How Do Shell and Liquid Forces Achieve Equilibrium in a Hemispherical Container?

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Homework Help Overview

The discussion revolves around the equilibrium of forces in a hemispherical container filled with liquid, particularly focusing on the interaction between the liquid and the shell. Participants explore the implications of rotating the shell and how it affects the pressure distribution within the liquid.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants discuss force balances and pressure calculations related to the liquid and shell. There is consideration of whether the liquid rotates with the shell and how that affects pressure. Some participants express uncertainty about the implications of a rotating liquid and the nature of pressure surfaces within the liquid.

Discussion Status

The discussion is active, with various interpretations being explored regarding the rotation of the liquid and the resulting pressure distributions. Some participants have offered insights into pressure calculations, while others are questioning the assumptions made about the system's setup.

Contextual Notes

There is an ongoing debate about the axis of rotation and its implications for pressure calculations. Participants are also considering the constraints of the problem, such as the assumption of a completely filled shell and the nature of the surfaces of constant pressure within the liquid.

  • #31
Ok . Thanks .
 
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  • #32
haruspex said:
The horizontal axis case for an arbitrary rotation rate is quite interesting.
I'm stumped on one aspect of this case. In order to know the pressure throughout, I first need to know the pressure at one point. In the rotation about a vertical axis we just assumed the pressure is zero at the top.

What if the pressure is zero at the top before any rotation, and then we start rotating about a horizontal axis? Is there a simple way to see what the pressure will be at some one point so I can determine the pressure everywhere else?
 
  • #33
TSny said:
I'm stumped on one aspect of this case. In order to know the pressure throughout, I first need to know the pressure at one point. In the rotation about a vertical axis we just assumed the pressure is zero at the top.

What if the pressure is zero at the top before any rotation, and then we start rotating about a horizontal axis? Is there a simple way to see what the pressure will be at some one point so I can determine the pressure everywhere else?
Yes, that's what makes it tricky.
I believe one is justified in saying that the minimum pressure is always zero. Below a threshold rotation that will always be at the top. In the limit it is at the centre.
 
Last edited:
  • #34
haruspex said:
Yes, that's what makes it tricky.
I believe one is justified in saying that the minimum pressure is always zero. Below a threshold rotation that will always be at the top. In the limit is at at the centre.
OK. That makes sense to me. Then I think the threshold is for ω = √(g/R). For larger ω the point of zero pressure will be located a distance r above the center of the sphere where ω2r = g. Is that what you find?
 
  • #35
TSny said:
OK. That makes sense to me. Then I think the threshold is for ω = √(g/R). For larger ω the point of zero pressure will be located a distance r above the center of the sphere where ω2r = g. Is that what you find?
Yes.
 
  • #36
OK, thanks. That clears it up.
 

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