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A coin on 2D membrane

  • Thread starter skrat
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  • #1
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Homework Statement


Find stationary state of a circular membrane, on which a coin is put.

Homework Equations




The Attempt at a Solution


I am not sure about the second boundary condition, so if any of you has the time, please check the following solution:
$$u_{tt}=c^2\nabla ^2u+\rho g$$ Where ##u## is the defelction from a horizontal ##x## axis, ##\rho ## is the density of the coin and ##g## is the gravity. In polar coordinates this means $$\frac 1 r \frac{\partial }{\partial r}(r\frac{\partial }{\partial r}u)=-\frac{\rho g}{c^2}$$ Solving this PDE should give me $$u(r)=-\frac{\rho g}{4c^2}r^2+Aln(r)+B$$ Now the first boundary condition is kind of the obvious one: $$u(r=R)=0=-\frac{\rho g}{4c^2}R^2+Aln(R)+B$$ if ##R## is the radius of the membrane.
Now the second boundary condition is a bit confusing for me... But here is my idea:
If I write Newton's law for the coin with radius ##a##, than the condition should be (I guess): $$mg=F_0\frac{du}{dr}$$ where I already did an approximation that the membrane will deflect by a really small angle ##\sin \varphi \approx \varphi =\frac{du}{dr}##. Where ##F_0## should be the tension on the membrane. - and this is the most confusing part, because in order to determine constants ##A## and ##B## I defined a new constant ##F_0## which is also unknown.

.. Hmmm? Is this ok or is the second boundary condition a complete nonsense?
 

Answers and Replies

  • #2
TSny
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Have you stated everything that is given in the problem? I would think that you would need to assume that the "surface tension" of the membrane is given. The surface tension is related to your constant Fo.

I believe you want to find the stationary displacement of the membrane between r = a and r = R. In this region, what is the density of the coin?
 
  • #3
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I stated almost everything yes. - The problem also says that I if I need any parameters (such as: coin radius and its mass, etc....) I should just assume that they are all known. But yeah, I think you are right. If I add surface tension as one of the parameters of the problem, than my idea in the first post should be just fine.
I believe you want to find the stationary displacement of the membrane between r = a and r = R. In this region, what is the density of the coin?
Yes.
 
  • #4
TSny
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In the equation ##u_{tt}=c^2\nabla ^2u+\rho g## should the last term be there? Are there any external forces acting on the membrane between r = a and r = R?
 
  • #5
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Hmm, good point.
Not if I would be interested in the stationary displacement of the membrane between ##r=a## and ##r=R##. Thank you!
 
  • #6
TSny
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Right, that last term should not be there. You can also check that the term does not have the same dimensions as the other terms.
 
  • #7
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Ok, hold on a second...

The general form is ##u_{tt}=c^2\nabla ^2u +f(x,t)##. And let's say that I want to know the displacement of the membrane from ##r=0## to ##r=R##. Ok, of course the displacement from ##r=0## to ##r=a## will be constant so this really isn't a new problem, but my question here is:
Until you mentioned it, I was sure that the last term is ok. (Despite the fact that I don't need it.) How would have to change it?

The last term should (as far as I know) describe the external forces on the membrane. In this case, that's the coin with its mass. So ##\rho g##. But like you said, the dimensions don't match o_O
 
  • #8
TSny
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##f(x,t)## is the force per unit area (or pressure) divided by the mass per unit area.
 
  • #9
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One (hopefully last) question. I am sorry to bring this up a few days later, but.

The question is am.. how to say... it is not directly related to this problem, but it I will be more thoroughly if I post it here.
In the first post I wrote a PDE $$\frac 1 r \frac{\partial }{\partial r}(r\frac{\partial }{\partial r}u)=-\frac{\rho g}{c^2}$$ which I later decided to solve by integrating both sides. However I also have all sorts of problem with $$\nabla ^2 u(r,t)= f(r,t)$$ where as a solution are Bessel and Neuman functions.

So my question here is... How do I know when to integrate and when the solution is Bessel function? Both problems are in polar coordinates, both problems have a nice symmetry that depends only on ##r## .... I can't see any difference between the problems - yet the solutions couldn't be me more different.
 

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