How do you find the pressure in a fluid?

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A solid surface can exert pressure, but fluids (i.e. liquids or gases) can also exert pressure. This might seem strange if you think about it because it's hard to imagine hammering in a nail with liquid. To make sense of this, imagine being submerged to some depth in water. The water above you would be pushing down on you because of the force of gravity and would therefore be exerting pressure on you. If you go deeper, there will be more water above you, so the weight and pressure from the water would increase too.

You probably wouldn't blow up since your body/skin/bones are strong enough to hold you together. Still, it would be really, really uncomfortable. Besides the lack of oxygen and possible direct radiation exposure from the sun, your eyes would bulge, your eardrums could pop, and the saliva on your tongue would probably boil since the boiling point of water decreases as pressure goes down. At zero pressure your body temperature is enough to boil the water on your tongue as well as the fluid in your eyes. So basically, don't ever get caught by space pirates.

Not only can the weight of liquids exert pressure, but the weight of gases can as well. For instance, the weight of the air in our atmosphere is substantial and we're almost always at the bottom of it. The pressure exerted on your body by the weight of the atmosphere is surprisingly large. The reason you don't notice it is because the atmospheric pressure is always there. We only notice a change in pressure above or below normal atmospheric pressure (like when we fly in an airplane or go underwater in a pool). We aren't harmed by the large atmospheric pressure because our body is able to exert a force outward to balance the air pressure inward. But this means that if you were to be thrown into the vacuum of outer space by space pirates, your body pressure would continue pushing out with a large force, yet no air would be pushing in.

This is one of the great mysteries of the universe. I doubt we will ever know. If you find out, please contact the Department of Physics Mysteries immediately.

Okay, so the weight of a fluid can exert pressure on objects submerged in it, but how can we determine exactly how much pressure a fluid will exert? Consider a can of beans that got dropped in a pool as seen in the following diagram.

The weight of the column of water above the can of beans is creating pressure at the top of the can. To figure out an expression for the pressure we'll start with the definition of pressure.

P=FA

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For the force

F

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we should plug in the weight of the column of water above the can of beans. The weight is always found with

W=mg

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, so the weight of the column of water can be written as

W=mwg

&#;

 

where

mw

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is the mass of the water column above the beans. We'll plug this into the equation for pressure above and get,

P=mwgA

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At this point it might not be obvious what to do, but we can simplify this expression by writing

mw

&#;

 

in terms of the density and volume of the water. Since density equals mass per unit of volume

ρ=mV

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, we can solve this for the mass of the water column and write

mw=ρwVw

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where

ρw

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is the density of the water and

Vw

&#;

 

is the volume of the water column above the can (not the entire volume of the pool). Plugging in

mw=ρwVw

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for the mass of the water column into the previous equation we get,

P=ρwVwgA

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At first glance this appears to have only made the formula more complex, but something magical is about to happen. We have volume in the numerator and area in the denominator, so we're going to try to cancel something here to simplify things. We know that the volume of a cylinder is

Vw=Ah

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where

A

&#;

 

is the area of the base of the cylinder and

h

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is the height of the cylinder. We can plug in

Vw=Ah

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for the volume of water into the previous equation and cancel the areas to get:

P=ρw(Ah)gA=ρwhg

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Good question. The original area

A

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in the denominator was the area upon which the force is exerted, which was the area of the top of the can. The area

A

&#;

 

in the numerator refers to the area of the column of water. Since the area of the column of water is equal to the area of the top of the can, these areas do in fact cancel.

Not only did we cancel the areas, but we also created a formula that only depends on the density of the water

ρw

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, the depth below the water

h

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, and the magnitude of the acceleration due to gravity

g

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. This is really nice since nowhere does it depend on the area, volume, or mass of the can of beans. In fact, this formula doesn't depend on anything about the can of beans other than the depth it is below the surface of the fluid. So this formula would work equally well for any object in any liquid. Or, you could use it to find the pressure at a specific depth in a liquid without speaking of any object being submerged at all. You'll often see this formula with the

h

&#;

 

and the

g

&#;

 

swapping places like this,

P=ρgh

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Just to be clear here,

ρ

&#;

 

is always talking about the density of the fluid causing the pressure, not the density of the object submerged in the fluid. The

h

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is talking about the depth in the fluid, so even though it will be "below" the surface of fluid we plug in a positive number. And the

g

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is the magnitude of the acceleration due to gravity which is

+9.8ms2

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.

OK, if you are really clever you might have realized that the bottom of the can is slightly lower in the fluid than the top of the can, and since the pressure gets larger the deeper you go (

Pgauge=ρgh

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) the upward pressure on the bottom of the can should be slightly larger than the downward pressure on the top of the can. This means that the overall effect of the pressure from the water is to crush the can and to exert a net upward force on it. This net upward force from the difference in pressure is the reason why there's a buoyant force on objects submerged in a fluid! But...we're getting a little ahead of ourselves so let's hold this thought for now.

Now you might think, "OK, so the weight of the water and pressure on the top of the can of beans will push the can downward right?" That's true, but it's only a half truth. It turns out that not only does the force from water pressure push down on the top of the can, the water pressure actually causes a force that pushes inward on the can from all directions. The overall effect of the water pressure is not to force the can downward. The water pressure actually tries to crush the can from all directions as seen in the diagram below.

P

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in the formula

ρgh

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is a scalar that tells you the amount of this squashing force per unit area in a fluid.

OK, so here is a subtle fact about pressure; it's defined to be a scalar, not a vector. So why do people seem to represent pressure in diagrams with arrows as if it were a vector with a particular direction?

Even though pressure is not a vector and has no direction in and of itself, the force exerted by the pressure on the surface of a particular object is a vector. So when people draw diagrams with pressure pointing in specific directions, those arrows can be thought of as representative of the direction of the forces on those surfaces exerted by the pressure from the fluid.

If there were no surface upon which the pressure could exert a force, it would make no sense to draw a direction for the force at that point inside the water. On the left hand side of the diagram below there are water molecules and pressure, but no well defined direction of force. The right hand side of the diagram below shows the well defined directions of forces on an ice cream cone submerged in the water.

While we're on the topic, we might as well make it clear that the force exerted on a surface by fluid pressure is always directed inwards and perpendicular (at a right angle) to the surface.

If it helps, you can think about it this way. When the can of beans fell into the water, it quite rudely displaced a large amount of water molecules from the region where the can is now. This caused the entire water level to rise. But water is pulled down by gravity which makes it want to try and find the lowest level possible. So the water tries to force itself back into the region of volume that it was displaced from in an effort to try and lower the overall height of the body of water. So, whether a can of beans (or any other object) is in the water or not, the water molecules are always being squashed into each other from the force of gravity as they try to lower the water level to the lowest point possible. The pressurein the formulais a scalar that tells you the amount of this squashing force per unit area in a fluid.

Many people think air has no mass and no weight, but that's not true. The narrow column of air with the same radius as a typical can of beans that stretches from sea level to the top of the atmosphere has a mass of around

30 kg

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(that's like the weight of 30 pineapples). The force from atmospheric pressure on the top of a chessboard would be comparable to the weight of a car.

You might wonder how we can pick up the chessboard so easily if the weight of a car is pushing down on it, but it's because the weight of a car is also pushing up on it. Remember that the force from fluid pressure does not just push down, it pushes inwards perpendicular to the surface from every direction. It may not seem like there is any air under the chessboard when placed on the table but the roughness and cracks of the chess board are enough to allow air underneath. If you could get rid of all the air underneath the chessboard and prevent air from being allowed to sneak back in, that board would be stuck to the table like a suction cup. In fact, that's how suction cups work. They push the air out to create less pressure inside than out. The smooth plastic of the suction cup prevents air from sneaking back in. The higher pressure outside air pushes the suction cup into the surface. (see the diagram below)

Once air sneaks back in, the inside pressure becomes the same as the outside pressure and the cup can easily be taken off the surface.

At this point, if you've been paying close attention you might wonder "Hey, there's air above the water right? Shouldn't the weight of the column of air above the column of water also contribute to the total pressure at the top of the can of beans?" And you would be correct. The air above the column of water is also pushing down and its weight is surprisingly large.

If you wanted a formula for the total pressure (also called absolute pressure) at the top of the can of beans you would have to add the pressure from the Earth's atmosphere

Patm

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to the pressure from the liquid

ρgh

&#;

 

.

Ptotal=ρgh+Patm

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ρairgh

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for the atmospheric pressure

Patm

&#;

 

since our depth in the Earth's atmosphere is pretty much constant for any measurements made near land.

A problem with trying to use

ρairgh

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to find the pressure at a certain depth in the atmosphere is that unlike the water example, the density of the air in the atmosphere is not the same at all altitudes. As you go higher in the atmosphere the density of air decreases so we can't treat

ρair

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as a constant.

We typically don't try to derive a fancy term likefor the atmospheric pressuresince our depth in the Earth's atmosphere is pretty much constant for any measurements made near land.

This means that the atmospheric pressure at the surface of the Earth stays relatively constant. The value of the atmospheric pressure at the surface of the Earth is stuck right around

1.01×105Pa

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. There are small fluctuations around this number caused by variations in weather patterns, humidity, altitude, etc., but for the most part when doing physics calculations we just assume that this number is a constant and stays fixed. This means, as long as the fluid you're finding the pressure for is near the surface of the Earth and exposed to the atmosphere (not in some sort of vacuum chamber) you can find the total pressure (also called absolute pressure) with this formula.

Ptotal=ρgh+1.01×105Pa

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The

ρgh

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corresponds to the pressure created by the weight of a liquid, and the

1.01×105 Pa

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corresponds to the pressure of the Earth's atmosphere near sea level.

Pressure

Everyone&#;s been under pressure at one time or another, or in certain circumstances have really &#;felt the pressure.&#; From a scientific perspective, however, pressure has a very specific definition, and its exploration leads to some very important applications.

In physics, pressure is the effect of a force acting upon a surface. Mathematically, it is a scalar quantity calculated as the force applied per unit area, where the force applied is always perpendicular to the surface. The SI unit of pressure, a Pascal (Pa), is equivalent to a N/m2.

All states of matter can exert pressure. When you walk across an ice-covered lake, you are applying a pressure to the ice equal to the force of gravity on your body (your weight) divided by the area over which you&#;re contacting the ice. This is why it is important to spread your weight out when traversing fragile surfaces. Your odds of breaking through the ice go up tremendously if you walk across the ice in high heels, as the small area contacting the ice leads to a high pressure. This is also the reason snow shoes have such a large area. They are designed to reduce the pressure applied to the top crust of snow so that you can walk more easily without sinking into snow drifts.

Fluids, also, can exert pressure. All fluids exert outward pressure in all directions on the sides of any container holding the fluid. Even the Earth&#;s atmosphere exerts pressure, which you are experiencing right now. The pressures inside and outside your body are so well balanced, however, that you rarely notice the 101,325 Pascals due to the atmosphere (approximately 10N/cm2). If you ride in an airplane and change altitude (and therefore pressure) quickly, you may have experienced a &#;popping&#; sensation in your ears &#; this is due to the pressure inside your ear balancing the pressure outside your ear in a transfer of air through small tubes that connect your inner ear to your throat.

 

Question: Air pressure is approximately 100,000 Pascals. What force is exerted on this book when it is sitting flat on a desk? The area of the book&#;s cover is 0.035 m2.

Answer:

 

 

Question: A fisherman with a mass of 75kg falls asleep on his four-legged chair of mass 5 kg. If each leg of the chair has a surface area of 2.5×10-4 m2 in contact with the ground, what is the average pressure exerted by the fisherman and chair on the ground?

Answer: The force applied is the force of gravity, therefore we can write:

 

 

Question: A scale which reads 0 in the vacuum of space is placed on the surface of planet Physica. On the planet&#;s surface, the scale indicates a force of 10,000 Newtons. Calculate the surface area of the scale, given that atmospheric pressure on the surface of Physica is 80,000 Pascals.

Answer:

 

 

Question: Rank the following from highest pressure to lowest pressure upon the ground:

• The atmosphere at sea level
• A -kg elephant with total area 0.5 m2 in contact with the ground
• A 65-kg lady in high heels with total area 0.005 m2 in contacting with the ground
• A -kg car with a total tire contact area of 0.2 m2

Answer: From highest pressure to lowest pressure:

• The elephant (137,000 Pa)
• The lady in high heels (127,000 Pa)
• The atmosphere (100,000 Pa)
• The car (78,400 Pa)

 

The pressure that a fluid exerts on an object submerged in that fluid can be calculated almost as simply. If the object is submersed to a depth (h), the pressure is found by multiplying the density of the fluid by the depth submerged, all multiplied by the acceleration due to gravity.

This is known as the gauge pressure, because this is the reading you would observe on a pressure gauge. If there is also atmosphere above the fluid, such as the situation here on earth, you can determine the absolute pressure, or total pressure, by adding in the atmospheric pressure (P0), which is equal to approximately 100,000 Pascals.

 

Question: Samantha spots buried treasure while scuba diving on her Caribbean vacation. If she must descend to a depth of 40 meters to examine the pressure, what gauge pressure will she read on her scuba equipment? The density of sea water is kg/m3.

Answer:

 

 

Question: What is the absolute pressure exerted on the diver in the previous problem by the water and atmosphere?

Answer:

 

 

Question: A diver&#;s pressure gauge reads 250,000 Pascals in fresh water (ρ= kg/m3). How deep is the diver?

Answer:

 

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