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Physics 213

Waves, Fluids and Thermal Physics

Summer 2007Lecturer: Mike Kagan (mak411@psu.edu, 322 Whitmore)

Today’s Discussion:Today’s Discussion:

Fluids� Pressure and Pascal’s principle

� Bouyancy, Archimedes principle

� Bernoulli’s equation

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Fluids. Fluids. Description.Description.

So far we have only considered motion of So far we have only considered motion of point particles.point particles.

Fluid = Fluid = too manytoo many particles (particles (e.g. ~10e.g. ~101919 molecules in 1cmmolecules in 1cm33 of airof air))

Need Need newnew collectivecollective description, new physical quantitiesdescription, new physical quantities

But! We shall use the But! We shall use the samesame physical laws:physical laws:

NewtonNewton’’s Laws, Conservation Laws etc.s Laws, Conservation Laws etc.

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Fluids. Fluids. Parameters.Parameters.

DensityDensity

(for homogeneous fluid)(for homogeneous fluid) 10-17Lab vacuum

1.21Air

1019Black hole (solar mass)

1018Neutron star

1017Uranium nucleus

21400Platinum

19600Gold

13600Mercury

1000Water

900Ice

10-20Space

Densities (kg/m3)

Sample problem:Sample problem:

What is the mass of the air

in this room?

Compare with the mass

of people in this room.

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Fluids. Fluids. Parameters.Parameters.

PressurePressure

�Different physical mechanism

(collisions of molecules – will learn in 2 weeks)

�Stress: something pooling/stretching

Pressure: something pushing out

(in other words: stress = -pressure)

- similar to stress, BUT!

Pressure is a scalar (F is the force magnitude)

Vector force perpendicular to given plane,

NO shear stress in (non-viscous) fluids!

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Fluids. Fluids. Parameters.Parameters.

DensityDensity

(for homogeneous fluid)(for homogeneous fluid) 10-17Lab vacuum

1.21Air

1019Black hole (solar mass)

1018Neutron star

1017Uranium nucleus

21400Platinum

19600Gold

13600Mercury

1000Water

900Ice

10-20Space

Densities (kg/m3)

Sample problem:Sample problem:

What is the mass of the air

in this room?

Compare it with the mass

of people in this room.

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Pressure. Pressure. Units.Units.

SI: Pascal, 1 Pa = 1 N/m2

Torr: named after Evangelista Torricelli (Galileo’s apprentice, first to measure the atmospheric pressure)

Equal to the millimeter of mercury, or mmHg

American: pounds/in2 or psi

Conversions:

1 atm = 1.01 x 105 Pa = 760 Torr = 14.7psi (where 1 atm is the average pressure at sea level due to the large fluid

mass of the atmosphere above and around us)

Sample problem:You inflate the front tires on your car to 28 psi.

Later you measure your blood pressure, obtaining a reading of 120/80,

the reading s being in mm Hg. In kilopascals, what are

(a) your tire pressure and

(b) Your blood pressure?

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Pressure. Pressure. Observed phenomena.Observed phenomena.

Fluids exert pressure on their surroundings.

Need to explain:

1) Pressure increases under water

- your years feel this effect

2) Pressure decreases at high altitudes

- harder to breathe on mountaintops

- ears “popping” in airplanes

3) 4) Pascal vases (demo)

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1) Isolated fluid (no external

forces)

Pressure. Pressure. Pascal’s Law.Pascal’s Law.

2) Fluid in gravitational fieldpp=const=const

h

A

Free body diagram

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Pascal’s Law.Pascal’s Law. Sample problem.Sample problem.

Q: In which container is pressure highest at depth h?

A: None. They are all the same.

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Pascal’s Law.Pascal’s Law. Applications.Applications.

Hydraulic press

Gain in force!

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Archimedes principle.Archimedes principle. BouyancyBouyancy..

h

A

"any body partially or completely submerged in a fluid"any body partially or completely submerged in a fluid

is buoyed up by a force equal to the weight of the fluid is buoyed up by a force equal to the weight of the fluid

displaced by the body."displaced by the body."

Block ( ) in water ( )

Indeed, the bouyant force (see previous slide):

Free body diagram

floats

sinks

BouyantBouyant force = weight of liquid within volume displaced by bodyforce = weight of liquid within volume displaced by body

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Archimedes principle.Archimedes principle. Sample problems.Sample problems.

1.1. Block of ice is floating in water. Block of ice is floating in water.

What fraction of the block is submerged?What fraction of the block is submerged?

2. Block of ice is floating in water covered 2. Block of ice is floating in water covered

with a thick layer of oil (density 700 kg/mwith a thick layer of oil (density 700 kg/m33). ).

What fraction of the block is submerged into What fraction of the block is submerged into

water? (Water and oil do not mix.)water? (Water and oil do not mix.)

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Conservation laws.Conservation laws. Mass.Mass.Cross section changing

in the middle:

AAll that flows in flows outll that flows in flows out(f(fluid incompressible luid incompressible –– density constdensity const))

Incoming volumeIncoming volume

Outgoing volumeOutgoing volume

=R=R

raterate

Continuity equationContinuity equation

“ the narrower “ the narrower –– the faster”the faster”

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Continuity equation.Continuity equation. Sample problem.Sample problem.

What is the volume flow rate from the faucet?What is the volume flow rate from the faucet?

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Conservation laws.Conservation laws. Energy.Energy.

Bernoulli equationBernoulli equation

oror

work work

per unit per unit

volumevolume

kinetic energy kinetic energy

per unit volumeper unit volume

potential energy potential energy

per unit volumeper unit volume

For constant “y”, faster flowing fluids have

lower pressure than slower flowing fluids

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Bernoulli equation. Bernoulli equation. Example.Example.

Spinning:

Curve Ball

Figure taken from

Georgia State University

Physics Department website

Not spinning:

Not a Curve Ball

Top view

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Bernoulli equation. Bernoulli equation. Applications.Applications.

Lift force on airplane wing

Curves represent velocity field lines:

(analogously to electric field lines)

denser lines imply greater speed,

hence regions of lower pressure

Velocity of air above wing

greater than that under wing

Pressure difference exerts

Lift Force on wing

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What we learned:What we learned:

� Pressure & Pascal’s Lawfluid pressure at all points in a connected body of an incompressible fluid at rest,

which are at the same absolute height, are the same

� Archimedes’ principle & buoyancyBouyantBouyant force = weight of liquid within volume displaced by bodyforce = weight of liquid within volume displaced by body

� Equation of continuity (the narrower the narrower –– the fasterthe faster)

A1v1=A2v2

� Bernoulli’s equation (faster flowing fluids have lower pressure)

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Next TimeNext Time

�� SoundSound

�� Beats, , and shock wavesBeats, , and shock waves

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Creative problemsCreative problems (next Monday’s recitation).

1) Find the density of a cork, using a hard (massless) wire and a

graduated jar.

2) Explain how the sprinkler works (Fig 1).

3) A tank is filled with water to a height H. A hole is punched in

one of the walls at a depth h below the water surface (see

figure below). What value of h would maximize the distance

x? Check your prediction using a plastic bottle and water.

Fig 1 Fig 2