Slope rotating around a vertical axis.

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Discussion Overview

The discussion centers on the forces acting on a mass positioned between two springs on a slope that rotates around a vertical axis. Participants explore the implications of centrifugal and Coriolis forces in this context, considering the constraints of motion along the slope and the coordinate system used.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant seeks to understand the forces acting on a mass on a rotating slope, questioning how to account for the components of angular velocity that affect the centrifugal and Coriolis forces.
  • Another participant asks whether the mass is constrained to move only in the xz-plane or if the spring attachments can rotate freely.
  • A different participant challenges the initial assumption about the plane of rotation, asserting that the rotation occurs in the xy-plane and that the mass can only oscillate along the x-axis.
  • There is a clarification regarding the coordinate system, with one participant confirming that their axes are rotated by an angle alpha.
  • Another participant describes the motion of the mass, suggesting it must move in the positive-z direction to remain on the slope while moving outward.
  • One participant disputes this, insisting that the mass can only move along the x-axis, prompting further discussion about the implications of the slope's geometry.
  • Another participant notes that the slope's nature implies a relationship between changes in x and z, indicating that movement in one direction affects the other.

Areas of Agreement / Disagreement

Participants express differing views on the constraints of the mass's motion and the interpretation of the coordinate system. There is no consensus on whether the mass can move freely in the z-direction or is limited to the x-axis.

Contextual Notes

Participants have not fully resolved the assumptions regarding the coordinate system and the nature of the slope, which may affect the analysis of forces acting on the mass.

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)?
 

Attachments

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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.
 

Attachments

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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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