Piston oscillating in insulated tube with gas on both sides

[C.E]

I got if from when you said "$(v-Ax)^{\gamma}\approx v^\gamma-Ax\gamma$" in post 8.

 

[Mapes]

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.

 

[C.E]

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?

 

[Mapes]

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 [itex](1+b)^\gamma\approx 1+\gamma b[/tex] (and I messed up).[/itex]