Why does pressure not depend on the shape of the container?

In summary, the conversation discusses the concept of pressure in a fluid, specifically in relation to the height and shape of a container. The equations P = \frac{F}{A} and P = ρgh are mentioned as ways to calculate pressure. The question is raised about why pressure in water only depends on the height, and the answer is found by considering a column of water and the forces acting on it. It is then explained that in a truncated cone container, the pressure force perpendicular to the slanted wall creates a downward force that compensates for the missing weight of water, resulting in equal pressure at the base of both the cylindrical and conular containers.
  • #1
ollien
6
0
Hey there. I'm trying having a bit of trouble wrapping my head around this.

Homework Statement


That pressure is equal at all the same height no matter the shape of the container.

Homework Equations


[itex]P = \frac{F}{A}[/itex]
[itex]P = ρgh[/itex]

The Attempt at a Solution



I'm trying to understand this from my physics class, but I can't seem to get a hold on it. Why does pressure in water only depend on the height? I can understand that pressure at a depth is caused by the weight of the water on top of it, as shown by [itex]P = \frac{F}{A}[/itex]

Now, I know that the following is also true. [itex]P = ρgh[/itex] Now, this equation shows directly that it is related to height, and it can also be rearranged to [itex]P = \frac{F}{A}[/itex]. However, this is where to start to run into issues.

Let's say we have a cylindrical container of fluid, like so.

OG3v99D.jpg


This cylinder, filled with fluid, will have a pressure on the bottom. Now, let's assume we have a conular container, with the same height, and equal base area, like so.

S6zXEWI.png


From my knowledge of [itex]P = ρgh[/itex], I know that the pressures are equal at the base. However, I can't begin to understand why. There is a lower volume of water in the cone, and therefore a lower mass. This means that there's a lower force pressing down from the water than in the cylinder. Because of this, shouldn't there be two different gauge pressures? What's the reasoning for there being equal gauge pressures? I can't begin to figure it out for the life of me.

Thanks so much.
 
Physics news on Phys.org
  • #2
Consider a column of water whose base is near the outside of the cone. It is short, so the base only has a small weight of water above it. Now think about the top cc of that column. On one side of that cc the container wall is pushing on it. What is pushing on the other side, and how much water is above that?

You're right though, it is unintuitive. I use siphoning regularly to put grey water from my washing machine onto my garden, and every time I do it I can't quite believe it is happening.
 
  • #3
I'm sorry, I don't quite understand what you mean. What's a cc?
 
  • #4
ollien said:
I'm sorry, I don't quite understand what you mean. What's a cc?
cc = cubic centimeter
 
  • #5
Ollen,

The pressure would be the same even if you had a narrow cylinder like a capillary tube. All the equation is saying is the pressure at a certain dept ONLY depends on the depth of the water at a depth your interested in. And also, the pressure is the same in all directions at a certain depth. Does that help?
 
  • #6
-Robert said:
Ollen,

The pressure would be the same even if you had a narrow cylinder like a capillary tube. All the equation is saying is the pressure at a certain dept ONLY depends on the depth of the water at a depth your interested in. And also, the pressure is the same in all directions at a certain depth. Does that help?

Why? I can't figure that out.

andrewkirk said:
Consider a column of water whose base is near the outside of the cone. It is short, so the base only has a small weight of water above it. Now think about the top cc of that column. On one side of that cc the container wall is pushing on it. What is pushing on the other side, and how much water is above that?

You're right though, it is unintuitive. I use siphoning regularly to put grey water from my washing machine onto my garden, and every time I do it I can't quite believe it is happening.

Now knowing what a cc is, pushing on the other side, would it be the other walls?
 
  • #7
ollien said:
Now knowing what a cc is, pushing on the other side, would it be the other walls?
No. Remember that we are talking about a thin column of water whose base is close to the wall. So where that column hits the wall, the side of the column opposite to the side that is against the wall cannot be against another wall. If its not pushing against (and being pushed by) a wall, then it's pushing against ... what?
 
  • #8
andrewkirk said:
No. Remember that we are talking about a thin column of water whose base is close to the wall. So where that column hits the wall, the side of the column opposite to the side that is against the wall cannot be against another wall. If its not pushing against (and being pushed by) a wall, then it's pushing against ... what?

Is it the atmosphere?
 
  • #9
No. That would mean that the column of water nearer the wall is taller than the column of water next to it but further from the wall, which would mean the surface of the water is not horizontal. Draw a picture, showing the column, hitting the sloping container wall at the top, then draw a column next to it, which will hit the container wall higher up.
 
  • #10
Got it. That makes more sense.

Thanks!
 
  • #11
In your truncated cone example, fluid pressure acts perpendicular to the slanted wall, and the slanted wall pushes back on the fluid. Since this pressure force is perpendicular to the slanted wall, there is a component of this force in the downward direction. So the slanted wall is exerting a downward force on the fluid. This turns out to exactly compensate for the missing weight.

Chet
 
Last edited:
  • Like
Likes Ravi Singh choudhary

Related to Why does pressure not depend on the shape of the container?

1. Why does pressure not depend on the shape of the container?

Pressure is defined as force per unit area. The shape of the container does not affect the force being applied, only the area on which it is being applied. Therefore, the pressure remains constant regardless of the shape of the container.

2. How does Boyle's Law explain the relationship between pressure and container shape?

Boyle's Law states that at a constant temperature, the pressure and volume of a gas are inversely proportional. This means that as the volume of a container decreases, the pressure of the gas inside increases, and vice versa. However, the shape of the container does not affect the volume of the gas, so the pressure remains constant.

3. Is the shape of the container ever a factor in determining pressure?

In certain cases, such as with a closed container containing a liquid or gas, the shape of the container may impact the pressure. This is because the weight of the liquid or gas may distribute differently depending on the shape of the container, resulting in a different pressure at the bottom of the container.

4. How does the ideal gas law relate to the shape of the container?

The ideal gas law, which states that the pressure, volume, and temperature of a gas are all directly proportional, does not take into account the shape of the container. This is because it assumes that the gas particles are in constant motion and will evenly distribute throughout the container, regardless of its shape.

5. Can the shape of the container affect the pressure in a liquid?

Yes, the shape of a container can affect the pressure in a liquid. This is due to the weight of the liquid being distributed differently in different shaped containers. For example, a tall and narrow container will have a higher pressure at the bottom compared to a short and wide container with the same amount of liquid.

Similar threads

  • Introductory Physics Homework Help
Replies
18
Views
621
  • Introductory Physics Homework Help
Replies
1
Views
521
  • Introductory Physics Homework Help
Replies
3
Views
288
  • Introductory Physics Homework Help
Replies
9
Views
1K
  • Introductory Physics Homework Help
Replies
8
Views
2K
  • Introductory Physics Homework Help
Replies
8
Views
787
  • Introductory Physics Homework Help
Replies
15
Views
1K
  • Introductory Physics Homework Help
Replies
7
Views
1K
  • Introductory Physics Homework Help
Replies
2
Views
804
  • Introductory Physics Homework Help
Replies
4
Views
3K
Back
Top