# Potential energy of a cylinder inside a tube filled with compressed air

1. May 2, 2010

### vwjoek

Let the cross sectional area of the tube be given by A and the position of the cylinder by s, and assume that the gas is ideal so that the pressure times the volume (Ω) is a constant. i.e. pΩ = constant. Show that the potential energy of this compressed gas is given by V = -(p_0)(s_0)(A)ln(s/s_0). p_0 is the initial pressure and s_0 is the initial value of s. The forces opposing motion are negligible

Okay so here is my train of thought, the equation from the book states: (without any nonconservative forces)
T_1 + V_1 = T_2 + V_2, where T is 1/2 mv^2 and V is the potential energy,

but i know that the force due to pressure is PA = F and isn't dV/ds = F?

So if that were the case, wouldnt V=PA(s-s_0)
(just for clarity V is potential not velocity or any other variable)
thank you