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Ph
PPT
Developer Notes
Version
Date
01
2003/06/25
02
2003/08/14
Who
SC
DK
What
Initial version
Rewritten - added the POEs, prediction, reading,
challenge
1. DK - Stacey needs to answer exercise 8.
2. DK - we need an activity here besides the POEs. Or do we just skip it? It would
be nice if they could play with changing pressure due to depth a bit.
Goals
1. Understand pressure increases with depth within a fluid
2. Discuss blood pressure
Concepts & Skills Introduced
Area
Physics
Physiology
Concept
In fluids, pressure increases with depth
Ph
Blood pressure
Time Required
Warm-up Questions
Why do healthcare professionals take your blood pressure at your upper arm (rather than
your feet)?
[Because your upper arm is at the level of your heart, so represents the pressure at your
heart. Lower in your body, the pressure increases.]
Aside from drowning, why is SCUBA diving considered such a hazardous activity?
[Because of the increased pressure, but more importantly, the changes in pressure.]
Presentation
We're looking at the change of pressure with depth in a liquid. The first thing is to get a
qualitative feel for it. This POE should help.
POE
P - Multi-hole cup. Punch two identical holes in the side
of an aluminum drink can. Punch the holes with an ice
pick or smallish nail. Punch one hole very near the
bottom, and one hole halfway up. Offset the holes a
sideways a little so the streams of water don't hit. Tell the
students that you will fill the container with water. Have
them predict whether the water coming out of the holes
will go the same distance, if the top one will go farther, or
if the bottom one will go farther.
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O - Fill the container with water and let the water run out. The water from the lower hole
will squirt farther.
E - Why do the streams of water have different paths, and why does the lower one go
farther? Pressure increases with depth. We know this from swimming and diving. Greater
depth results in greater pressure. The holes have the same area but different pressures.
Since P=F/A., the difference must be the force. A greater force leads to greater
acceleration, F=ma , and greater acceleration leads to greater velocity. We're back at
projectiles - the water leaves the can with a greater velocity from the lower hole; they
both accelerate down due to gravity.
Here's another POE that is similar to the cartesian diver that you may have seen. (An
eyedropper or ketchup packet in a 2 L soft drink bottle full of water, with the cap on.
Squeeze the bottle and you can make the object float or sink.)
Setup - Get a tall (0.3 m or more), clear container of water, like a 2 L soft drink bottle (it
will help to cut the top off where it starts to narrow) or a graduated cylinder. Get a glass
eyedropper with a rubber bulb. Fill it with water to the point that it just barely floats. Play
with the eyedropper so that when you push it near the bottom, it sinks, and when you pull
it part way up, it rises. A wire coat hanger with a little hook in the end works well here.
Start the POE with the dropper floating.
P - Let the students see the dropper floating in the cylinder. Have the students predict
what will happen if you push the dropper near the bottom. Will it sink or float?
O - Push it down so that it sinks to the bottom and stays there. Then pull it partway up
again so that it floats.
E - Pressure in a liquid increases with depth. As the dropper goes farther down, the
pressure increases so that the air in the dropper compresses, thus increasing its density.
When its density becomes greater than water's, it sinks. And vice versa.
Continue the POE with another one. They should get this one correct after the previous
one, but it's good reinforcement.
Setup - The setup is almost the same as the previous one except you start with the
dropper at the bottom and then drain water off until it floats. It's handy to have a way to
drain the cylinder. You can punch a small hole near the bottom of the soft drink bottle to
drain it. A siphon works, but is not good lab practice. A valve at the bottom would be
very nice if you have the equipment. Worst case, you can pour the water off a little at a
time.
P - Let the students see the dropper at the bottom of the cylinder. As you remove water
from the cylinder, will the dropper stay at the bottom or rise?
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O - Remove water from the cylinder. The dropper will eventually rise.
E - Since the height of water above the dropper is less, the pressure is less, so the air in
the dropper expands, lowering its density so that it rises.
Pressure in a liquid of uniform density varies directly with depth. We can derive an
equation which shows this!
P = F/A
Pressure = Force/ Area
F = mg, substitute for F.
Force = mass  gravity
P = mg/A
D = m/V, so m = DV, substitute for m.
Density = mass/ Volume
P = DVg/A
V = Ah, substitute for V.
Volume = Area/ height
P = DAhg/A
The A's cancel, leaving.
P = Dhg
3
2
[units: (kg/m ) (m) (m/s ) = (kg) (m/s2)/ (m2) = N/m2 = Pa]
Look at the formula P = Dhg. If D is constant (because the fluid is all the same) and g is
constant (g doesn't change much over short distances), then pressure changes with
changing depth (height below surface).
P = Dgh
There is no hands-on activity here. The POEs, discussion, reading, and exercises should
help the students understand the concept of increasing pressure with depth in a fluid.
As a summary activity, do a quantitative prediction. This is a nice little activity because it
combines density, Pascal's Principle, and changing pressure with depth. Oil is less dense
than water and does not mix with it, so it floats on top.
Prediction
Setup - Get a glass U-tube
and a ring stand. (Or use a
How high
0.3 m length of clear
oil
will the
water
ht
flexible tubing with an ID
10 cm high
water be?
of about 1 cm, bent into a
U shape. You can support
the tube with masking
tape and a 4x4 block.).
before
after
Fill the tube part way with
water. Also have some
vegetable oil in a beaker. Tell the students that you will pour oil into one side of the Utube. The oil will float on the water because it is less dense, and oil and water don't mix.
(You can do a small demonstration in a beaker first.)
Prediction(s) -
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First, a knowledge question: What is the density of water?
[1 g/cm3 ]
Second, some information: The density of vegetable oil is about 0.8 gm/cm3.
Qualitative: After you pour the oil in, will the column of liquid on the water side be
higher or lower than the column of liquid on the oil side?
[Lower]
Quantitative: How high will the water rise if 10 cm of vegetable oil is poured into the
other side?
[The water will rise about 8 cm. You can measure the height by measuring both
sides from the bottom of the tube, then subtracting the height to the bottom of the
oil from each. The pressure of the water at any height in the U-tube is equal to the
pressure of the water on the other side at the same height. (The pressure increases
at a different rate through the oil.) So the column of oil has to be exerting the
same pressure at its bottom as the water column on the other side is. Since
P=Dgh.
The pressure on each side is the same, so
Doilghoil=Dwaterghwater g is the same on both sides and cancels out.
To get the height of the water, divide both sides by Dwater.
You're left with
hwater=(Doil/Dwater)(hoil)
hwater= (0.8 g/cm3 / 1.0 g/cm3)(10 cm) = 8 cm]
Assessment
Writing Prompts
Relevance
Answers to Exercises
1. The pipes will have the same water pressure at the bottom. P = Dgh. Pressure is
due to density, g, and the height of water, not its volume or shape.
2. Pressure increases with depth, so the blood pressure in your legs is greater than in
other parts of your body. The greater pressure can lead to swollen veins.
3. The force on the bottom of the tank is about 300,000 N.
F = ma. (1000 kg/m3  3 m  5 m  2 m)  9.8 m/s2 = 300,000 N.
The pressure on the bottom of the tank is about 20,000 Pa.
P = F/A. 300,000 N / 3 m  5 m = 20,000 Pa
4. The pressure of the mercury would be 13.5 times as much. P = Dgh. h and g are
the same, but mercury's density is 13.5 times greater, so its pressure is 13.5 times
greater.
5. No, in a weightless condition, pressure in a fluid does not change with depth.
Pressure requires a force against an area, but since everything is falling at the
same speed, there is no area of support force.
6. The bag needs to be 1.3 m above the arm.
h = P/Dg. 13,300 Pa / (1050 kg/m3  9.8 m/s2) = 1.3 m.
7. The difference in blood pressure between arteries in the heart and foot is about
13,300 Pa, just like the previous problem.
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13,300 Pa is equivalent to 100 mm Hg. (1 mm Hg/133 Pa)  13,300 Pa = 100 mm
Hg.
The total blood pressure in the foot artery is 200 mm Hg. 100 mm Hg + 100 mm
Hg.
8. There is no difference in the blood pressure between the heart and head because
there is no height difference when a person is lying down.
If the person is sitting up, the pressure difference is 31 mm Hg. (100 mm Hg/ 1.3
m)  0.4 m = 31 mm Hg.
The average pressure in the head is 69 mm Hg. 100 mm Hg - 31 mm Hg = 69 mm
Hg
If you sit up too fast, the pressure in your head will
Answers to Challenge/ Extension
1. The balloon will expand as it rises because the air pressure decreases. Unless it
reaches a level in the atmosphere where its density equals the air's density, it will
expand until it pops.
2. A floating object is pushed up by a force equal to its weight. If not, it would
accelerate up or down. The force comes from the liquid displaced, so a floating
object displaces its weight of liquid.
3. A sinking object displaces its volume of liquid. It is buoyed up by the weight of
the liquid displaced.
4. An object in a liquid is buoyed up by the weight of liquid it displaces.
5. The blood pressure in your ankle should be about 100 mm Hg higher than your
heart, about double.
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Background
Problem
Materials
Procedure
Summary
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Reading
We’ve seen what happens when pressure is applied to liquid in a closed container
(Pascal’s principle). But is that the only cause for change in pressure? What happens
when you’re swimming and you swim deeper and deeper? The pressure increases and
you have to equalize it by blowing. How about when you ride in an elevator or airplane?
The pressure changes and you have to "pop" your ears. Pressure in a liquid changes with
depth.
We’ve looked at how the heart applies pressure to the blood in order to move it around.
The term blood pressure refers to the pressure that blood exerts on the walls of the blood
vessels. Blood pressure can be an indicator of health. Do you think that your blood
pressure is the same throughout your body? Why or why not? If you think blood
pressure is different in different places in your body, where would it be highest?
Starting with the formula P=F/A, and using the formulas F=mg, and V=Ah, and D=m/V,
we can derive the formula
P=Dgh
Pressure = density of the fluid  g  the height of the fluid.
(Your teacher can show you the derivation, or as a challenge, you can work it out.)
Look at the formula P=Dgh. If the height (h) of a fluid increases, the pressure (P) will
also increase as long as D and g remain constant. Density (D) remains constant in a liquid
because liquids don't compress (much), and gravitational acceleration (g) is nearly
constant over short distances, so
P=Dgh
A change in height of a fluid means a change in pressure. In short, if D and g are constant
Ph
So, the deeper you go in a fluid, the higher the pressure. That's the reason fluids are selfleveling. Just like the multi-hole cup, the more pressure there is, the more force there is.
If the liquid is deeper on one side of a container, there will be more force generated on
that side than the other side - unbalanced forces mean acceleration, and the liquid will
move - "Water seeks its own level."
Blood in your body works the same way. Since the pressure in a fluid varies with depth,
blood in the lower body has a higher pressure than in the upper body. Blood pressure is
usually measured at the same level as the heart in order to measure the pressure of the
blood leaving the heart. Many other factors also influence blood pressure, including
friction encountered in blood vessels and dilation or constriction of arteries. Postural
effects on blood pressure result from the Ph relationship (e.g. feeling light-headed when
you stand up too quickly).
Another unit for pressure is mm Hg (millimeters of mercury). It refers to how far the
pressure would raise a column of mercury. This is the unit of pressure used to measure
blood pressure. 1 mm Hg = 133 Pa = 0.02 psi. (Mercury has a density of 13,500 kg/m3.
Since 1 mm = 0.001 m, 1 mm Hg is 13.5 kg/m2, and 13.5 kg/m2  9.8 m/s2 = 133 Pa.)
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You can see changing pressure with depth if you dive in the ocean. From 20 feet or so
down, release some bubbles and rise with them. If you watch carefully, you can see them
expand as the pressure decreases.
Exercises
1. Two water pipes, of the same diameter, go from the top of a building to the
bottom. One is zig-zag while the other is straight. When filled, which one will
have higher water pressure at the bottom?
2. People who have professions that require them to stand for long periods of time
often develop problems with swollen veins in their legs (called varicose veins).
Why?
3. What is the force on the bottom of a big fish tank that is 3 m wide and 5 m long,
and has a water depth of 2 m? (Density of water = 1000 kg/m3). What is the
pressure? (Ignore pressure of the atmosphere).
4. Mercury has a density that is 13.5 times greater than water. How does the
pressure 1 m below the surface of a sample of mercury compare to the pressure 1
m below the surface of water?
5. In a weightless condition, does pressure in a fluid change with depth? Explain.
6. How high above the arm does a bag of transfusion blood need to be in order to
enter the vein with a pressure of 13,300 Pa (100 mm Hg)? The density of blood is
1050 kg/m3.
7. Calculate the difference in blood pressure between an artery in a foot and in the
main artery leaving the heart (called the aorta). The foot is 1.3 m below the heart.
Convert your answer to mmHg. The average pressure in the aorta is 100 mmHg.
What is total blood pressure in the foot artery?
8. What is the difference in blood pressure between the heart and the head when a
person is lying flat? What is the difference in blood pressure when the person is
sitting up straight, if the head is 0.4 m above the heart? If the average pressure at
the heart is 100 mmHg, what is the pressure in the head (in mmHg)?
Normally your body makes adjustments to compensate for this difference in
pressure. Explain what happens if you sit up too fast.
Challenge/ Extension
One of the effects of increasing pressure with depth is buoyancy - things weigh less in a
fluid. Because pressure increases with depth in a fluid, the pressure on the bottom of an
object in a fluid is greater than the pressure on the top, so the object is pushed up by the
difference in forces. If its density is low enough it will rise. If its density is too great it
will sink, but it will still feel lighter, as a rock does in water. Air is a liquid, so you
actually weigh a little less in air than you would in a vacuum.
1. When a child loses a helium-filled rubber balloon and the balloon floats away into
the air, what do you think happens to the balloon as it rises? What eventually
happens?
2. How much force pushes up on a floating object? How much weight of liquid does
a floating object displace?
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3. How much weight of liquid does a sinking object displace? How much force
buoys it up?
4. In general, how much force buoys up an object in a liquid?
5. In a weightless condition, would a cork float in water? Explain.
6. Check the blood pressure in your arm, like normal. Predict the pressure in your
ankle and test it to see if you're right. You'll need blood pressure testing
equipment.
Glossary
1. Pressure in a liquid equals the density of the liquid times gravity times the depth.
In a liquid of uniform density, the pressure varies directly with depth. P = Dgh
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