How Does the Spring Constant Affect Piston Equilibrium?

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SUMMARY

The discussion focuses on the relationship between the spring constant (k) and piston equilibrium in a closed cylinder system. The equilibrium condition is established when the pressure (p0) in the left chamber and the spring compression (delta L) are balanced. The derived formula for delta L is expressed as delta L = p0A/k, where A is the cross-sectional area of the piston. Additionally, the net force on the piston is calculated using the equation p1 = p0L0/(L0 + x), leading to the force expression pA = Ap0L0/(L0 + x).

PREREQUISITES
  • Understanding of basic physics concepts such as pressure, force, and equilibrium.
  • Familiarity with Hooke's Law and spring constants.
  • Knowledge of gas laws and their applications in closed systems.
  • Ability to manipulate algebraic expressions and equations.
NEXT STEPS
  • Study Hooke's Law in detail to understand spring behavior under various loads.
  • Explore the Ideal Gas Law and its implications for pressure and volume changes.
  • Learn about dynamic equilibrium in fluid systems and its mathematical modeling.
  • Investigate the effects of varying spring constants on system stability and response.
USEFUL FOR

Students of physics, engineers working with mechanical systems, and anyone interested in understanding the dynamics of piston systems in closed environments.

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Homework Statement


The closed cylinder in the figure has a tight-fitting but frictionless piston of mass M. The piston is in equilibrium when the left chamber has pressure p0 and length L0 while the spring on the right is compressed by delta L. The spring constant of the spring is k.

a)What is delta L in terms of p0, L0, A, M, and k?

b) Find an expression for the net force on the piston. Assume all motions of are slow enough for the gas to remain at the same temp.

Homework Equations





The Attempt at a Solution



a) I didn't include all of the variables, so it seems that this part is too simple. Here's what I did: F=pA

so the force of the gas on the piston = the force of the spring on the piston

p0A=kdeltaL
deltaL= p0A/k

b) p1v1=p2v2

p0(L0)(A) = p1(L0+x)(A)

p1= p0L0/(L0 + x)

So the force is pA = Ap0L0/(L0+x) is this right?
 
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apistonpressure.jpg
here's the figure
 

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