Coriolis force on rotating plane

[Ayotte]

A particle of mass m is confined to move, without friction, in a vertical plane, with axes x horizontal and y vertically up. The plane is forced to rotate with constant angular velocity $$\Omega$$ about the y axis. Find the equations of motion for x and y, solve them, and describe the possible motions.



F_cor=2m v $$\times$$ $$\Omega$$



This is listed in a section of the homework titled "The Coriolis Force." However, when I think about the problem, the only fictitious force that I can see applying here is centrifugal. Any motion in the y direction will be parallel with Omega, and therefore the cross product will be zero, and any motion in the x direction will cause the cross product to be in the z direction. I'm confused, because the problem is in the section where the problems should involve the Cor. force, so I would like confirmation that I'm correct.

 

[ehild]

You are right, the Coriolis force has no effect in the x,y plane.

ehild

 

[Cleonis]

A particle of mass m is confined to move, without friction, in a vertical plane, with axes x horizontal and y vertically up. The plane is forced to rotate with constant angular velocity $$\Omega$$ about the y axis.

Take a simpler case first:
A horizontal tube, in which an object can slide without friction, the tube is rotating with constant angular velocity. Let the object be released at some distance to the central axis of rotation.

Obviously in that case the object will start sliding away from the axis of rotation.
The tube constitutes a rotating system. As the object gains radial velocity the wall of the tube must exert a force upon the object to keep it co-rotating.

The formula $$F = 2mv \times \Omega$$ expresses how much tangential acceleration (wrt the rotating system) there will be if there is no force upon the object.

Without a force an object moving away from the central axis will start to fall behind the rotating motion. But in this case the tube prevents the object from "falling behind"; buildup of tangential velocity (wrt the rotating system) is prevented, so the force that the tube exerts upon the object will be equal to and opposite in direction to the fictitious Coriolis force.The problem you describe can be seen as an object that can slide frictionless between two sheets. The two sheets are vertical.

Release an object from the top, from a point that is some distance away from the central axis. Gravity will accelerate the object downward, and the constraint of co-rotating with the vertical, rotating plane will cause the object to accelerate outward.