Pressure of liquid given radius, help please

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    Liquid Pressure Radius
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Discussion Overview

The discussion revolves around calculating the pressure of a liquid in a test tube being spun in an ultracentrifuge, focusing on the relationship between pressure, radius, and angular velocity. The scope includes mathematical reasoning and conceptual clarification regarding fluid dynamics in a rotating system, while ignoring gravity and atmospheric pressure.

Discussion Character

  • Mathematical reasoning, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant presents a formula for pressure at a radius r within the test tube, involving the density of water and angular velocity, but expresses uncertainty about deriving the complete equation.
  • Another participant suggests that pressure at a depth must account for the force needed to accelerate the fluid above it and hints at using an integral to find this relationship.
  • A later reply questions the specifics of the integral setup and requests clarification on what should be integrated to find the pressure at any point in the tube.
  • Further clarification is requested regarding writing an expression for the net force on an infinitesimal slice of fluid and how to integrate this to determine pressure.

Areas of Agreement / Disagreement

Participants are engaged in a collaborative exploration of the problem, with no consensus reached on the method to derive the pressure formula. Multiple viewpoints and approaches are presented, indicating an unresolved discussion.

Contextual Notes

Participants have not fully defined the assumptions regarding the fluid dynamics involved, and there are unresolved steps in the mathematical derivation of pressure as it relates to the rotating system.

harlow_barton
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A test tube filled with water is being spun around in an ultracentrifuge with angular velocity. The test tube is lying along a radius and the free surface of the water is at radius r(o).

Show that the pressure at radius r within the test tube is:

p = .5(p)(angular velocity)^2(r^(2) -r(o)^2)

where p is the density of the water. Ignore gravity and atmospheric pressure.


p = p - g(density)(height)


gravity or centripetal acceleration, a= r(angular velocity)^2

height or depth of water, h = r- r(o)

this only gets me to p= p + density*r*angular velocity^2(r-r(0))

I'm not sure where the rest comes from!
 
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At any depth 'below' the surface, the pressure has to provide enough force to accelerate all the fluid 'above' it. Hint: Set up an integral.
 
Doc Al, I'm not sure I understand what I should be taking the integral of. Could you explain further?
 
harlow_barton said:
Doc Al, I'm not sure I understand what I should be taking the integral of. Could you explain further?
Write an expression for the net force on an infinitesimal slice (thickness dr) of the fluid in the tube; then integrate from r(0) to r to find the total force, and then the pressure, at any point along the tube.
 

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