Rotation
October 13, 2020 at 4:47 pm
(This post was last modified: October 13, 2020 at 6:11 pm by Smaug.)
Here's a question that's been bugging me for a long time. It's actually been rather hard to formulate it. What is the fundamental property that defines rotation from translation? That may be a stupid question but here's the reasoning:
Let us assume two reference frames O1 x1 y1 z1 and O2 x2 y2 z2. Axes z1 and z2 are parallel to each other and points O1 and O2 remain at a constant distance. Let's assume then that the second frame is spinning at a constant angular velocity about z2 axis relative to the first frame.
If we view the problem exclusively from the point of kinematics there should be no way to tell which frame is rotating: you can take either one of them as an inertial frame (non-rotating) then the other one would be non-inertial (either just spinning or orbiting & spinning).
But if we bring dynamics into play things have to get different. Let us assume that each of the above frames is fixed to a body with a z-axis symmetry. Then the spinning body must experience centrifugal forces. This begs an assumption that centrifucal forces may serve as a clue to differentiate rotation from translation on a fundamental level. But these forces themselves are 'fictitious' (introduced to account for normal accelerations in a rotating frame) so this reasoning leads to circular logic. We can also introduce another mass moving across the spinning body to use Coriolis force as a sign of rotation but Coriolis force is also a 'fictitious' one so that's no better.
A simple illustration would be shape of Earth: we can formally take a fixed reference frame associated with Earth as an inertial one yet there will still be centrifugal forces acting upon Earth and flattening it slightly and Coriolis forces acting upon objects moving upon Earth's surface.
So, long story short: what is the fundamental way to tell if something is rotating?
Let us assume two reference frames O1 x1 y1 z1 and O2 x2 y2 z2. Axes z1 and z2 are parallel to each other and points O1 and O2 remain at a constant distance. Let's assume then that the second frame is spinning at a constant angular velocity about z2 axis relative to the first frame.
If we view the problem exclusively from the point of kinematics there should be no way to tell which frame is rotating: you can take either one of them as an inertial frame (non-rotating) then the other one would be non-inertial (either just spinning or orbiting & spinning).
But if we bring dynamics into play things have to get different. Let us assume that each of the above frames is fixed to a body with a z-axis symmetry. Then the spinning body must experience centrifugal forces. This begs an assumption that centrifucal forces may serve as a clue to differentiate rotation from translation on a fundamental level. But these forces themselves are 'fictitious' (introduced to account for normal accelerations in a rotating frame) so this reasoning leads to circular logic. We can also introduce another mass moving across the spinning body to use Coriolis force as a sign of rotation but Coriolis force is also a 'fictitious' one so that's no better.
A simple illustration would be shape of Earth: we can formally take a fixed reference frame associated with Earth as an inertial one yet there will still be centrifugal forces acting upon Earth and flattening it slightly and Coriolis forces acting upon objects moving upon Earth's surface.
So, long story short: what is the fundamental way to tell if something is rotating?
