Where Did I Go Wrong in Proving Harmonic Motion in an Adiabatic Process?

In summary, the article explores the challenges faced in proving harmonic motion within an adiabatic process, highlighting key misconceptions and the complexities of applying classical mechanics principles to thermodynamic systems. It emphasizes the importance of accurately accounting for system constraints and the interplay between energy conservation and the characteristics of harmonic oscillators in adiabatic conditions.
  • #1
pedrovisk
7
0
Thread moved from the technical forums to the schoolwork forums
TL;DR Summary: Problem said that the ball moves in a harmonic motion and asked to prove it. The process is adiabatic

Problem said that the ball moves in a harmonic motion and asked to prove it. The process is adiabatic.

I did the development, but at certain point I'm having a problem. The right answer is that the P(Vx) and Vx are P/V (the initial pressure made by the gas and the initial volume), but my equations leads to P(Vx) and Vx. Where is my mistake?

Does the final volume and the final pressure are very close to the original volume and pression that is safe to assume they're almost equal?

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  • #2
pedrovisk said:
The right answer is that the P(Vx) and Vx are P/V (the initial pressure made by the gas and the initial volume), but my equations leads to P(Vx) and Vx.
I don't understand what this means.
 
  • #3
vela said:
I don't understand what this means.
Elaborate, please.

But got the answer in another forum, so no need for reply.
 
  • #4
pedrovisk said:
Elaborate, please.
For example, I would interpret "##P(V_x)## and ##V_x## are ##P/V##" to mean ##P(V_x)=V_x=P/V##, which doesn't make sense.
 
  • #5
vela said:
For example, I would interpret "##P(V_x)## and ##V_x## are ##P/V##" to mean ##P(V_x)=V_x=P/V##, which doesn't make sense.
The exercise is based on a real experiment named "Ruchhardt experiment". This experiment aims to calculate \gamma (Cp/Cv). The ball in the bottle follows SHM. So it is possible calculate \gamma finding the period of this SHM.

The equation of the force that makes the ball go back to equilibrium is ##dF = -\gamma .A^2.(P/V).dx##. P and V are the pressure and volume when the acceleration of the ball is 0.

As you can see, I got the equation "right". Problem is that instead o P/V (which are constants), I got ##P(V_x)/V_x##.

The question, that is the part you did not understand, is why the equation has ##P/V## instead of ##P(V_x)/V_x)##. It happens that the variation of ##P(V_x)## and ##(V_x)## are so small that it can be considered as equal to the initial pressure and volume, just like I said in the end of the original post.
 
  • #6
I can't see the figure. Just a big black screen.
 
  • #7
Chestermiller said:
I can't see the figure. Just a big black screen.
The figure has been deleted, which is why we ban the use of external image servers.

Thread locked.
 

FAQ: Where Did I Go Wrong in Proving Harmonic Motion in an Adiabatic Process?

Why is my assumption of constant energy incorrect in an adiabatic process?

In an adiabatic process, there is no heat exchange with the surroundings, but the internal energy of the system can change due to work done on or by the system. For harmonic motion, if you're assuming constant energy, you might be neglecting the work interactions that can alter the internal energy, leading to an incorrect proof.

How does the adiabatic condition affect the equations of motion?

The adiabatic condition implies that there is no heat transfer (Q = 0). This impacts the first law of thermodynamics, which simplifies to ΔU = W. In harmonic motion, this means the internal energy changes are directly related to the work done by the system, affecting the potential and kinetic energy terms in your equations of motion.

Did I account for the changing volume and pressure correctly?

In an adiabatic process, the relationship between pressure and volume is governed by the adiabatic equation \(PV^\gamma = \text{constant}\), where γ is the heat capacity ratio (Cp/Cv). Incorrectly accounting for these variables can lead to errors in deriving the correct harmonic motion equations.

Is my use of the ideal gas law appropriate in this context?

The ideal gas law \(PV = nRT\) can still be used in an adiabatic process, but you must correctly incorporate it with the adiabatic condition \(PV^\gamma = \text{constant}\). Misapplying the ideal gas law without considering the adiabatic constraints can lead to incorrect conclusions about the system's behavior.

Did I properly include the effects of non-conservative forces?

In harmonic motion, non-conservative forces such as friction or damping can alter the energy dynamics. In an adiabatic process, if these forces are present and not accounted for, they can cause discrepancies in your proof. Ensure you consider all forces acting on the system to accurately describe the motion.

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