Deriving motion of a displaced piston on a gas cylinder

[Agrasin]

First, I'm a second year high school physics student so my thinking may be half-baked. I was wondering what motion a piston would exhibit if it was the frictionless lid of an ideal gas cylinder with a constant amount of air, and if it was pushed or pulled a little bit.

There would be a restoring force exerted by the atmospheric pressure - pressure inside the cylinder. At first I thought it would simple harmonic motion, but then I visualized a real-world experiment and thought that the motion would be more like y = 1 - e^-x.

I set f = ma = m*(second derivative of y(t)) equal to the restoring force, which is a constant times (1/ h+y) where h is the equilibrium height of the piston.

I don't know how to solve second order differential equations. I reasoned that the second derivative of y(t) is equal to a function of the form 1/y. Therefore y(t) is of the form of the second antiderivative of y^-1, which is the integral of ln(x), which is xln(x)-x.

The graph of xln(x)-x looks nothing like the motion the piston should exhibit.

Could you help me understand this?

 

[abitslow]

There are a lot of (mental) tools you probably don't have access to in order to solve this problem. There are 3 ways to 'write down' the (Classical) Laws of Physics: Newtonian, LaGrangian, and Hamitonian. I went thru 2 years of college physics without ever hearing about the last two. IDK if now days its different, but suffice to say they are advanced topics. So, yeah this problem is challenging for you. Aristotle thought that energy had to be (continually) added to keep an object in motion, obviously he lived on the Mediterranean, not in Scandanavia. We know better: In a non-expanding Universe, energy is conserved. [That's not quite right (unless you add momentum and pressure-stress to your concept of energy (and I don't mean E= E+mv+P!)), but close enough for Classical Physics.] So, energy is conserved. YOU have posed this problem as if the energy you give to the system (in moving the piston from its equilibrium position) will do what? disappear? wrong. So, actually you know that with constant energy all that can happen to it is it 'slosh' back and forth between P.E. and K.E. If this is not harmonic motion, then I don't know what is. (I'm not saying that a real system will do this, friction does 'rob' a system of K.E. as will thermal effects - a change in pressure means a change in temperature, 'ceterus paribus' (all other things being equal). So, since both of these are problems at the walls of your piston, they aren't part of the simple case of a frictionless piston and an ideal gas, with no heat transfer. So, I expect the motion will be a cosine wave. Highest distance at time 0, then oscillates (forever) back and forth. Dampening is really an advanced topic but you could model it by e^(-kt)*cos(k't). The progression of Mechanics (The Physics of motion of objects) is first you get a good grounding with the Newtonian Way of Thought, then LaGrangian methods are taught - by that time you can solve some simple ODEs and you're experienced enough to be able to encapsulate the critical elements of a problem. Hamiltonian methods are similar to LaGrangian, but are a great lead into Quantum Mechanics. The way I would solve this is to set up the equation for the total energy of the system, E = P.E. + K.E. as a function of position (y) and momentum (mv = m dy/dt) and use the Euler-Lagrange equation(s). After I figured out the 'ideal' case, I'd assume that E(t) = E(0) - εt where ε is very small (reversible, near equilibrium, ε²~0) and see how that affects the result. If you are interested, Susskind has 10 lectures on Classical Mechanics available (Stanford) on itunes or youtube that deal with similar (simpler) problems. You'd have to take both that course (no homework!) as well as his course on statistical mechanics (ideal gas law) to have a firm grasp of how to solve this, imho. One thing you need to understand (as wikipedia points out) is that for the Euler-Lagrange method, given j = g(x)+h(v) even though v = f(x), you differentiate with respect to x and v separately (but only for a particular 'step' in the method)...it can be confusing.
BTW d(cos(x))/dx =-sin(x), and d(-sin(x))/dx = -cos(x) so y''+y = 0 if y=cos(t)
most ODEs are either e^ or sin or cos (if they aren't simple polynomials). But nobody can solve most ODEs, we have to pick problems which have 'easy' solutions, LOL.

 

[Agrasin]

Wow, never heard of Lagrangian and Hamiltonian before your post. Thanks for leading me to the Susskind lectures. I think what you were talking about was a 9 lecture series in classical mechanics (found it on iTunes). Does the course involve math beyond easy MV calc?

Also, you're right, I thought I could neglect the changes in internal energy. Silly me. But the changes in internal energy from external work done wouldn't effect the piston height, would it? Just the temperature?