How Does the Spring Constant Affect Piston Equilibrium?

[bcjochim07]

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?

 

[bcjochim07]

here's the figure

 

[Moetasim]

Thank you for sharing your attempt at solving the problem. Your approach seems to be on the right track, but there are a few things that need to be clarified. Firstly, in part a), you are correct in using the equation F=pA to equate the force of the gas on the piston to the force of the spring on the piston. However, the equation you used to solve for delta L is incorrect. The correct equation should be delta L= p0A/k, where p0 is the initial pressure, A is the cross-sectional area of the piston, and k is the spring constant. This is because the force of the spring is equal to the change in length of the spring multiplied by the spring constant.

In part b), you correctly used the equation p1v1=p2v2 to relate the initial and final pressures and volumes of the gas. However, the equation you used to solve for the final pressure is incorrect. The correct equation should be p1=p0(L0/(L0+x)), where p1 is the final pressure, p0 is the initial pressure, L0 is the initial length, and x is the change in length of the piston. This is because the volume of the gas is changing as the piston moves, and the pressure is inversely proportional to the volume.

To find the net force on the piston, you can use the equation F= p1A-p0A, where p1 is the final pressure and p0 is the initial pressure. This will give you the net force exerted by the gas on the piston, taking into account the change in pressure and the area of the piston.

I hope this helps clarify your understanding of the problem. Keep up the good work!