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I am a mathematician wanting to understand the differences between the concepts of angular momentum and centrifugal force.

The following two ideas are clear to me from a physical point of view, but I have a difficult time discerning the difference between them as people tell me they are different but do not give me any explicit reason as to why.

  1. Angular momentum is a vector quantity (taken in the physical sense) of a mass's rotational velocity about some axis.

  2. Centrifugal force is defined on the axis of a rotational reference frame, which depends on the inertia of the object.

My question: What is the subtle difference between this notion of "rotational reference frame" and the notion of the vector quantity of a mass's rotational velocity?. Are they not physically the same point (vector quantity) of rotation about an axis, thus making the meaning of centrifugal force a relative way to speak about the meaning of angular momentum?

I hope this question isn't too naive. I have been really hoping to understand the subtle difference between these physical concepts in a straightforward way.

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    $\begingroup$ An object moving on straight line has a linear velocity $v$, telling us how many m/sec the object makes. For a rotating object is more relevant to speak of angular velocity $\omega$ instead of the linear velocity, i.e. with which angle/sec does the object rotate. Now, in physics we have conservation laws, about quantities remaining constant during the evolution of the object. In some conservation laws, when regarding linear motion another quantity appears instead velocity: linear momentum $p=mv$ where $m$ is the mass of the object. By analogy, for rotating objects we have angular momentum, $\endgroup$ – Sofia Dec 23 '14 at 22:50
  • $\begingroup$ (cont.) $L = I \omega $, where $I$ is here an analog of the mass. Let's now go to the centrifugal force. If you place an object on a gyroscope and the gyroscope begins to rotate, the object will fly away from the gyroscope, unless it is glued to the gyroscope. This is the centrifugal force, it pushes the object away from the center of rotation. In fact, this force is apparent, no force acts on the object, it only doesn't rotate together with the gyroscope. So, the two concepts are different. $\endgroup$ – Sofia Dec 23 '14 at 22:59
  • $\begingroup$ @Sofia Thank you very much! I enjoyed the idea about a gyroscope, as it helped me get a solid picture to think about. $\endgroup$ – cmn1 Dec 24 '14 at 1:31
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I can't quite fathom the source of your confusion (I think it might have something to do with a focus on the notion of rotation here---angular momentum does not require rotational motion), so I'm having trouble writing a really clear response. For the moment I would rather offer a program for practicing the right skills rather than reinforcing the mistaken thinking.

  1. Stop trying to do physics is non-inertial frames until you are totally comfortable doing physics in inertia frames. That means there is no centrifugal (pseudo-)force, only a centripetal component to the forces acting on the body.

  2. Work a lot with angular momentum, get used to the idea that you get the same physics from it no matter what point you chose as the "axis" (though the values of $L$ and $I$ change) and that you can chose a notional axis that does not correspond to a physical pivot.

  3. When you start again with non-inertial reference frames do a non-rotating one first. Get used to the idea that pseudo-forces emerge from using a "wrong" coordinate system and that you can get the results in a "right" coordinate system instead.

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