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  1. Here is a typical definition of air pressure:

Air pressure is caused by the weight of the air molecules above. Even tiny air molecules have some weight, and the huge numbers of air molecules that make up the layers of our atmosphere collectively have a great deal of weight, which presses down on whatever is below.

  1. And yet, all sources I've seen state that air pressure is equal in all directions.

1 & 2 seem contradictory.

Related question:

Why does air pressure from above not crush us? The answer I see consistently given is that an equal air pressure from below balances it out. But if a car were resting on me from above and crushing me, then another car pressing against me from below would not relieve that pressure -- it would only increase the pressure I would feel! If I were in an enclosed closet, and one of the walls were to press in against me, and then the opposite wall would also press in against me, the second wall would not "balance" things out, but rather only increase the pressure I would feel!

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  • $\begingroup$ Consider the flowing property of pressure in a fluid. Pressure is dependent ONLY on depth of fluid. For example: A 1 ince diameter pipe one mile high has exactly the same pressure at the bottom as a 500 foot diameter pipe of the same height. This isn't an answer, just something you should consider to understand this a little more. $\endgroup$ – Bassinator Apr 11 '15 at 15:44
  • $\begingroup$ The short answer is that you are confusing a gradient with an anisotropy. Pressures change from one location to another, but not from one orientation to another. Fluids can't support shear without deforming so as to relieve that shear. Without shear along the boundary of a control volume, any net pressure difference would either cause the entire volume to accelerate, or cause it to deform in shape. Both of these result in kinematic energy appearing to account for the work imbalance between the high pressure PV work and the lesser low pressure PV work. $\endgroup$ – Phil Sweet Sep 27 '18 at 2:41
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Why is air pressure equal in all directions?

Imagine what it would mean for a thin flat piece of metal if the pressure were not equal from above and below. There would be more pressure pushing down from the top than pushing up from below which would equate to a net force. This force would start to accelerate the piece of metal downwards; there would be no equilibrium. Now forget about the piece of metal. Without it there would be air molecules rushing down from the pressure gradient. They would actually rush down until they equalized the pressure gradient and stopped moving.

Why does air pressure from above not crush us? The answer I see consistently given is that an equal air pressure from below balances it out.

This isn't quite correct. The pressure is not simply equal from above and below with your body being a zone of different pressure. Rather, your entire body is at the same pressure as the surroundings. To understand the difference, think about a tank from which some of the air can be evacuated (a vacuum tank). When the tank is full of air at equal pressure to the surroundings the lid can be removed easily. If you seal the container, pump some of the air out, and then try to remove the lid you will find that it is very tightly stuck. This is because there is a strong force on the lid caused by the pressure gradient between inside and outside.

The fact that your body is at atmospheric pressure is actually very important to the way it functions. If you were to be thrown out of a spaceship where the pressure is near zero, all of the gases (oxygen being an important one) would evaporate out of the fluids in your body.

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Air pressure is exerted on the surface of a body by air molecules hitting the surface and being reflected. Each of these reflections (gazillions of which happen per second) transfers a little impulse on the surface, which macroscopically means a permanent force (per unit of area). Why do the air molecules bounce and hit all the time? Either because the air is moving at large (aka. "wind"), or because they bounce around irregularly (aka. "temperature"). The latter kind of movement knows no preferred direction and therefore the pressure is the same no matter what orientation the test surface has. The very fact that there is no net movement (wind) is expressed by the fact that the same force acts on the back side of a thin surface as on the front side (so there is no net force).

Then how come the air pressure is related to the weight of air above us? In equilibrum the force caused by air pressure from below on an imaginary horizontal surface is just enough to keep the air column above it "in place", whoich means that it equals the weight. We need not always have equilibrum, but if we don't then the stringer of the forces causes accelaration and movement - until equilibrum is reached.

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I've tried to split up the questions a bit, drop a comment if I've missed something.

Air pressure is caused by the weight of the air molecules above.

This is indeed correct. The air pressure is proportional to how much air is above it: You have less on a high mountain than at sea level. The diagram shows this in practise.

atmosphere pressure

Air pressure is equal in all directions.

This is also true: It will push equally in all directions. If it would be unequal, it would try to reach equilibrium. The air molecules will be subject to both the force of gravity pulling it towards earth (compressing it) and the force of other molecules, pushing it away.

Sour![][2]

Source

And yet, all sources I've seen state that air pressure is equal in all directions.

For some little point in the atmosphere, this would be true. There would be equal force acting on it in all directions.

1 & 2 seem contradictory.

There is a very small difference for, say, a small cubed container since the bottom will have a tiny bit higher pressure from the air above it than the top side and the pressure will be marginally higher. However that decrease in pressure with altitude will take place both inside and outside the box. In general, the pressure difference can be ignored for almost all applications.

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Pressure is given by the formula,

${P = {\rho}gh}$

Where:

  • ${\rho}$ = density

  • ${g}$ = gravitational constant

  • ${h}$ = height/depth

Pressure at any point below the upper boundary of fluids, such as air and water, is uniform in all directions due to the fluid molecules being in constant motion and continually bumping into one another. Pressure increases with the depth of the fluid due to the amount of fluid above it, but any point on a horizontal plane will have the same pressure.

Compare this to rock in the Earth's crust and mantle. Ignoring tectonic stresses, the pressure in the vertical direction is still given by

${P = {\rho}gh}$

However, because of the solid nature of rock, molecules are not rapidly moving and they do not continually bump into one another. Consequently, pressure in the lateral direction is not equal to the pressure in the vertical direction and pressure/stress in rock is not uniform in all directions.

This source gives the lateral pressure/stress as being related to the vertical pressure/stress.

${{\sigma}_h = k{\sigma}_v = k{\rho}gh}$

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Pressure is the average outward force that molecules exert on their surrounding.

If you take the case of air molecules bouncing around hitting everything they push outwards to the sides equally, but as you mentioned their weight means that they do indeed push downward harder than they push upwards. As the weight of air in a small space is very small, this difference can usually be neglected. However, without this difference balloons wouldn't float. This tiny difference adds up in the atmosphere till the pressures down here on the surface are actually pretty significant.

The reason the cars would squish you is that when the car pushes down with a high pressure, it will move your surface inward until you push back with the same pressure. Unfortunately for you, as your internal pressure increases that makes your sides at higher pressure than the air around you, so your sides squish out since the air doesn't push back as hard. So, it's not enough to just be pushed from the top and bottom, or even from four sides. You have to be pushed from all directions, including up your nose and inside your lungs in order for your internal pressure to be able to comfortably push back against the high pressure.

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Both statements are correct. The best way to understand how these two statements can coexist is to understand the concept of a gas pressure.

Now to understand pressure we look at a container full of gas molecules. Gas molecules do not behave at all like solids or liquids. In a gas the molecules are not attracted to one another so they fly around at extreme speeds bouncing into objects and other gas molecules. These collisions are elastic so no energy is lost during collisions.

enter image description here

Every time a collision occurs some kind of energy transfer takes place between molecules. However, on a macroscopic level, there are so many collision taking place that they average out to zero energy transferred. Imagine that a gas molecule is about to hit the wall off the above container. We know that when the molecule hits it will bounce off and head in the other direction just like a bouncy ball. The wall will also feel a force due to Newton's second law. However on the other side of the container the exact same thing is happening. In fact the same thing is happening on the outside of the container as well. All these collisions exert a force but they all cancel each other out.

Now let us apply this to your first deffinition. As you stated air pressure is caused by the weight of the air molecules above. Gas molecules are attracted by gravity towards the earths surface. As a gas molecule is pulled toward the surface of the earth chances are it will hit another gas molecule and bounce off of it in another direction. Now lets say that in this particular collision the first molecule hits the top of the second molecule. This causes the second molecule to travel downwards even faster than the first molecule. This happens again and again until the molecule bounces of the surface of the earth. This is how your first definition is derived. The key is to remember that this is a gas pressure and thus is from all sides.

This is the hardest concept to grasp because when someone hears that there are hundreds of pounds of air above them they imagine hundreds of pounds of steel plates on their shoulders. Do not think of it like that. If a bouncy ball falls on your head it pushes you down. However if it misses, hits the floor, bounces up and hits you the two forces cancel each other out. The trick is to realize that so many collisions are occurring at such a minute scale that you do not "feel" the pressure of the atmosphere.

Solid objects are very good at resisting an even force from all directions. Have you ever heard that you can't crush an egg if you squeeze it from all directions? The same concept applies to to your body. The atmosphere is pushing very hard from all directions (even from inside your lungs!) but they all cancel out.

To contrast this imagine a steel drum with just a few gas molecules in it what would happen?

enter image description here

Now despite this being cool notice that the sides of the barrel collapse as well. This means that air molecules were pushing from the side but there was nothing to push back from the inside. We can see from the imploding barrel that the atmosphere is compressing us with enough force to crumple a steel drum. However because this pressure is exerted from every direction the forces are cancel out and we don't feel a thing.

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I'd like to add my understanding, in case it helps anyone comprehend the reason. The reason there is pressure from all sides in these situations is due to the properties of fluids in equilibrium. In the atmosphere, for example, the air molecules being "weighted down on" from above would squeeze out the sides of the air column, if that were possible. It is not, of course, because the air column adjacent is under the same force and thus they are no better off. Gas molecules are energetic in every direction, or in other words, fluid pressure cannot exist in one direction if in equilibrium, as any difference in pressure would yield motion (wind).

The reason that you use the weight of the fluid above you (air, ocean, etc.) to know the horizontal pressure you would feel is because you are assuming you are in an area under equilibrium and thus you know from the above reasons that the "horizontal" pressure is equal to the "vertical" pressure.

Another intuition I like is the idea of a pneumatic piston. The cylinder that contains the fluid needs to be strong to keep from bursting. If you replaced the fluid with a metal rod and put piston force on that instead, the cylinder walls wouldn't feel anything.

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