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Transcript
PPT
Depth and Pressure
Developer Notes
 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.
Version
01
02
Date
2003/06/25
2003/08/14
Who
SC
DK
03
2004/06/09
dk
Revisions
Initial version
 Rewritten - added the POEs, prediction, reading,
challenge
 Added new material
 Reformatted
Goals
1) Students will know that pressure increases with depth.
2) Students will know and be able to use P = Dgh.
Concepts & Skills Introduced
Area
physics
physics
physics
physiology
Concept
In fluids, pressure increases with depth
Ph
P = Dgh
Blood pressure
Standards Addressed
Time Required
Warm-up Question
 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 POEs, discussion, reading,
and exercises should help the students understand the concept of increasing pressure with depth
in a fluid. The first thing is to get a qualitative feel for it. This POE should help.
POE - Multi-hole cup
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Depth and Pressure
Setup – Punch two identical holes in the side of an
aluminum drink can or plastic cup. 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.
P – Show the container to the students. Tell them you will
fill the container with water and let it drain out through the
holes. 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.
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.
POE: Cartesian Diver
Set up – Partially fill a glass eyedropper with a rubber bulb with water. Put it in a 2 L soft drink
bottle full of water, with the cap on. Squeeze the bottle and you can make the eyedropper float or
sink.
P - What will happen when I squeeze the bottle?
O - When you squeeze the bottle, the eyedropper sinks. When you release the bottle, it rises.
E – [When you squeeze the bottle, the pressure inside increases (because V decreases while n and
T remain constant). By Pascal’s Principle, it is transmitted through the water. The increased
pressure compresses the air in the diver so that its density increases until it is greater than the
water, and it sinks.]
POE – Sinking Cartesian Diver
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?
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Depth and Pressure
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.
POE – Rising Cartesian Diver
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?
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. Here’s a derivation that 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
[units: (kg/m3) (m) (m/s2) = (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
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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.
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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Depth and Pressure
Background
Pressure increases with depth. But in what pattern?
Problem
Find out how pressure increases with depth.
Materials
1
tall container filled with water
1
depth gauge (made with a thistle tube or funnel)
Procedure
1) Use the lab worksheet.
2) Identify the controls in the experiment.
3) Identify the independent variable.
4) Identify the dependent variable.
5) Insert the depth gauge into the container of water so that the large part of the gauge is
completely under water. Take a reading of the depth gauge and the depth of the water. Lower
the gauge and take at least 5 readings. Make a table of your data.
Summary
1) Are depth and pressure directly or inversely related?
2) Graph your data.
3) Are depth and pressure linearly or exponentially related?
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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 self-leveling.
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
 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?
 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?
 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).
 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?
 In a weightless condition, does pressure in a fluid change with depth? Explain.
 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.
 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?
 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
1. 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
 Pressure - 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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