I was recently solving a problem which had rotation of rotating axis. The answer was pretty simple: just vector addition of their angular velocities. After searching up a bit about rotation of axis of rotation, I came across Euler rotational theorem. But I was a bit confused with the statement that the resultant axis of rotation is not necessarily fixed and can change with time. What does that even mean? The theorem clearly states that there can be only one axis of rotation, but if that axis is itself changing, won't it result in some new axis of rotation and so on? I know I am missing something but have no idea what. (Please try to keep the language simple as I am a high school student.)
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$\begingroup$ Not strictly related : Velocity in a turning reference frame. $\endgroup$– VoulkosCommented Jun 6, 2021 at 22:15
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$\begingroup$ google.com/… , see picture of rotating fan, as time happens , the axis which blades spin rotates $\endgroup$– BrianCommented Jun 7, 2021 at 1:39
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2 Answers
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The is only one rotation axis at each instant in time, but as time evolves the axis can change. In fact, it can change abruptly and its location/direction cannot be determined from the values at previous times.
If you were looking at a single frame in time (like a movie frame) and know the velocities of each object, then you can definitely calculate the position, direction, and magnitude of rotation.
But in the next frame, the same calculation might result in different values, as velocities can change abruptly (like during impacts for example).
answered Jun 6, 2021 at 21:08
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1$\begingroup$ so it will be an extremely chaotic motion with axis of rotation just changing from time to time $\endgroup$ Commented Jun 7, 2021 at 7:57
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$\begingroup$ It can be chaotic, or smooth or constant or fixed. Anything is possible from a physics standpoint. $\endgroup$ Commented Jun 7, 2021 at 12:12
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It is a consequence of the hypothesis of a rigid body. A simple example is a dumbell, with two masses $M$ linked by a rod with negligible mass compared to $M$.
For an observer in the center of mass, or both masses are at rest, or if they move, its instantaneous velocity must be perpendicular to the rod and one opposite to the other (otherwise its relative distance changes and the body is not rigid). But the path followed by the masses can be any curve, since that kinematical constraint is fulfilled, what means any path in a sphere centered in the COM.
In this example, it can only be achieved by applying an external torque. Without it, the movement is a constant rotation in the same plane. But for a general geometry, an external torque is not required for a non planar path.
answered Jun 6, 2021 at 22:11