# Fluids wrap-up

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24 Οκτ 2013 (πριν από 4 χρόνια και 6 μήνες)

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680

AP Physics

Fluids Wrap Up

Here are the equations that you get to play around with:

This is the equation for the pressure of something as a function of depth in a fluid. You
would use it to figure out the pressure acting at

a depth of 25.0 m in a lake for example.

This is the equation for the buoyant force. It is also an equation that calculates the weight
of an object as a function of its density, volume, and the acceleration of gravity.

This equation represents flow rate, which is the cross sectional area,
A
, multiplied by the
velocity of the fluid,
v
. This is set up for two locations in a flow system. The flow rate for
a fluid that is incompressible must stay c
onstant, so this equation allows you to calculate the
linear speed of the fluid as a function of the cross sectional area of the system.

This is a Bernoulli’s equation. This allows you to calculate pressure, linear speed, &tc
. For
a system at different places within the system.

Here is the stuff you need to be able to do.

A.

Fluid Mechanics

1.

Hydrostatic pressure

a)

You should understand that a fluid exerts pressure in all directions.

This is basic. For example, atmospheric press
ure goes in all directions about an object

under it, over it, on the sides, &tc. Good old Pascal’s Principle.

b)

You should understand that a fluid at rest exerts pressure perpendicular to any surface that it
contacts.

This is also an application of Pasc
al’s principle. The pressure is everywhere throughout
the liquid. The direction of the force acting on a surface is always perpendicular to the
surface.

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c)

You should understand and be able to use the relationship between pressure and depth in a
liquid,

.

The actual equation that is provided you is

where
would be some
initial pressure. We did a bunch of these problems. Guage pressure is based on the idea that
the atmo
spheric pressure is zero pressure. Absolute pressure uses a perfect vacuum

zero
pascals

as its zero pressure. So guage pressure differs from absolute pressure by one
atmosphere. The pressure at a certain depth would be give by
. For an
absolute pressure you would set
equal to the atmospheric pressure. For a gage pressure
you would drop the

term.

2.

Buoyancy

a)

You should understand that the difference in the pressure on
the upper and lower surfaces of
an object immersed in a liquid results in an upward force on the object.

We went through this when the Physics Kahuna derived the buoyancy equation for you.
Because the pressure depends on depth, the pressure increases wit
h the depth. So if the
top of a regular object is 10 m below the surface and the bottom of it is 15 m below

five
meters deeper, the force, which is pressure times area, must be greater. Thus there is a
larger force pushing up on the bottom of the body t
han the pressure pushing down on the
top of the body. The net force is upward and is given the name of ‘buoyant’ force.

b)

You should understand and be able to apply Archimedes’ principle; the buoyant force on a
submersed object is equal to the weight of th
e liquid it displaces.

Well, the statement gives you Archimedes’ principle and tells you to understand it. So do
that.

3.

Fluid flow continuity

a)

You should understand that for laminar flow, the flow rate of a liquid through its cross section is
the same at

any point along its path.

So okay, do that too.

b)

You understand and be able to apply the equation of continuity,
.

Actually the equation that you are given is:

the density part isn’t in the
equatio
n. This is because in the type of problem that you’ll be doing, the density won’t change
and will remain constant. Because of that, it cancels out of the equation. The Physics Kahuna
is not at all sure why statement b) above had a different form of the eq
uation. Probably some
miscommunication at the College Board.

Anyway, we did a bunch of problems where you
got to use the equation. It is all pie.

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4.

Bernoulli’s equation

a)

You should understand that the pressure of a flowing liquid is low where the velo
city is high,
and vice versa.

Simple principle, simple stuff. Hey you can do it!

b)

You should understand and be able to apply Bernoulli’s equation,

The Physics Kahuna is not sure what sort of questions you can expect. He prov
ided you
with several of them, but is only guessing. So it goes.

Well. That’s it for what you need to know.

In Flanders Fields

In Flanders field the poppies grow

Between the crosses, row on row,

That mark our place; and in the sky

The larks, still bravely singing fly

Scarce heard amid the g
uns below.

We are the Dead. Short days ago

We lived, felt dawn, saw sunset glow,

Loved and were loved, and now we lie

In Flanders fields.

Take up our quarrel with the foe;

To you from failing hands we throw

The torch; be yours to hold it high.

If ye bre
ak faith with us who die

We shall not sleep, though poppies grow

In Flanders fields.

---

LtCol John McCrae

John McCrae, a noted poet, was a member of the first Canadian
Contingent and served on the front lines for four years in France during the
Great W
ar (as it was called until WW II). He was killed in combat on 28
January, 1918.

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From 2002:

In the laboratory, your are given a cylindrical beaker containing a fluid and you are asked to determine
the density

of the fluid. You are to use a spring of negligible mass and unknown spring constant
k

attached to a stand. An irregularly shaped object of known mass
m

and density
D

(
D

>>
)
hangs from the spring. You
may also choose from among the following items to complete the task.

A metric ruler

A stopwatch

String

(a)

Explain how you could experimentally determine the spring constant
k
.

The mass of the weight is known, suspend the mass from the spring in air, meas
ure the
displacement of the spring and calculate k from the equation

where F
s

is the
mg, the weight of the thing.

(b)

The spring
-
object system is now arranged so that the object (but not the spring) is immersed in the
unknown flu
id, as shown above. Describe any changes that are observed in the spring
-
object
system and explain why they occur.

The mass will have less weight in the fluid because of the buoyant force. It will decrease
by the amount of the force which is

(c)

Explain how you could experimentally determine the density of the fluid.

Knowing k we can calculate the apparent weight of the object in the fluid. The
difference between its weight in the air and in the fluid will equal the buoyant fo
rce.

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The volume of the object could be calculated using the equation for density:

The volume of fluid displaced will be the same as the volume of the object.

Knowing the buoyant force, we can use the buoyant force to ca
lculate the density of the
fluid.

(d)

Show explicitly, using equations, how you will use your measurements to calculate the fluid density
. Start by identifying any symbols you use in your equations.

Symbol

Physical quantity

This is the force that stretches the spring. In air, it will be the

weight of the object.

The spring constant

The spring displace
ment

The weight of the object

The mass of the object

g

The acceleration of gravity

The buoyant force

The density of the fluid

Volume of fluid displaced by the object

Acceleration of gravity

Measure the displacement of the spring by the object in air. Calculate the weight of the object
using
. Using
this weight, calculate the spring constant from
.
Calculate
the volume of the object using the equation for density. This will be the same as the volume of
the fluid displaced.

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Note the spring displacement. From this calcul
ate the weight of the object in the fluid using the
buoyant force equation. The difference in the two forces is the buoyant force. Using the volume
displaced, the buoyant force, and the acceleration of gravity, calculate the density of the fluid.
(Using

the buoyant force equation.)

So there it is. Your’re all set fluid mechanicswise.

FLUID CONCEPTS

Matter is made up of atoms. Every atom of an element is alike.

We can model atoms as tiny, hard “billiard balls.”

In a substance, atoms are combine
d to form molecules.

When atoms or molecules collide, they collide elastically (no loss of kinetic energy).

There are three forms of matter that we encounter in this class: solids, liquids, and gases. In all cases,
the atoms or molecules are in consta
nt motion in a material.

In a solid, the average position of the atoms or molecules remains constant. That is why solids tend to
maintain their shape, in general.

In a liquid, the atoms or molecules are free to move around, but the distance between at
oms or
molecules remains relatively small. For this reason, a liquid takes the shape of its container, but does
not expand to fill it.

The molecules of a liquid feel a mutual attraction. This creates surface tension, which acts something
like a “skin”
on the surface of the liquid.

In a gas, the molecules are widely separated and feel little mutual attraction. This is why a gas tends to
expand to take the shape of its container.

We will treat liquids as incompressible. It is true that the compressibil
ity for liquids in the situations we
consider is negligible.

Pressure is force per unit measure of area. The SI unit of pressure is the Pascal (Pa) or N/m
2
.

Since a fluid is not rigid, it can only exert a force perpendicular to a surface. Picture the par
ticles as
little balls colliding with the surface, as in when you bounce a ball off a wall. Can you see that the
direction of the velocity change is always perpendicular to the wall? (Try drawing it).

A fluid has weight. This is the reason why it exerts

a pressure.

The pressure of a fluid is proportional to the density of the fluid, the gravitation constant (for earth,
average g=9.8 N/kg), and the depth of the fluid. Thus the pressure exerted by a fluid only varies with
the depth, not with the volume,
the shape of the container, or any other factors.

The atmosphere exerts considerable pressure on objects at the surface of the earth.

Many simple devices depend on atmospheric pressure: drinking straws, suction cups, mercury
barometers. Be sure you can e
xplain how these and others work using the concepts in this section.

Gauges measure atmospheric pressure as 0 Pa. Atmospheric pressure must be added to gauge
pressure to give the actual pressure (called “absolute pressure”).

Since a liquid is incompressi
ble, a pressure exerted on it is transmitted throughout the liquid (Pascal’s
Principle).

An object floats b/c it displaces a volume of water whose weight equals the object’s weight
(Archimedes’ Principle). The sum of the vertical forces is zero. Altern
ate version: an object floats b/c
its average density is less than the surrounding fluid. Second alternate version: an object floats b/c the

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pressure at the bottom of the object is higher than the pressure at the top (the resulting force must be
equal to
the weight of the object).

We will treat only fluids that flow in thin sheets, i.e. laminar flow. The thin sheets can be modelled in
two dimensions as streamlines.

Fluids flow from high pressure to low pressure.

Because a fluid is incompressible, the vo
lume flow rate is constant, even if the diameter of the pipe
changes (Continuity Principle). The product of Area and velocity of a fluid is a constant for a particular
pipe. This means the fluid speeds up in a constriction and slows down in a wider part
of the pipe.

Bernoulli made a conservation of energy statement for a fluid in laminar flow: the energy per volume is
constant in a pipe (Bernoulli’s Principle). This is true even if the diameter of the pipe and therefore the
velocity change.

Fast
-
moving f
luids are at lower pressure than slow
-
moving fluids (Venturi Effect). This is a
consequence of Bernoulli’s Principle (i.e., conservation of energy).

Airplanes fly b/c of Bernoulli’s Principle and the momentum change of the air (due to the “attack angle”

of the wing). Can you explain this?

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"On Les Aura!"
--

Soldat Jacques Bonhomme loquitur:

See you that stretch of shell
-
torn mud spotted with pools of mire,

Crossed by a burst abandoned trench and tortured s
trands of wire,

Where splintered pickets reel and sag and leprous trench
-
rats play,

That scour the Devil's hunting
-
ground to seek their carrion prey?

That is the field my father loved, the field that once was mine,

The land I nursed for my child's chil
d as my fathers did long syne.

See there a mound of powdered stones, all flattened, smashed, and torn,

Gone black with damp and green with slime?
--

Ere you and I were born

My father's father built a house, a little house and bare,

And there I brought

my woman home
--

that heap of rubble there!

The soil of France! Fat fields and green that bred my blood and bone!

Each wound that scars my bosom's pride burns deeper than my own.

But yet there is one thing to say
--

one thing that pays for all,

Whate
ver lot our bodies know, whatever fate befall,

We hold the line! We hold it still! My fields are No Man's Land,

But the good God is debonair and holds us by the hand.

On les aura!" See there! and there! soaked heaps of huddled grey!

My fields shall laug
h
--

enriched by those who sought them for a prey.

James H. Knight
-