needed to maintain circular orbit, we find that velocity decreases as radius increases.

Yes, this is the reasoning. In the disk case the two points are related by the same angular velocity. In the planetary orbit this is no longer the case.

Is angular momentum conserved along different points of the same orbit, or is it also conserved between two separate orbits ? Like, I know I can find the velocity at different points of an elliptic orbit due to conservation of angular momentum, but I can't compare two separate orbits that way, can I ?

This is also correct. Along a given orbit trajectory the angular momentum is conserved, since the force acting is central, and hence there is no external torque acting on the system. However, there is no reason to assume the angular momentum is the same between two orbits; not all orbits have the same angular momentum.

Something to also be careful of is comparing two points at different radii and actually moving from one radius to the other in a trajectory. In the disk case it looks like you are considering the former, but for the orbit case you are considering the latter. Despite this, it looks like you have a good handle on both systems and how to think about them.

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