Thermodynamics: Horizontal Tube w/ Sliding Piston - Q&A

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Homework Help Overview

The discussion revolves around a thermodynamics problem involving a horizontal tube with a sliding piston, where the gas is thermally insulated. The problem requires analyzing the forces acting on the piston and the nature of its motion, particularly under adiabatic conditions.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants explore the implications of adiabatic processes and question the applicability of ideal gas equations. There are discussions about the nature of work done during the displacement of the piston and the assumptions regarding the gas properties.

Discussion Status

Several participants have provided insights into the use of Taylor series for approximating expressions related to pressure and volume. There is ongoing exploration of how to relate the pressures on either side of the piston and how to derive the force acting on it. Some participants express confusion about the correct application of Taylor series and the implications of their results.

Contextual Notes

There is uncertainty regarding the ideal gas assumption and the specific equations that apply to the non-ideal gas scenario. Participants are also navigating the complexities of deriving expressions without complete information about temperature or the number of moles of gas.

  • #31
I got if from when you said "(v-Ax)^{\gamma}\approx v^\gamma-Ax\gamma" in post 8.
 
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  • #32
Ah, that was my own typo. Sorry about that. So you should now have

F=-kx=-\frac{2\gamma p A^2}{V}x

which doesn't quite match the given answer, but I suspect we're right and the given answer wrong. Please update if you find another solution.
 
  • #33
I have read about Taylor expansions on wikipedia and am still a little confused. What are the Taylor expansions I should be using for (v2-Ax)^gamma and (v1+Ax)^gamma? How did you find them?
 
  • #34
The idea here is that

f(a+b)\approx f(a)+b\,f^\prime(a)

for small b. In this case

f(a)=a^\gamma

So

f(a+b)\approx a^\gamma+\gamma b a^{\gamma-1}=a^\gamma\left(1+\frac{\gamma b}{a}\right)

as you figured out from your knowledge that (1+b)^\gamma\approx 1+\gamma b[/tex] (and I messed up).
 

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