Theory of CPU in CV Lab

Setup Overview

• Equipment:
  • Jar with Tube: Contains a metal cylinder that moves freely.
  • Assumption: No gas escapes around the cylinder.
  • Equilibrium: Gravity balances pressure differences, stabilizing the cylinder.

Cylinder Movement

• Perturbation Effects:
  • If moved up, pressure drops, gravity wins, cylinder falls.
  • If moved down, pressure increases, cylinder rises.
  • Result: Cylinder oscillates around equilibrium.
• Oscillation Characteristics:
  • Short period oscillation.
  • During oscillation, heat transfer is negligible (adiabatic process).

Adiabatic Process and Gamma

• Adiabatic Equation: ( PV^\gamma = \text{constant} ).
• Gamma (( \gamma )): Ratio ( \gamma = C_p / C_v ).
• Differentiation: Leads to a dynamic equation similar to harmonic oscillation.

Mathematical Derivation

• Cylinder Volume Change:
  • Volume change ( dV = \pi r^2 dH ).
  • Replace in equations to find force change ( dF ).
• Harmonic Oscillation Equation:
  • ( dF = -k \times dH ).
  • ( k ) is a constant derived from system parameters.

Period and Gamma Calculation

• Period Formula: ( T = 2\pi \sqrt{\frac{m}{k}} ).
• Gamma Equation: Derived from combined equations.
  • ( \gamma = \frac{64mV}{P_0D^4T^2} ).

Measurement Process

• *Oscillation Measurement:
  • Measured by pressure changes over time.
  • Ideal: Sinusoidal function.
  • Reality: Damped oscillation.*

Data Analysis

• Data Collection:
  • Use software tools to record pressure vs. time.
  • Fit data to damped oscillation model to find period (( T )).

Lab Procedure

• Setup Preparation:
  • Use an air pump to maintain stable pressure.
  • Avoid overpressure to prevent cylinder ejection.
• Data Recording:
  • Use sparkline platform to collect barometric pressure data.
  • Use curve fitting to analyze the pressure oscillation.

Alternative Gases

• Gas Tanks:
  • Use various gases (e.g., CO2, Argon) for experiments.
  • Gas tanks stabilize pressure instead of air pumps.

Conclusion

• Objective is to measure ( \gamma = C_p / C_v ) using oscillation data.
• Method involves precise control and analysis of oscillating cylinder behavior.