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

In summary, the potential energy of the compressed gas is given by V = -(p_0)(s_0)(A)ln(s/s_0), where p_0 is the initial pressure and s_0 is the initial value of s. The forces opposing motion are negligible.
  • #1
vwjoek
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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, wouldn't V=PA(s-s_0)
(just for clarity V is potential not velocity or any other variable)
thank you
 
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  • #2
:) Yes, you are correct that the force due to pressure is PA = F, so the potential energy of the compressed gas can be expressed as V = PA(s-s_0). However, since the gas is ideal, the pressure is related to the volume (pΩ = constant). Since the cross sectional area of the tube is given by A and the position of the cylinder is given by s, we can then express the volume of the gas as V = As. Thus, we can rewrite the potential energy as V = PA(s-s_0) = p_0A(s-s_0). Finally, if we take the natural logarithm of both sides, we get the desired result: V = -(p_0)(s_0)(A)ln(s/s_0).
 

1. What is potential energy?

Potential energy is the energy that an object possesses due to its position or configuration. It is stored energy that can be converted into other forms, such as kinetic energy.

2. How is potential energy related to a cylinder inside a tube filled with compressed air?

In this scenario, the potential energy is related to the amount of compressed air inside the tube. The more compressed the air is, the higher the potential energy of the system due to the increase in pressure.

3. How is the potential energy of a cylinder inside a tube filled with compressed air calculated?

The potential energy of this system can be calculated using the formula: PE = mgh, where m is the mass of the cylinder, g is the acceleration due to gravity, and h is the height of the cylinder above the ground (or the bottom of the tube).

4. Can potential energy be converted into other forms of energy?

Yes, potential energy can be converted into other forms of energy, such as kinetic energy. In the case of the cylinder inside the tube, the potential energy can be converted into kinetic energy when the compressed air is released and the cylinder moves.

5. How does the potential energy of the cylinder inside the tube change as the compressed air is released?

As the compressed air is released, the pressure inside the tube decreases and therefore the potential energy of the system decreases. This is because the height of the cylinder above the ground (or bottom of the tube) decreases as the air is released, resulting in a decrease in the potential energy of the system.

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