Classical Mechanics - Coriolis Force

[macrsp]

Homework Statement

This is a fairly general problem that came up while trying to model a system. Given a rotating disk and an inertially fixed object, how is the fictional coriolis force handled? For example, if there is a dot on the ground below a sheet of transparent plastic rotating at speed [itex]\omega[/itex], does an observer on the sheet of plastic observe the coriolis affect, and why or why not?

Homework Equations

The relevant subset of the relevant equation,

a_Rotating = - 2[itex]\omega[/itex] x V_Rotating - [itex]\omega[/itex] x ([itex]\omega[/itex] x X_Rotating)

The Attempt at a Solution

Well, the motion can be correctly described by the
- [itex]\omega[/itex] x ([itex]\omega[/itex] x X_Rotating)
portion of the equation.

However, because there is apparent rotation, there is a V_Rotating, so
- 2[itex]\omega[/itex] x V_Rotating
is non zero, which makes no sense. I'm probably making a very simple mistake somewhere, and I suspect that it has to do with V being the velocity in the fixed frame, not the rotating frame, but all the explanations of the coriolis equation seem to state it with the velocity being from the rotating frame.

 

[mark726]

Hello, thank you for bringing up this interesting problem. The Coriolis force is a result of the combination of the rotational motion and the linear motion of an object. In this case, the rotating disk is providing the rotational motion and the inertially fixed object is providing the linear motion. The Coriolis force is given by the equation you have provided, -2\omega x VRotating - \omega x (\omega x XRotating).

To understand why the observer on the rotating disk does not observe the Coriolis effect, we must first understand how the velocity vector is defined in this scenario. In this equation, V is the velocity of the object in the inertial frame, not the rotating frame. This means that the observer on the rotating disk will see the object moving with a velocity V, while an observer in the inertial frame will see the object moving with a velocity of V + \omega x XRotating.

This means that the Coriolis force, which is dependent on the difference in velocities, will be zero for the observer on the rotating disk. This is because the observer on the disk is already accounting for the rotational motion of the disk, and therefore does not see any difference in velocities between the object and the disk.

In conclusion, the observer on the rotating disk will not observe the Coriolis effect because their frame of reference already takes into account the rotational motion of the disk. I hope this helps clarify the issue. Please let me know if you have any further questions.