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Hello I am still confused about rotating coordinate frames and want to ask a question about it. Is it correct that strictly speaking the mass must be connected with the axis of rotation in the rotating coordinate frame? For example by a rod, or a rotary disk, or whatever. I mean otherwise I do not see how an Euler force or a Coriolis force can act on a particle? (I think I am confused because this connection is usually not showed in figures in the textbooks)

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A rotating coordinate system is a non-inertial coordinate system. For general motion there is translational plus rotational motion. The motion of any object can be evaluated using either an inertial or a non-inertial system. There is no requirement for any fixed relationship between the object and any axes of any coordinate system; however, for some problems this can be the case, such as using a rotating coordinate system for uniform circular motion of a particle, it is convenient to take the particle at a fixed position in the rotating system, and located along one of the rotating axes. In a non-inertial system "fictitious" forces (and torques) are present that must be accounted for in evaluating the motion.

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Is it correct that strictly speaking the mass must be connected with the axis of rotation in the rotating coordinate frame? For example by a rod, or a rotary disk, or whatever. I mean otherwise I do not see how an Euler force or a Coriolis force can act on a particle?

There is no need for a physical connection. These fictitious forces are artifacts of the coordinate system and any object described by that coordinate system is affected regardless of whether it is mechanically connected or not.

Consider, for example, a disconnected isolated object at rest in the inertial frame. In the rotating frame it “orbits” the axis in the anti-spin direction. There is no physical connection, so the only possible forces are the fictitious centrifugal and Coriolis forces. These sum to produce the observed motion, completely in the absence of a physical connection.

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    $\begingroup$ Indeed, assuming the rotation uniform, the sum of centrifugal and Coriolis forces is centripetal as is due for a uniformly rotating body... $\endgroup$ Jul 3, 2019 at 20:24

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