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Changes in volume change total force?

  1. Mar 25, 2013 #1

    If you had a cube, with a volume of one cubic inch, that means that each side would be one inch. As a cube, it would be like a die, and therefore have 6 sides, and 6 square inches.

    800 lb/sqinch=force/6sqinches



    Force equals 4800 pounds

    So now remove a side of the die, and put another die on. Four sides with 2 square inches, and 2 sides with 1 square inch is 10 square inches.

    Pressure=4800 pounds/10 sqinches

    Pressure then equals 480 PSI


    However, according to Boyle's law, as you increase volume, pressure decreases inversely. So since the volume doubled, the pressure should have been halved.

    Can someone explain why this didnt work out? I am putting money being a problem with my force value, since that force is only a mass, without an acceleration. However, there are no other units in that equation above, so I am confused. Is force changed with volume when it comes to pressure? And if so, is that because the acceleration is somehow changed during compression/decompression?

  2. jcsd
  3. Mar 25, 2013 #2

    Simon Bridge

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    Welcome to PF;
    Boyles law applies to a gas inside a container.
    If you double the volume, but do not change the amount of gas, then the pressure must decrease - since the same amount of gas has more wall area to go around.

    What you are doing above is totally different.

    If you applied 800lbs/sqin to every face of the 1in cude, the total force on the cube would be zero.
    (A cube has opposite pairs of sides.)
    Thus the force calculation cannot be correct.

    You have noticed that if you double the volume you don't double the surface area.
  4. Mar 25, 2013 #3
    The amount of gas wouldnt be changing. Basically, it would be taking a container that has 1 cubic inch of volume inside of it (I used a cube/die as a better visual aid, and easier math than a sphere) and contains 800 PSI of air in it. Then change the volume of that container to 2 cubic inches.

    According to pressure=force/area, that 800 PSI should now be 480 PSI. But Boyle's law says it should be 400. (unless I am misunderstanding Boyle's law... But as I said, the gas wont be changing at all. Same amount of gas. Just doubling the volume)

    Edit: I just saw that I never stated that the 800 PSI was inside the die, and not pressing on it from the outside. That is my mistake. Sorry about that. The cube is a container.
  5. Mar 25, 2013 #4

    Simon Bridge

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    But the total force on the walls of the container is zero.
  6. Mar 25, 2013 #5
    Increasing "sides" doesn't really matter with Boyle's Law from how I understand it.
    piVi = pfVf

    If you increase the volume of your die, the internal pressure of the die will decrease.

    Are you talking about force being applied externally?
  7. Mar 25, 2013 #6
    Yes, sorry, I had forgotten a bit of information in my haste. The pressure is internal. The die is a container.
  8. Mar 25, 2013 #7
    So I am now assuming that pressure=force/area is actually not for fluids in a container. Is that correct? More like for a brick on a sheet of glass? If so, then what would be the correct formula for finding the pressure in a container?
  9. Mar 25, 2013 #8
    I think you're making things complicated by making them simple. Imagine a sphere! As volume increases, the pressure decreases. The particles are bouncing evenly off every part of the inside of the container. It's not comparable to a single force being applied. It's an infinite of equal forces being applied in all directions.

    PV = nRT, right?
  10. Mar 25, 2013 #9


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    For fluids, pressure = rho * g * h, where
    rho - mass density of fluid (kg/m^3)
    g - acceleration due to gravity (9.81 m/s^2)
    h - depth of fluid at point where pressure is measured (m)

    Pressure has units of kg-m^2/(m^3s^2), which by rearranging becomes:
    (kg-m/s^2)*(m/m^3) = N/m^2, which is units of force / area.
  11. Mar 25, 2013 #10
    Sorry, I havent really done any fluid mechanics. I learned the pressure formula (looks like it is from a single force, as you say) and thought that I would play with it, and I was told about Boyle's law by someone else. When I found the discrepancy, I wasnt sure why things werent added up. The idea of there being multiple forces makes sense now, but in my head at the time, the pressure seemed like a single force.

    Thank you very much for the help.
  12. Mar 25, 2013 #11

    The actual forces here are movements of individual particle. Every time a tiny little particle bounces off the inside of the container, you have a force. Those millions of millions of particles add up to give your total PSI, and it's pretty much evenly distributed in all directions.
  13. Mar 25, 2013 #12

    Simon Bridge

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    Like tolove says, you seem to be making things confusing by trying to simplify.

    The concept of pressure is a force distributed over an area.
    An surface of area A under pressure P acts as though it is under a force perpendicular to the surface, through the center of mass, magnitude F=PA.

    For a unit cube oriented so the x,y,z axis pass through the center of the appropriate faces, filled with a fluid that exerts pressure P, the effective force on each face will have magnitude F=P(A=1)=P. Since there are six sides, it is tempting to say the total force is 6P.

    But force is a vector and the direction of the effective forces will be different for the different faces.

    i.e. there are two faces perpendicular to the x axis. One at x=-1/2 and one at x=+1/2.
    The force from the pressure acting on the x=-1/2 face acts in the -x direction, and the force from the pressure on the x=+1/2 face acts in the +x direction. Thus, the net force in the x direction is zero. Repeat for the other three axis and the total force is zero.

    This should make sense to you: if the pressure inside the container added up to give a large force, like that, then the container would accelerate - but you have seen containers of gas before and they don't accelerate just because their contents are under pressure.

    I think that what you meant to ask is something like this: "since pressure, generally, is inversely proportional to area, how is it that the pressure of a gas is inversely proportional to volume."

    For a sphere radius R, containing a gas held at constant temperature, doubling R increases the volume 8x and the area 4x.
    Since pressure comes from the molecules of gas bouncing off the inside of the sphere - the molecules have 4x more area to bounce off, so you'd expect the pressure to go down to 1/4.
    But by ##P_1V_1=P_2V_2## it is actually 1/8 ... so this description of pressure seems to be incomplete. What's missing?

    That about right?
    Last edited: Mar 25, 2013
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