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Say we have a spinning ring of mass $M$, rotating at $W_0$, at a radius $r$ from some pivot point. This ring has massless spokes extending out to a length of $2r$.

enter image description here

From this, we can calculate the initial angular momentum:

$L_0 = IW_0$

$I_0 = Mr^2$

$L_0 = Mr^2W_0$

While it's spinning, we drop on top of the massless spokes another ring of twice the radius, $2r$, but the same mass, $M$. Friction between the spokes and the ring makes sure the two rings spin together.

enter image description here

Conservation of angular momentum should allow us to find the final angular velocity of the two rings together:

$L_0 = L_f$

$I_fW_f = I_0W_0$

$I_f = Mr^2 + M(2r)^2 = 5Mr^2$

$5Mr^2W_f = Mr^2W_0$

$W_f = \frac{W_0}{5}$

However, I've always learned that conservation of angular momentum doesn't break conservation of momentum. That is, conservation of momentum should still hold.

However, if I try to solve this same problem using only conservation of linear momentum, I get a different result. Let $P$ stand for momentum, $V_{inner}$ stand for the final linear velocity of little segments of the inner disk, and $V_{outer}$ stand for the final linear velocity of little segments of the outer disk:

$P_0 = MV = MrW_0$

$P_0 = P_f$

$MV_0 = MV_{inner} + MV_{outer}$

$MrW_0 = MV_{inner} + MV_{outer}$

$MV_{inner} = MrW_f$

$MV_{outer} = M(2r)W_f$

$MrW_0 = MrW_f + M(2r)W_f$

$MrW_0 = 3MrW_f$

$W_f = \frac{W_0}{3}$

As you can see, conservation of linear momentum gives me a different result...what am I doing wrong?

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  • $\begingroup$ Please do not respond to answers in edits, and in particular do not edit your question after an answer has been given that makes that answer no longer an answer to your question. If you have a new question, please ask a new question. $\endgroup$ – ACuriousMind Dec 27 '18 at 14:15
  • $\begingroup$ @ACuriousMind ok agreed, I will $\endgroup$ – Joshua Ronis Dec 27 '18 at 14:16
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Momentum is a vector quantity. The total momentum of a uniform spinning disc is zero, so the momentum begins at zero and ends at zero.

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  • $\begingroup$ knzhou thanks for your answer, I just liked it and its true, I hadn't thought of that. But I'm still conceptually really confused. Could you please look at my edit? It's a bit long, and asks a slightly different question, so perhaps I'll just repost the entire thing tonight under a different title and close this one, but before I do that I'd really appreciate if you could take a look at it. Thanks! $\endgroup$ – Joshua Ronis Dec 27 '18 at 14:09
  • $\begingroup$ @JoshuaRonis Honestly, I'm having trouble picturing what you mean with the lines. You might want to add a picture or make a separate question for that. $\endgroup$ – knzhou Dec 27 '18 at 14:14
  • $\begingroup$ Hi @knzhou. I just asked a separate question: I'll post the link here: physics.stackexchange.com/questions/451758/… $\endgroup$ – Joshua Ronis Jan 2 at 19:33
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Let's consider following Situation: A ball is hit with a stick off-center. It starts sliping. There is friction on ground. This ensures that ball starts rolling after some time. Now,you can calculate from conservation of angular momentum it's velocity after it starts rolling( By taking origin at ground,so that friction applies no torque). Now since there is friction force,you can't apply conservation of momentum. Hence:
Conservation of Angular momentum[math]\centernot \iff [/math] Conservation of Momentum

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