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Find the hydrostatic force on one end of the tank if it is filled to a depth of 8 ft.

  1. Mar 23, 2012 #1
    1. The problem statement, all variables and given/known data
    A large tank is designed with ends in the shape of the region between the curves y =(1/2)x^2 and y = 12, measured in feet. Find the hydrostatic force on one end of the tank if it is filled to a depth of 8 ft with gasoline. (Assume the gasoline's density is 42.0 lb/ft^3).

    This is a section on applications.

    2. Relevant equations

    I'm not sure if this is precisely all I need.

    Force F = ρgdA (g gravity, d depth, A area)

    3. The attempt at a solution

    (42.0)(9.8) ∫ sqrt(2y) (8-y) dy on [0,8]

    823.2 ∫ 8 sqrt(2y) - ysqrt(2y) dy on [0,8]

    823.2 [ 16sqrt(2)/3 y^(3/2) - 2sqrt(2)/5 y^(5/2)] on [0,8]

    which evaluates to 56, 197.12 lb
     
  2. jcsd
  3. Mar 23, 2012 #2

    gneill

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    Re: Find the hydrostatic force on one end of the tank if it is filled to a depth of 8

    You're mixing metric with imperial if you use 9.8 for g.
     
  4. Mar 23, 2012 #3
    Re: Find the hydrostatic force on one end of the tank if it is filled to a depth of 8

    Oh, right! It's 32ft/sec^2, right?
     
  5. Mar 23, 2012 #4

    gneill

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    Re: Find the hydrostatic force on one end of the tank if it is filled to a depth of 8

    32.174 ft/sec2 :smile:
     
  6. Mar 23, 2012 #5
    Re: Find the hydrostatic force on one end of the tank if it is filled to a depth of 8

    Is there any way that you can tell me if my solution is in the right direction?
     
  7. Mar 23, 2012 #6

    gneill

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    Re: Find the hydrostatic force on one end of the tank if it is filled to a depth of 8

    Sure. It looks like you've got the right idea for the integration, and the equation being integrated looks appropriate.
     
  8. Mar 23, 2012 #7
    Re: Find the hydrostatic force on one end of the tank if it is filled to a depth of 8

    Hey gneill and sushifan.

    Really good problem.


    How did you guys get dA as xdy?

    My diagram suggests something else.May be am making a mistake :-(
     
  9. Mar 23, 2012 #8
    Re: Find the hydrostatic force on one end of the tank if it is filled to a depth of 8

    I thought of my radius as x units as we traveled vertically on the y-axis, so I rewrote the given function in terms of y.

    Could you share your set up of the problem? Maybe it's me who's making the mistake! I'm actually unsure of my solution.
     
  10. Mar 23, 2012 #9

    gneill

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    Re: Find the hydrostatic force on one end of the tank if it is filled to a depth of 8

    The contour is that of a parabola, y = (1/2)x2 . So for a given y (corresponding to some height up from the vertex), the half-width is ##\sqrt{2y}## . The area element dA is then the total width by the height, dy, thus forming a thin rectangular area element. The "2" in "2y" must have units "feet" associated with it in order to keep units consistent.
     
  11. Mar 23, 2012 #10

    Firstly, I would take pressure as Po (Atmospheric pressure)
    + phg as its not written to neglect the atmosphere. (Do this only if the answers not matching or not coming twice of what you got with the correct units).

    Secondly, my area is coming exactly twice of what you got.(But it maybe wrong.Although I don't know why!)



    So the area of the small rectangle should be
    2*(2y)^1/2*dy .

    Right?

    And really good observation on the units gneill :-)
     
    Last edited: Mar 24, 2012
  12. Mar 24, 2012 #11
    Re: Find the hydrostatic force on one end of the tank if it is filled to a depth of 8

    Ah, I have to take the integral twice to cover both sides of the parabola. That's what I forgot.
     
  13. Mar 24, 2012 #12
    Re: Find the hydrostatic force on one end of the tank if it is filled to a depth of 8

    Does the answer match?
     
  14. Mar 24, 2012 #13
    Re: Find the hydrostatic force on one end of the tank if it is filled to a depth of 8

    Well I'm not sure what numerical value you got for your answer. You said you got twice my answer; do you mean twice my answer (56, 197.12) from my very first post?

    Because that answer was wrong, since I used 9.8 for gravity and not 32.

    So, I'm redoing the entire calculation.

    It would help if you actually provided your set up or at least your final answer so I can compare.
     
    Last edited: Mar 24, 2012
  15. Mar 24, 2012 #14
    Re: Find the hydrostatic force on one end of the tank if it is filled to a depth of 8

    No i just got my expression twice as what you got in terms of variables.

    I was asking if your answer matches with that of text book ?

    I will post my answer in few mins .Let me calculate :-)
     
  16. Mar 24, 2012 #15
    Re: Find the hydrostatic force on one end of the tank if it is filled to a depth of 8

    My textbook unfortunately does not come with an answer key. :[

    And I'm doing my calculation right now, too.
     
  17. Mar 24, 2012 #16
    Re: Find the hydrostatic force on one end of the tank if it is filled to a depth of 8

    See, i would refrain from posting the answer as its against the Guideline :-)

    However,My expression is exactly double of your original post (with proper units of g offcourse)
     
  18. Mar 24, 2012 #17
    Re: Find the hydrostatic force on one end of the tank if it is filled to a depth of 8

    Didn't know that. I'm new here.

    And apologies for the late reply, I fell asleep.
     
  19. Mar 24, 2012 #18

    gneill

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    Re: Find the hydrostatic force on one end of the tank if it is filled to a depth of 8

    It's against the guidelines for those offering help to do the work for, or to simply give answers to the questioner. It's quite okay if the questioner does the work and comes to a correct answer :wink:
     
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