AP Physics 2UNIT 1 – FLUID MECHANICS

Fluid Mechanics - Hydrostatics

States of MatterBefore we begin to understand the nature of a Fluid we must understand the nature of all the states of matter:

The 3 primary states of matter - solid - Definite shape and volume. - liquid -Takes the shape of its container, yet has a definite volume.

- gas - Takes the shape and volume of its container.

Special "states - Plasma, Bose-Einstein Condensate

DensityThe 3 primary states have a distinct density, which is defined as mass per unit of volume.

Density is represented by the Greek letter, “RHO”, r

What is a Fluid?By definition, a fluid is any material that is unable to withstand a static shear stress. Unlike an elastic solid which responds to a shear stress with a recoverable deformation, a fluid responds with an irrecoverable flow.

Examples of fluids include gases and liquids.

Why fluids are useful in physics?Typically, liquids are considered to be incompressible. That is once you place a liquid in a sealed container you can DO WORK on the FLUID as if it were an object. The PRESSURE you apply is transmitted throughout the liquid and over the entire length of the fluid itself.

PressureOne of most important applications of a fluid is it's pressure- defined as a Force per unit Area

ExampleA water bed is 2.0 m on a side an 30.0 cm deep.

(a) Find its weight if the density of water is 1000 kg/m3.

(b) Find the pressure the that the water bed exerts on the floor. Assume that the entire lower surface of the bed makes contact with the floor.

mgWV

m

V

m

Va

1000

30.022)

1.2 m3

1200 kg

11760 N

24

11760)

m

N

A

mg

A

FPb 2940 N/m2

Hydrostatic PressureSuppose a Fluid (such as a liquid) is at REST, we call this HYDROSTATIC PRESSURE

Two important points

• A fluid will exert a pressure in all directions

• A fluid will exert a pressure perpendicular to any surface it compacts

Notice that the arrows on TOP of the objects are smaller than at the BOTTOM. This is because pressure is greatly affected by the DEPTH of the object. Since the bottom of each object is deeper than the top the pressure is greater at the bottom.

Pressure vs. DepthSuppose we had an object submerged

in water with the top part touching the atmosphere. If we were to draw an FBD for this object we would have three forces

1. The weight of the object

2. The force of the atmosphere pressing down

3. The force of the water pressing up

mg

Fatm

Fwater

Fwater= Fatm + mg

Pressure vs. DepthBut recall, pressure is force per unit area. So if we solve for force we can insert our new equation in.

ghPP

AhgAPPA

AhV

VgAPPA

VmV

m

mgAPPA

mgFFA

FP

o

o

o

o

atmwater

Note: The initial pressure in this case is atmospheric pressure, which is a CONSTANT.

Po=1x105 N/m2

A closer look at Pressure vs. Depth

ghPP o

ABSOLUTE PRESSURE

Initial Pressure – May or MAY NOT be atmospheric pressure

Depth below surface

ghP Gauge Pressure = CHANGE in pressure or the DIFFERENCE in the initial and absolute pressure

Examplea) Calculate the absolute pressure at an ocean depth of 1000 m. Assume that the density of water is 1000 kg/m3 and that Po= 1.01 x 105 Pa (N/m2).

b) Calculate the total force exerted on the outside of a 30.0 cm diameter circular submarine window at this depth.

P

xP

ghPP o

)1000)(8.9)(1000(101 5

22 )30.0(F

r

F

A

FP

9.9x106 N/m2 2.80 x 106 N

A closed systemIf you take a liquid and place it in a system that is CLOSED like plumbing for example or a car’s brake line, the PRESSURE is the same everywhere.

Since this is true, if you apply a force at one part of the system the pressure is the same at the other end of the system. The force, on the other hand MAY or MAY NOT equal the initial force applied. It depends on the AREA.

You can take advantage of the fact that the pressure is the same in a closed system as it has MANY applications.

The idea behind this is called PASCAL’S PRINCIPLE

Pascal’s Principle

Another Example - BrakesIn the case of a car's brake pads, you have a small initial force applied by you on the brake pedal. This transfers via a brake line, which had a small cylindrical area. The brake fluid then enters a chamber with more AREA allowing a LARGE FORCE to be applied on the brake shoes, which in turn slow the car down.

shoepadbrake

shoepadbrake

pedalbrake

pedalbrake

A

F

A

F

PP

/

/

21

BuoyancyWhen an object is immersed in a fluid, such as a liquid, it is buoyed UPWARD by a force called the BUOYANT FORCE.

When the object is placed in fluid is DISPLACES a certain amount of fluid. If the object is completely submerged, the VOLUME of the OBJECT is EQUAL to the VOLUME of FLUID it displaces.

Archimedes' Principle" An object is buoyed up by a force equal to the weight of the fluid displaced."

In the figure, we see that the difference between the weight in AIR and the weight in WATER is 3 lbs. This is the buoyant force that acts upward to cancel out part of the force. If you were to weight the water displaced it also would weigh 3 lbs.

Archimedes' Principle

Fluidobject

FluidB

FLUIDB

VV

VgF

VmmgF

)(

)(

ExampleA bargain hunter purchases a "gold" crown at a flea market. After she gets home, she hangs it from a scale and finds its weight in air to be 7.84 N. She then weighs the crown while it is immersed in water (density of water is 1000 kg/m3) and now the scale reads 6.86 N. Is the crown made of pure gold if the density of gold is 19.3 x 103 kg/m3?

object

objectobject

object

object

fluid

fluidfluidFluidB

B

buoyantwaterobjectairobject

V

m

mass

V

V

gVmgF

F

FFF

)(

86.684.7

)()(

0.98 N

0.0001 m3

0.0001 m3

0.80 kg

8000 kg/m3

NO! This is NOT gold as 8000<19300

Fluid Dynamics

Fluid FlowUp till now, we have pretty much focused on fluids at rest. Now let's look at fluids in motion

It is important that you understand that an IDEAL FLUID:•Is non viscous (meaning there is NO internal friction)•Is incompressible (meaning its Density is constant)•Its motion is steady and NON – TURBULENT

A fluid's motion can be said to be STREAMLINE, or LAMINAR. The path itself is called the streamline. By Laminar, we mean that every particle moves exactly along the smooth path as every particle that follows it. If the fluid DOES NOT have Laminar Flow it has TURBULENT FLOW in which the paths are irregular and called EDDY CURRENTS.

Mass Flow Rate

A

A

v

v

L

L

Consider a pipe with a fluid moving within it.

The volume of the blue region is the AREA times the length.

Length is velocity times time

Density is mass per volume

Putting it all together you have MASS FLOW RATE.

What happens if the Area changes?

A1

A2

v1

v2

L1=v1t

L2=v2t

The first thing you MUST understand is that MASS is NOT CREATED OR DESTROYED!IT IS CONSERVED.

The MASS that flows into a region = The MASS that flows out of a region.

Using the Mass Flow rate equation and the idea that a certain mass of water is constant as it moves to a new pipe section:

We have the Fluid Flow Continuity equation

ExampleThe speed of blood in the aorta is 50 cm/s and this vessel has a radius of 1.0 cm. If the capillaries have a total cross sectional area of 3000 cm2, what is the speed of the blood in them?

2

22

22121

2211

)3000()50()1(

v

v

vAvr

vAvA

0.052 cm/s

Bernoulli's PrincipleThe Swiss Physicist Daniel Bernoulli, was interested in how the velocity changes as the fluid moves through a pipe of different area. He especially wanted to incorporate pressure into his idea as well. Conceptually, his principle is stated as: " If the velocity of a fluid increases, the pressure decreases and vice versa."

The velocity can be increased by pushing the air over or through a CONSTRICTION

A change in pressure results in a NET FORCE towards the low pressure region.

Bernoulli's Principle

Funnel

Ping pong Ball

Constriction

Bernoulli's PrincipleThe constriction in the Subclavian artery causes the blood in the region to speed up and thus produces low pressure. The blood moving UP the LVA is then pushed DOWN instead of down causing a lack of blood flow to the brain. This condition is called TIA (transient ischemic attack) or “Subclavian Steal Syndrome.

One end of a gopher hole is higher than the other causing a constriction and low pressure region. Thus the air is constantly sucked out of the higher hole by the wind. The air enters the lower hole providing a sort of air re-circulating system effect to prevent suffocation.

Bernoulli's EquationLet’s look at this principle mathematically.

Work is done by a section of water applying a force on a second section in front of it over a displacement. According to Newton’s 3rd law, the second section of water applies an equal and opposite force back on the first. Thus is does negative work as the water still moves FORWARD. Pressure*Area is substituted for Force.

X = L

F1 on 2

-F2 on 1

Bernoulli's Equation

A1

A2

v1

v2

L1=v1t

L2=v2t

y2

ground

Work is also done by GRAVITY as the water travels a vertical displacement UPWARD. As the water moves UP the force due to gravity is DOWN. So the work is NEGATIVE.

y1

Bernoulli's EquationNow let’s find the NET WORK done by gravity and the water acting on itself.

WHAT DOES THE NET WORK EQUAL TO? A CHANGE IN KINETICENERGY!

Bernoulli's Equation

Consider that Density = Mass per unit Volume AND that VOLUME isequal to AREA time LENGTH

Bernoulli's Equation

We can now cancel out the AREA and LENGTH

Leaving:

Bernoulli's Equation

Moving everything related to one side results in:

What this basically shows is that Conservation of Energy holds true within a fluid and that if you add the PRESSURE, the KINETIC ENERGY (in terms of density) and POTENTIAL ENERGY (in terms of density) you get the SAME VALUE anywhere along a streamline.

ExampleWater circulates throughout the house in a hot-water heating system. If the water is pumped at a speed of 0.50 m/s through a 4.0 cm diameter pipe in the basement under a pressure of 3.0 atm, what will be the flow speed and pressure in a 2.6 cm-diameter pipe on the second floor 5.0 m above?

2

222

2221

21

2111

)026.0(50.0)04.0(

v

v

vrvr

vAvA

1.183 m/s

1 atm = 1x105 Pa

P

Px

ghvPghvP ooo

)5)(8.9)(1000()183.1)(1000(2

1)0)(8.9)(1000()50.0)(1000(

2

1103

2

1

2

1

225

22

2.5x105 Pa(N/m2) or 2.5 atm