# Slope rotating around a vertical axis.

• peripatein
In summary, the conversation revolves around determining the forces acting on a mass set between two springs on a rotating slope. The mass can only move along the slope in between the springs and the coordinate system is rotated at an angle. The centrifugal and coriolis forces are calculated, taking into consideration the vertical components of angular velocity. There is confusion about the orientation of the axes and the direction of movement for the mass. It is clarified that the mass can only move along the x-axis and the spring forces and gravity force are also directed along the slope.
peripatein
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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• untitled.JPG
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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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• untitled.JPG
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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.

## 1. What is slope rotating around a vertical axis?

Slope rotating around a vertical axis is a phenomenon in which a slope or inclined surface is rotated around a vertical axis. This can be seen in many natural and man-made structures, such as bridges, towers, and dams.

## 2. How does slope rotation occur?

Slope rotation occurs due to various factors, including the weight of the slope material, the angle of the slope, and external forces such as wind and water. Over time, these forces can cause the slope to shift and rotate around a vertical axis.

## 3. What are the effects of slope rotating around a vertical axis?

The effects of slope rotation can be significant and potentially hazardous. It can lead to landslides, erosion, and structural damage to buildings and other structures located on or near the slope. It can also impact the stability of roads and other infrastructure.

## 4. How is slope rotation measured and monitored?

Slope rotation is typically measured and monitored using various techniques, such as surveying, inclinometers, and tiltmeters. These methods help track the movement and rotation of the slope over time and can provide valuable data for predicting potential hazards.

## 5. Can slope rotation be prevented or controlled?

Slope rotation can be prevented or controlled through various methods, including slope stabilization techniques such as retaining walls, drainage systems, and vegetation control. Regular monitoring and maintenance can also help prevent excessive slope rotation and mitigate potential hazards.

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