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Yes, I've gone back and corrected that algebraic error.Chetty said:The sign of Patm must agree with the sign of mg.
So you see that the work done on the gas, the compression of the gas ##(l+H)-x(t)##, and the kinetic energy of the piston are all interacting functions of time during the compression. What do you think happens to the kinetic energy of the piston at very long times when the system equilibrates thermodynamically?I see you have taken x as measured from the bottom of the cylinder making the terms mg + Patm A always negative which makes sense for work done by the gas but am wrestling a bit with the positive piston kinetic energy but find no other issue with the equation. Interesting approach.