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Pressure is equal or force is equal at the bottom?

  1. May 30, 2016 #1
    upload_2016-5-30_19-45-21.png
    Problem Statement:
    Both containers are just equal in volume and having same height; you can assume one is upside down of another. What is pressure at the bottom of the containers. Lets say left side is container 1 and one on right side is 2.

    Relevant equation:
    upload_2016-5-30_19-49-56.png h is the depth of fluid from the top surface of fluid.

    My attempt:
    Thought 1
    : Pressure will be equal at bottom for both the cases; and force to the bottom we can find with multiplying the respective bottom surface areas.

    Thought 2: As volume and density of both the liquid is same; so the mass. When we draw free body diagram of the liquid, it appears the forces will be equal at bottom will be equal not the pressure.
     

    Attached Files:

  2. jcsd
  3. May 30, 2016 #2

    Paul Colby

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    I like your second thought. What is the total upward force on the bottom in terms of the pressure?
     
  4. May 30, 2016 #3
    Let's try this; I am connecting piezometer to the both tank near bottom sideways. What will be the level of liquid in the piezometer; it is same or different.
     
    Last edited: May 30, 2016
  5. May 30, 2016 #4

    ogg

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    For a fluid at or near equilibrium, pressure in any horizontal plane (actually, a surface of equal g) will be equal. What that will be depends on the weight (force). Consider a nail (spike) and a 20 kg weight. Which would you prefer: 1). The weight on top of the spike on top of your toe OR 2) the spike on top of the weight on top of your toe? Would you argue that it makes no difference to your toe?
     
  6. May 30, 2016 #5

    TSny

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    Yes.

    When you draw the free body diagram of the fluid, don't forget the forces on the fluid exerted by the sloping sides.
     
  7. May 30, 2016 #6
    I am not getting what you are saying.
     
  8. May 30, 2016 #7
    Weight of bodies are equal, so the forces. How pressure can be equal.
     
  9. May 30, 2016 #8

    TSny

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    Yes, the weight of the fluid is the same in each container. When you speak of force, what particular force are are you referring to?
     
  10. May 30, 2016 #9

    CWatters

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    See post #5.

    For the right hand vessel the force acting on the base is NOT just the pressure multiplied by the area of the base. There is also pressure acting vertically on the sloping sides and these also press on the base.
     
  11. May 30, 2016 #10
    I agree with TSny. In your force analysis, you neglected to include the upward and downward forces exerted on the fluid by the sloping sides. Don't forget that pressure acts perpendicular to all surfaces.
     
  12. May 30, 2016 #11

    Ray Vickson

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    Right. But in the left-hand case, the force on the sides points up and out, but does NOT make the whole thing weigh less!

    I think there is a difference between the case where the container has sides only (but no bottom) and the container with rigid sides and bottom. In the former case the force on the table under the liquid is just area times pressure; in the latter case, it is the weight of the water plus container.
     
  13. May 30, 2016 #12

    TSny

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    Hopefully, everyone agrees that the two systems weigh the same. However, I believe some of the confusion is coming from not specifying clearly the "force" that one is talking about. There are at least three distinct "forces at the bottom" :

    (1) The force that the fluid exerts against the flat bottom of the container
    (2) The force that the flat bottom of the container exerts on the fluid
    (3) The force that the flat bottom of the container exerts on the table

    In post #8, I was trying to get the OP to clarify his use of the word "force".
     
  14. May 30, 2016 #13
    Hi Ray. I'm a little confused, but I think we are saying the same thing. In the left-hand figure, the sides are pushing down and in on the fluid, and this downward component adds to the weight of the fluid, so a larger area is required at the base to push up on the fluid. In the right-hand figure, the sides are pushing up and in on the fluid, so a smaller area is required at the base to support the weight of the fluid. So, in the left-hand figure, the upward pressure of the base produces a force which is greater than the weight of the fluid, and, in the right-hand figure, the upward pressure of the base produces a force which is less than the weight of the fluid. In both cases, the weight of the fluid is the same.
     
  15. May 30, 2016 #14

    Paul Colby

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    This is a really good problem. My 2 cents, pressure is isotropic (independent of direction) and both thought 1 and 2 of the OP are correct. Now, while pressure is isotropic the force on the walls of the container are along the normal to the surface. Draw a free body that includes these forces.
     
  16. May 30, 2016 #15
    Thought 2 is not correct. Several responders have pointed out that thought 1 is correct. Both the force and the pressure exerted by the base on the fluid cannot be the same in the two figures. It is the pressure that is the same in the two figures, not the force.
     
  17. May 30, 2016 #16

    Paul Colby

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    A free body diagram is always correct provided all the forces are correctly represented and the body in question is known. There are forces applied to the fluid by the container. There are forces applied to the container. The force of the ground on the container must counter the weight. This does have to equal the net forces applied by the water to the inside of the container which includes the side walls.
     
  18. May 30, 2016 #17
    You do realize that the OP's original question was not about the force that the ground exerts on the container, right?
     
  19. May 30, 2016 #18

    Paul Colby

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    "You do realize that the OP's original question was not about the force that the ground exerts on the container, right?"

    yes

    The integral of the force of the water on the inside of the container is equal to the downward weight. The pressure times the area is the normal force on each area element. The pressure is given by the equation the OP quoted which yields the same value in both cases.

    The pressure on the outside of the container is the weight divided by the area.
     
  20. May 30, 2016 #19

    Paul Colby

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    An upward force is applied to the bottom of each container originating from the pressure of the fluid. The point of the problem is possibly that one should just apply the laws of fluids and not worry too much about the seeming paradox caused by the identical weights and differing areas. Now, having noticed that the pressure applied by the ground is in fact different you can resolve this by understanding how the slanted sidewalls effect each case. The original problem says "pressure at the bottom". This could be taken both ways as inside and out. These are not equal given a rigid container.
     
  21. May 30, 2016 #20
    Here is the FBD of fluid only; neglect the black lines resembling the walls
    upload_2016-5-31_10-10-51.png

    I am very much satisfied with TSny; When I would put weighing machine below the container; I would have same weight for both the container.

    My thought is; I would connect a piezometer to the both of the container, it would have same fluid column height that means same pressure; since childhood we are reading pressure is scalar and independent of shape of the container. Horizontal component of forces of due to side walls will be nullified in both the cases.

    What is the difference between two condition is the component of force that is only acting to the flat bottom. The net force on bottom is same for both the cases.

    Moral of the story: net force to the bottom and pressure at bottom will be equal, the difference is the normal force acting only to the flat bottom.
     
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