Can I get Surface Density from Volume Density?

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Discussion Overview

The discussion revolves around the relationship between volume density and surface density in the context of a thin, circular aluminum membrane used in a vibration frequency problem. Participants explore how to derive surface density without a specified thickness, particularly focusing on the implications of the membrane's geometry.

Discussion Character

  • Exploratory
  • Debate/contested
  • Technical explanation

Main Points Raised

  • One participant suggests that surface density can be calculated from volume density using the formula \(\sigma = \rho^{2/3}\).
  • Another participant argues against this approach, stating that if surface density is defined as \(\sigma\), stacking multiple thin sheets would lead to an infinite mass if treated as having finite thickness, thus complicating the relationship between volume and surface density.
  • This participant proposes that as thickness approaches zero, the surface density would also approach zero, using a hypothetical scenario involving a 1cm thick piece of material to illustrate their point.
  • A different participant assumes the membrane is spherical rather than circular and mentions that volume density provides the mass, implying that further details are needed to solve the problem.

Areas of Agreement / Disagreement

Participants express differing views on the relationship between volume density and surface density, with no consensus reached on how to derive surface density from volume density in this specific context.

Contextual Notes

The discussion highlights the complexities involved in transitioning from volume density to surface density, particularly in the absence of a defined thickness. There are unresolved mathematical implications regarding the limits of thickness and the assumptions made about the membrane's geometry.

Farina
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I'm working on a vibration frequency problem
involving a thin, circular aluminum membrane
with a radius of 0.01m.

I know the volume density of Al.

How do I arrive at a surface density for this circular
membrane -- especially since I'm not given the
thickness (I'm told that frequencies for thin membranes
are independent of thickness).

I could see how to do this if I had a rectangular membrane,
but I have a circular membrane instead.

??
 
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Yes, you can calculate the surface density from the volume density. It's just [itex]\sigma = \rho ^ {2/3}[/itex]
 
No, I don't think you can. If something has surface density [itex]\sigma[/itex], and you stack 3 thin sheets on top of each other, the total mass will be the mass of the three sheets. Now, something with finite thickness would be like having an infinite number of thin sheets stacked on top of each other, so the mass would be infinite (and so would the volume density).

Conversely, assume something has volume density [itex]\rho[/itex]. Let's say that we take a very bad approximation of it's surface density by taking a 1cm thick piece of the substance, and approximating it's surface density to be it's mass/surface area = mass/(volume/1cm) = 1cm * [itex]\rho[/itex]. Now, the "true" surface density would be this number as the thickness approches zero. If we start with a thickness t = 1cm, then we have that it's "bad-approximate" surface density is [itex]t\rho[/itex]. What we need to do, obviously, is evaluate the limit as t approaches zero, and since [itex]\rho[/itex] is just some positive finite number, the limit is zero, so it's surface density is zero, which is what we have in real life (because objects are 3-d).

I'm not sure how to go about solving your problem, but the best suggestion I can give is to treat "thin" as having the thickness of the atomic radius of aluminum. You can then treat the membrane as a zero-thickness membrane with surface density (approximated to) [itex]t\rho[/itex], where t is the radius of aluminum atom, and [itex]\rho[/itex] is its density.

Gokul is saying something else, I'm not sure where he's getting that from.
 
I assume you mean a sperical membrane not circular -- the volume density then tells you the mass of the membrane -- thickness assumed at some value -- so you have the details the rest is up to you -- I would not know how to solve this offhand.
 

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