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Or more specificly: Is it possible to have a system where angular momentum is conserved,but not linear?

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They are independent. This of a mass roating around a central point in a circular orbit. L is conserved but P is not.

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    $\begingroup$ Why isn't P conserved? If there are no external forces then it should be. $\endgroup$ – sammy gerbil Dec 20 '18 at 20:08
  • $\begingroup$ there is an external force, the centripetal force by the rope, or whatever is making the object's motion circular. v is constantly changing direction, and so is P. $\endgroup$ – Wolphram jonny Dec 21 '18 at 0:42

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