Cylinder Piston

Cylinder Piston

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Elastic particle sildes without friction inside a cylinder between the bottom of the cylinder and the piston?

Initial kinetic energy of the particle is 1Joule
and distance between the bottom and
the pistion is Do = 1m.

What will be kinetic energy of the particle if the
piston in slowly pushed inside until D1 = 1/2m?

Let u be the speed of the particle, v be the speed of the piston, L be the height of the piston. After colliding with the piston moving in at speed v, the new speed of the particle is u + 2v. This happens at 2L/u intervals so the time derivative of u averaged over many cycles is:

du/dt = vu/L

Using E = m u^2/2 and v = -dL/dt:

dE/dt
= m u du/dt
= m u^2 v/L
= -(m u^2/L)(dL/dt)
= -2E * (1/L)(dL/dt)
-->
dE/E = - 2 dL/L
-->
E = constant/L^2

The adiabatic theorem in quantum mechanics seems to give the same result: If the piston is moved slowly enough, an energy eigenstate (a wavefunction of the form sin(2pi n/L) will remain so with the same number (n) of nodes so the momentum p ~ 1/L and E = p^2/2m ~ 1/L^2

L --> L/2
-->
E --> 4E = 4 joules.

I'm ignoring the effect of gravity if any (it probably would have some effect if the potential energy were the same order of magnitude as the particle energy)

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leyland l60 6 cylinder 12 piston engine (monster)