- #36
Red_CCF
- 532
- 0
Chestermiller said:Yes, so far. The acceleration can be expressed in terms of x as d2x/dt2. So:
[tex]m\frac{d^2x}{dt^2}=P_IA[/tex]
If we multiply both sides of this equation by dx/dt, and integrate with respect to time, we get:
[tex]W(t)=\frac{1}{2}mv^2(t)[/tex]
where W(t) is the work done by the gas on the piston up to time t, v(t) = dx/dt, and the right hand side is recognized as the kinetic energy of the piston.
I got a little mixed up, how did you go from the first to the second equation? Also, for this integral to work one has to assume that a and PI are non-zero for at least some time during the time interval; how is this known?
Chestermiller said:Here are some questions to consider:
What happens when the piston collides with the far end of the cylinder (if the collision is elastic)? Does the piston keep moving forever (even if the piston is frictionless)? If the frictionless piston eventually stops, what causes it to slow down and stop?
Chet
If the piston collides at at the end of the cylinder, I would assume that some energy is transferred to the cylinder even if the collision is elastic (i.e. if the cylinder was on a frictionless surface it would gain kinetic energy). If there are no irreversibilities within the gas itself, I think the piston would initially move back until the gas pressure slows it down and pushes it towards the far wall again, and this behaviour should oscillate with decreasing range of motion for the piston. I think this behaviour would continue as t → ∞, so maybe the piston never stops?
Thanks very much