Why Does the Textbook Solution Use Average Pressure for Piston Work Calculation?

[theBEAST]

Homework Statement

Here is the solution from the textbook:

I don't understand how they can assume a linear relationship. In fact I don't even understand the math... Why does the solution take the average pressure? Isn't the relationship between pressure and volume PV^n = constant?

 

[ehild]

The pressure increases linearly with the volume because of the spring. If the cross section of the piston is A, and the spring gets shorter by x, the change of the volume is Ax, and the spring force kx balances the increment of pressure.

ehild

 

[theBEAST]

The pressure increases linearly with the volume because of the spring. If the cross section of the piston is A, and the spring gets shorter by x, the change of the volume is Ax, and the spring force kx balances the increment of pressure.

ehild

Can this be represented mathematically? For example can you solve for x and A and have the math work out at the end? I am trying to do this because I am still a bit confused.

 

[ehild]

Can this be represented mathematically? For example can you solve for x and A and have the math work out at the end? I am trying to do this because I am still a bit confused.

Yes, it can be done. Write up the volume in terms of x, change of length of the spring.
Write the equation of balance between the increment of pressure of the steam and the pressure corresponding to the spring force.

ehild

 

[theBEAST]

Yes, it can be done. Write up the volume in terms of x, change of length of the spring.
Write the equation of balance between the increment of pressure of the steam and the pressure corresponding to the spring force.

ehild

My friend and I tried doing what you said but we are still pretty stuck at where to go next.

We did come to agreement on the fact that the spring is linear since you get some relationship P(x)*A = kx + mg.

So you see P(x) is linear.

 

[ehild]

Yes, P(x) is linear in x. Find the work done by the steam. You need the P(V) function. How is the volume related to x?

ehild

 

[theBEAST]

Yes, P(x) is linear in x. Find the work done by the steam. You need the P(V) function. How is the volume related to x?

ehild

So we know:

P(x)*A = kx + mg

W = ∫P(x)dV = = ∫P(x)Adx = ∫(kx + mg) dx

However what is k and m? We are not sure how to figure these two unknowns out.

 

[ehild]

You know the initial and final volumes, so what are the limits of integration with respect to x?

You can get k from the initial and final pressures.

ehild

 

[theBEAST]

You know the initial and final volumes, so what are the limits of integration with respect to x?

You can get k from the initial and final pressures.

ehild

So with initial and final pressures I can get two equations (using force balance) with two unknowns (m and k) and solve for both k and m?

 

[ehild]

mg=Pi, the initial pressure. You have three unknowns, A, k and x(final), but A will cancel.

ehild