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In Goldstein's Classical Mechanics, in the chapter on Rigid Body Mechanics he establishes the fact that no matter what point you choose on the Rigid Body, the angular velocity remains the same. The argument is based on the fact that the angular velocity field in a rigid body must be continuous. My questions are,

1)What does continuity here mean?

2)Are there physical scenarios where the continuity of angular velocity field is violated?

3)If there are such discontinuities if at all, then to what degree do they affect the motion of the body in a qualitative sense?

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A continuous velocity field means that in an infinitesimal change of position within the object, the velocity changes an infinitesimal amount, related by a finite rate of change. By contrast, a discontinuous field means that infinitesimal position changes give a finite velocity change, i.e. an infinite rate of change.

A rigid body cannot have a discontinuous velocity field because that means different parts of the object are moving in ways different from each other. Imagine if you had a wheel where the rim made more revolutions per second than the tire;you couldn't call that wheel a "rigid body" because the rim and tire are sliding past each other. Of course the angular velocity increases with radius, but it increases smoothly with radius, with no sudden jumps. If there is a sudden change in angular velocity between the outside of the rim and the inside of the tire, which are adjacent to each other, that constitutes a discontinuity in the velocity field.

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  • $\begingroup$ How can this argument be used to conclude that angular velocity is the same no matter what point you choose? $\endgroup$
    – physicscircus
    Nov 10, 2017 at 4:11
  • $\begingroup$ @Abhikumbale if it were not the same everywhere, different parts of the object would be rotating at different angular rates. Remember that angular velocity is measured in angle per time - radians per second in SI units, revolutions per minute is common in industrial settings, etc. Obviously if a rigid object is rotating $x$ times per second, all parts of it rotate at that rate. $\endgroup$
    – Asher
    Nov 10, 2017 at 4:19
  • $\begingroup$ Thank you for your explanations. I found a link that accurately answers my question,physics.stackexchange.com/questions/246950/… $\endgroup$
    – physicscircus
    Nov 10, 2017 at 4:31

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