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What is relative angular velocity of A wrt B and that of A wrt C in the figure given below?

Description:

  • A, B and C lie on a solid cylinder(rigid body) rotating with a constant angular velocity $\vec{\omega}$ about z axis
  • A and C lie on the plane perpendicular to $\vec{\omega}$
  • B lies directly below A with the same distance from the axis as A i.e. they lie in the line parallel to axis of rotation(z axis)

What I think are correct answers:

a) Relative angular velocity of A wrt C is $\vec{\omega}$ only (even direction is same as the original $\vec{\omega}$). This is proved in the answer of this question:Relative angular velocity of point with respect to another point

b) Relative angular velocity of A wrt B is zero because relative velocity of A wrt to B is zero.

Are my answers correct?

enter image description here

What would be the interpretation of non-zero angular velocity of A with respect to C and that of zero angular velocity of A wrt B? If i sit on C and observe A do i see A moving wrt C, but if i sit on B and observe A will i see A still? If i see A moving do i need to move my head also to continually see A?

dark_prince
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1 Answers1

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I would say that your answers are correct. Note that point A makes one revolution around point C for each revolution around the axis.

R.W. Bird
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  • just one more doubt, while standing at C and observing A do I see A revolving around me? do I need to move my head to continually observe A? if not, then what is the interpretation of non zero angular velocity? – dark_prince Sep 01 '20 at 15:10
  • If point C is moving with the cylinder, sit on the cylinder at point C, facing outward toward point A. You will see things in the background (not moving with the cylinder) moving relative to point A. After the cylinder, you, and points C and A have made one revolution, the background will be back to where it started relative to point A. Every point in the cylinder has the same angular velocity relative to every other point in the cylinder. – R.W. Bird Sep 01 '20 at 18:33
  • @R.W.Bird: Wouldn't that also be true if you were sitting at B and facing A? But I think from the derivation in the linked thread that the relative angular velocity of A with respect to B was shown to be 0. – Not_Einstein Sep 01 '20 at 18:39
  • Maybe the resolution of my last comment is that A making one rotation relative to the background, as seen from B or C, for each rotation of the body just means A has angular velocity with respect to the background (or the "lab frame"), but not with respect to either B or C. – Not_Einstein Sep 01 '20 at 21:46
  • @R.W.Bird but that would be true even if we sit on B and observe A – dark_prince Sep 02 '20 at 01:15
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    How about if you sit on B and observe A in line with a distant star? But I do need to qualify my statement: Each point in a rotating rigid object will have the same angular velocity relative to any line, passing through and moving with the object, which is parallel to the axis of rotation (unless the point is on that line). Note that any such line passes through many other points. – R.W. Bird Sep 02 '20 at 15:08