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Work done on piston by steam

  1. Feb 25, 2013 #1
    1. The problem statement, all variables and given/known data
    vTaJDgy.png

    Here is the solution from the textbook:
    KO2zHHR.png

    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?
     
  2. jcsd
  3. Feb 25, 2013 #2

    ehild

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    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
     
  4. Feb 25, 2013 #3
    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.
     
  5. Feb 25, 2013 #4

    ehild

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    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
     
  6. Feb 28, 2013 #5
    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.
     
  7. Feb 28, 2013 #6

    ehild

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    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
     
  8. Feb 28, 2013 #7
    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.
     
  9. Feb 28, 2013 #8

    ehild

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    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
     
  10. Feb 28, 2013 #9
    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?
     
  11. Feb 28, 2013 #10

    ehild

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

    ehild
     
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