4.5.5.1 Pressure in a fluid

Most people know that liquids are fluids; we often hear the phrase "drink your fluids".
But most people don't know that gases are also fluids !

Why are they both "fluids"?
Its because of their particle structure; they are both made up of particles that are able to move around fluidly (meaning: over and around each other, unlike the particles of a solid which are fixed in positions, free only to vibrate). Hence the term "fluid".

OK. So, now we know that both liquids and gases are fluids, what causes them both to exert pressure on objects?

Well, again, it is due to their particle nature.
Their particles are constantly moving and quite literally bashing into whatever gets in their way. So, if an object, such as a person, is "in a fluid", like water, it is bombarded by the fluid's particles causing it to "feel a pressure".

Why don't we notice the pressure of the air when we are "in it"?
It's because in a gas, the particles as so free to move that they bombard us equally on all sides and in every direction, including top and bottom, so although the pressure is there we don't notice any single direction of pressure.
So, the first thing to learn is - in a fluid Pressure acts in every direction. See diagram 1.

Because the pressure acts in every direction over our body, we tend not to notice it.

The situation when we get into a liquid, however, is different!

Pressure DIFFERENCES in liquids cause objects to float.

In any volume of a liquid there are vastly more particles, compared to in air, so there is a greater pressure (due to more bombardment).
But, there is also a greater pressure DIFFERENCE over an object, such as a person, that is partly in and partly out of the liquid.

In diagram 2 you can see that the pressure due to the Water, even just below the surface is greater than the pressure due to the Air.

And as you look deeper in the Water you see that the pressure arrows get bigger indicating a larger pressure.

It is this Pressure DIFFERENCE that is the cause of floating.

Pressure, Force and Area

If you look again at diagram 2, you should be able to agree that all of the left pointing pressure arrows cancel with all of the right pointing pressure arrows, leaving just a single black downward Air pressure arrow and a single blue upward Water pressure arrow.
In diagram 3 we have deleted the left and right arrows leaving just the single up and down arrows since these are the ones that are causing the overall pressure DIFFERENCE.

Now, can you see that the DIFFERENCE of these pressures will cause a single resultant force and it will be upward, won't it?

This upward force is called Upthrust and if the Upthrust is big enough to balance the person's weight then he or she will float.

OK, so far so good, but what has Area to do with this?

Well, the water particles have to bombard the object (the person) to produce the Upthrust. If they only have a small area over which to act (just the feet), then it makes sense that the final upward force, or Upthrust, will be relatively small compared to if, for example, the person was lying down!!

In diagram 4 the person is lying flat in the water and so many more upward pressure arrows are able to act.

Since this pressure is acting over a larger area, it produces a larger resultant upward force or Upthrust. This is why it is easier to float lying down.

Note: It is not the Pressure or Pressure DIFFERNECE that is greater due to the increased area; it is simply that a given Pressure DIFFERENCE acting over a larger Area must inevitably produce a larger Upthrust force.

So objects with a large Area will be more likely to float.

Thankfully, we can sum all of this up very neatly with a simple bit of mathematics:

So, the Force is proportional to both the Pressure or Pressure DIFFERENCE and to the Area of the surface; the bigger the area the bigger the Upthrust.
So, to float on water easily, lie down to increase your contact Area!

If the upward force, F, the Upthrust, generated by the Pressure DIFFERENCE is greater than or equal to your Weight, then you will float!

By now you should begin to realise how it is that huge ships like ocean liners or oil tankers can float!

The clue is - the above equation - and specifically, the word Area.

Some of the world's largest ships have masses exceeding half a million metric tons and they are made from steel, but they float!
If you took such a lump of steel (or even just 1 Kg of steel) and simply dropped it in the water it would sink, no doubt.
The reason that a gigantic steel ship floats, is that the steel is spread out into a hull that covers a huge surface area.

The characteristic triangular cross sectional shape of a typical hull is also important because it allows a greater area to be in contact with the water as more cargo is loaded onto the ship and the ship drops lower and lower into the water. (Of course, there is a limit to how low the ship can "sink down" and all ships have a line around their hull which marks the maximum level to which they can safely be loaded.)

So, if you ever have to make a raft out of pieces of wood and perhaps some empty barrels or crates and some rope, the main design principle that you would adhere to would be to try to produce as large a surface (or base) area as possible. This would produce the largest "force" or "upthrust" from the water and increase your chance of floating.

In a moment we will look at how we know that the water pressure is 10,000 Pa at a depth of 1m (ie it wasn't just a guess), but before that, since we have introduced an equation, we need to explore it a little further, and like most equations, we can rearrange it. So, here is another example question involving the equation, but rearranged.

So, the equation rearranged for Pressure is:

Both forms of the equation are important and you must be able to recall and use them.

OK, so how do we KNOW that the water pressure is 10,000 Pa at a depth of 1m, which is what we stated in Example 1?

How to calculate the pressure at any depth in a liquid

To calculate the pressure at a point below any depth of any liquid (or below a "column of liquid"), not just water, we use the following equation:

So, Pressure, p, in a liquid increases with depth (height of the column), h.

You should be able to see from the equation that the pressure above a point in a liquid is proportional to the column height or "depth of liquid", and to the density of the liquid.
You need to make sure that you understand why these are so.

First, why is pressure proportional to depth or why does pressure increase with depth?
Well, the liquid is made from particles; there will be more particles in a higher/deeper column of water above a point and so there will be a greater weight of liquid above the point and thus a greater pressure.

Imagine 3 fish at 3 different depths of water, as shown below.

The first fish has less water particles above it, so less weight of water above it and so less Pressure; hence shorter pressure arrows compared to the third fish in the deepest water.

Second, why is pressure proportional to the density of the liquid?
This is also due to the particle nature of the liquid; a more dense liquid has more particles in a given space and so, once again, a greater weight of liquid above the point and so a greater pressure.

Don't make the mistake of thinking that the pressure due to the column of fluid above a point always acts downwards and so causes a downwards force; it doesn't!
Remember what we said at the start of this section - in a fluid, pressure acts in every direction.

So, if we put an object 1m below the water it doesn't get forced downwards by the water, instead we might feel it being buoyed up by the water and it might even want to float!
This is because the pressure lower down will always be greater than the pressure higher up, making an object float or making it feel less heavy.

A heavy bag is lowered into water on a rope.
It is "buoyed" up due to a greater pressure at the bottom compared to the top.
But take note that the pressure acts all over the bag.

A good example of how water pressure increases with depth is a dam built across a lake to make a reservoir. Engineers know that the base of the dam has to be built stronger than the top to cope with the higher pressure.

4.5.5.2 Atmospheric Pressure

Air is a Gas and we have already stated that a gas is a Fluid, just like a liquid.

Reminder: gases and liquids are called fluids due to the nature of their particles - they are able to move around or to "flow", fluidly.
A solid, on the other hand is not a fluid because its particles are not able to flow; (they do move, but only in the sense of a vibration).

Since the air particles are particles of matter (again, like liquid particles) they are held by the force of gravity, forming a "blanket" or "belt" of particles around the Earth which we call the "atmosphere".

Here is a definition for Atmosphere:
The atmosphere of Earth is the layer of gases, commonly known as air, that surrounds the planet and is retained by Earth's gravity.

Living at the bottom of this layer of gases, as we humans and most animals do, we feel the maximum atmospheric pressure on a daily basis. So, what is the value of this Standard Atmospheric Pressure?

It is a huge value, approximately 100,000 Pa.

Two questions immediately arise the first time you see this number;
1) Why is it so large? and
2) Why aren't we crushed by such a huge pressure? (it is equivalent to a force of 1000N acting on every cm² of our body.

1. Why is atmospheric pressure so large?
The answer is pretty obvious when you think about it; we have said that the atmosphere is a layer of gases and we are at the bottom of it! So, as we walk about the Earth, there is always something like a 20Km thickness of gas above our heads weighing down on us. The atmospheric pressure is due to the weight of the long column of particles above us. So, its not surprising that the pressure is large.

2. Why aren't we crushed by such a huge pressure?
At the beginning of this section on Pressure, on the first outline diagram of a person, we showed AIR pressure arrows pointing inwards, but the diagram must be incomplete. (Let's repeat diagram 1 here.)

Now that you know the size of the indicated inward atmospheric pressure you should realise that the person would get crushed IF diagram 1 told the whole story.
So, what is missing from the diagram?
Answer- there MUST be arrows inside the body pointing outwards!
There must be air inside the body producing an outward pressure to balance the inward pressure due to the atmosphere.
This turns out to be true.
There is air inside our bodies, in particular inside our lungs, ears, throat, stomach etc. This air produce a pressure which balances the inward pressure due to the atmosphere.
Additionally, there is a lot of liquid in our bodies, and liquids, unlike gases are incompressible so they resist the atmospheric pressure and push back on it with the same pressure.

Now we know why we are not crushed by the atmosphere, let's move on.

Atmospheric Pressure Varies With Height

At sea level a person has about 20 Km of atmosphere (air) above him or her producing a maximum pressure of about 100,000 Pa. If the person goes higher and higher up, say by climbing a mountain (a very big one, like in the Himalayas) then it is obvious that there will be less atmosphere above him or her and hence less weight acting downwards and so, less pressure.

So, we can initially conclude:
Pressure is due to the action of particles bombarding or colliding with an object. The higher up one goes, the lower is the atmospheric pressure because there are less particles above the person or less weight of air above the person.

See diagram 2.

But there is a second reason why Air pressure gets lower with height (a reason not shared by liquids, our other "fluid").

Unlike water and other liquids, the density of air is not constant.
As height increases, the density of the air in our atmosphere decreases. We say, it "gets thinner". At the very highest points we say that the air is "too thin to breathe" and at that point people need to bring their own air with them in cylinders. We see RAF jet pilots and mountaineers on Everest doing this, wearing full face breathing apparatus.
A decrease in density means a decrease in number of particles in the column of air above the person and hence in the weight of the column and thus in the pressure.

See diagram 3

So, we can finally conclude:

Atmospheric pressure varies with height due to two reasons:
1. As one moves higher up into the atmosphere, there is a shorter column of air above a person, so the weight of the air above a person decreases and the pressure decreases.
2. As one moves higher up into the atmosphere, the density of the air decreases and so the number of particles decreases, causing a decrease in the weight of the air above a person and a decrease in the pressure.