Slope rotating around a vertical axis.

AI Thread Summary
The discussion focuses on analyzing the forces acting on a mass positioned between two springs on a slope that rotates around a vertical axis. The key points include the consideration of centrifugal and Coriolis forces, with an emphasis on the components of angular velocity relevant to the slope's orientation. There is confusion regarding the movement of the mass, specifically whether it can only oscillate along the x-axis or if it must also move in the z-direction due to the slope's incline. The participants clarify that the mass's movement is constrained to the slope, affecting how the forces are calculated. Overall, the interaction between the mass, springs, and the rotating slope is central to understanding the dynamics involved.
peripatein
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This is NOT a HW question. I'd appreciate an explanation of the following:
I would like to determine the forces acting on a mass set between two springs of constant k on a slope (the slope's angle is alpha). The slope revolves around the vertical axis with angular velocity w and the mass could only move along the slope in between the springs. Please see attachment. Suppose I choose my axes so that my x-axis is parallel to the slope. While calculating the centrifugal and coriolis forces acting on the mass, only the components of omega vertical to my x-axis should be taken under consideration (the cross product would otherwise yield zero). However, aren't there two components of omega vertical to the x-axis (projection of omega on z as well as its projection on y)?
 

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Do you consider the mass to be constrained so that it only moves in the xz-plane, or are the spring attachments free to rotate in any direction?
 
Why x-z plane? The rotation is in the x-y plane (or r-theta if you will)! And the mass can only oscillate along the x-axis.
 
Okay, how about if I ask this way: are you calling your vertical axis y? (Your OP mentions the projection of omega on z).
 
My coordinate system is simply rotated by an angle alpha counter-clockwise. See attachment.
 

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So your coordinate axes are rotating, with angular velocity ω parallel to the z-axis. The mass has to move in the positive-z direction to move outward (positive-x direction) while staying stuck to the slope, right?
 
Has to move in the positive-z direction? Why? It can only move along my x-axis. I am not following.
 
It's a "slope," implying that dz/dx > 0 along the slope. So, if x changes then z must too. Unless I'm misunderstanding the problem, the mass is not allowed to move straight outward (in the positive x-direction) and through the slope surface.

The spring forces are also directed along the slope, by the way, as is the (net) force due to gravity.
 

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