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Physics 1010:The Physics of Everyday Life

TODAY• Pressure, Boyancy

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Administrative Mattters• Mark is e-mailing grades to students not on the

clicker list; if you have not received your gradee-mail Mark (Eng-hiang.Yeo@colorado.edu)

• Clickers 137097, 275456, 169571, 222227,275422 still unclaimed. If you own one of theseclickers please e-mail Yu (Yu.Ye@colorado.edu)

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Today’s topics

• Pressure, density, and temperature• Ideal gas law• Archimedes’ principle & buoyancy force• Water pressure

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Gases

• Consist of molecules (about 10-9 m across) thatbounce around

• They do not fall to ground because they havekinetic energy [(1/2)mv2 ]

• More kinetic energy means more temperature• Bouncing off the wall causes it to exert a force

(concervation of momentum; Newton’s III)

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Pressure: bouncing off the wallsThe piston is free to

move. After the tinymolecule hits it, thepiston

a) shakesb) moves to the left at

constant velocityc) accelerates to the leftd) remains stationary

The piston moves to the left atconstant velocity, as it acquiredmomentum from the molecule(concervation of momentum).

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Pressure: bouncing off the wallsThe piston is free to move.

Molecule after moleculehits it, the piston

a) shakesb) moves to the left at

constant velocityc) accelerates to the leftd) remains stationary

The piston accelerates to the left aseach molecule gives it moremomentum, meaning…..THERE IS A NET FORCE ON THE PISTON

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Pressure: the force per unit area

• More area exposed to gas implies more collisions• More collisions implies more momentum transfer• More momentum transfer implies more force• Force = Pressure x Area• Pressure = Force/Area• Unit: 1 Pascal (Pa) = 1 N/m2

• Atmospheric pressure = 100,000 Pa = 105 Pa =14.7 lbs/sq. inch

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Pressure is in all directions

• Water comes outwhether cork is ontop, side, or bottom

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Mass density: mass per unit volume• Allows calculation of mass from volume• Mass = Density x Volume• Some common densities:

Water (humans and animals): 1000 kg/m3 (64 lbs/cubicfoot)

Lead: 11,400 kg/m3

Uranium: 19,000 kg/m3

Air: 1.239 kg/m3

• Squeezing air: more air in smaller volume, sodenser, more density

Mass density is denoted by ρ

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Anything can form a density• Particle density: # of particles / unit volume• People density: # of people /unit volume• What is the people density in this room? (room is

~20mx15mx6m, class is ~200 students)

People density = number of people/volume of room200 people/1,800 m3 = 0.11 people/ m3

a) 0.05 people/m3

b) 0.1 people/m3

c) 0.5 people/m3

d) 1 people/m3

Mass density is denoted by ρ, number density by n

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Pressure increases with density• Twice the molecules

means twice the collisions• Twice the collisions means

twice the momentumtransfer

• Twice the momentumtransfer means twice thepressure

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Pressure increases with average kineticenergy [(1/2)mv2]

• More speed means morecollisions with the wall

• More speed and mass, moremomentum that each collisiontransfers to wall

• Pressure proportional to v xmv ~ mv2

• Pressure proportional totemperature

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Ideal gas law

• Pressure proportional to density xtemperature (relative to absolute zero)Density: more dense means more molecules

hitting sides of wallTemperature: more temperature means

molecules hitting more rapidly, with moremomentum transfer

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Ideal gas law

p ∝ n Tabsolute pressure(not pressure relativeto atmosphere)

absolute temperature(use Kelvin, not F or C!)

molecule density(number of molecules per unit volume)

“is proportional to”

Constant of proportionality: “Boltzmann’s constant”k = 1.38 x 10-23 J/K = 1.38 x 10-23 N m/K = 1.38 x 10-23 Pa m3/ K

p = n k T

Amazingly, k is a “universal” constant:it is the same for any species of gas.

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Can you answer?• In higher temperature gases, the molecules move• A) faster or B) slower?• Does pressure push down as well as up?• A)True B) False• How does pressure relate to force?• A) F=mass x acceleration B) F=Pressure x Area• What is a Pascal?• What does mass density mean? Or any kind of density?• How does pressure change with mass density and

temperature?

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Properties of the earth’s atmosphere• Earth radius = 4,000 mi• Atmospheric thickness = 4 mi• Less than one pixel on screen

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Atmospheric pressure is due to weight of air• Force down on

separating membraneis weight of air above

• Manifestation ispressure

• Force up from airbelow is pressure tobalance

• Pressure is weight ofcolumn of air dividedby area

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Pressure under water increases morerapidly as you go down

• Pressure is the weight of waterabove one square meter

• Weight increases as you go down• For each meter down, adding 1

cubic meter, or 1000 kg = 10,000N

• Each 10 m then add 100,000 N orone atmosphere!

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Buoyancy: how ballons lift and ships float

• Archimedes principle: anobject in a fluid feels anupward force equal to theweight of the fluid itdisplaces

• Fluid falls down whereobject was

• Net weight = weight ofobject in vacuum minusweight of fluid havingvolume equal to that ofobject

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1m

1m1m

Empty Cube(with side 1 meter)Not even air inside

At sea level, atmospheric pressure is about 100,000 Pa. How muchforce does the atmosphere exert on the top side of an empty cube withside 1 meter? What mass, set on top of the cube, would exertapproximately the same force? Hint: 100,000 Pa = 100,000 N/m2

100,000 Pam=

atmospheric force on one side mass exerting equivalent forceA. 1,000 N 102 kgB. 10,000 N 102 kgC. 10,000 N 1020 kgD. 100,000 N 1020 kgE. 100,000 N 10200 kg

Answer: E. Force is pressure times area:100,000 Pa x 1 m2 = 100,000 N/m2 x 1 m2 = 100,000 NA mass of 10,200 kg (10 tons!) exerts mg = 100,000 N

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The area of your palm is 50 cm2 = 0.005 m2.If you hold out your hand, palm up, what downward forcedoes the atmosphere (at about 100,000 Pa) exert on thepalm of your hand?

A. 50 NB. 500 NC. 5,000 ND. 50,000 NE. 500,000 N

Answer: B. 100,000 Pa x 0.005 m2 = 100,000 N/m2 x 0.005 m2 = 500 NNote: a mass of 51 kg exerts about mg = 500 N = 110 pounds.

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The atmosphere exerts a force around 500 N on the palmof your hand. Where does this force come from?

The weight of air:

1m1m

100,000 Pa ~10,000 kg

= =

About 10,000 kgof air is ina 1 m2 columnabove the earth’ssurface

About 50 kgof air is ina 50 cm2 columnabove the earth’ssurface

Sev

eral

mile

s of

air

= ~1

0,00

0 kg

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Why don’t you feel 100 lbs of force on your palm?

A because you’re pushing back with 100 lbsB because it’s pushing on all sidesC because you’re used to itD because we’re strong

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Why don’t you feel 100 lbs of force on your palm?

your hand

Air pressure is the same all around your hand ⇒ no net force*

100 lbs downward

100 lbs upward

*Actually, there is a net force on your hand--buoyancy (thepressure below your hand is slightly greater than the pressureabove); however, the buoyant force (in air) is negligible for objectsmuch denser than air. Keep watching for more on buoyancy.

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The atmosphere exerts a force around 500 N on the palmof your hand, regardless of whether your palm is turnedupwards or downwards (or sideways).

Fluids exert the same pressure in every direction,because fluids flow when the pressure is different in different directions.

Cube of Solid with downward pressure:maintains shape without pressure onsides.

Cube of Fluid with downward pressure:flows out of shape unless an equal pressureis applied to other sides

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Water Pressure and GravityWater, unlike air, maintains nearly the same density regardless ofpressure (in-compressible): 1 g/cm3, or 1000 kg/m3

1m1m

1m

1m

1m

1m

A tall water trough on the surface of theEarth, 1 m x 1m x 4m.Each cubic meter of water weighs 10,000 N.What is the pressure at the bottom of thetrough, 4 meters below the surface of thewater (which is at atmospheric pressure)?A. 0B. 40,000 PaC. 100,000 PaD. 140,000 PaE. 400,000 Pa

Atmosphericpressure:100,000 Pa

Clicker Question:

10,000 N

10,000 N

10,000 N

10,000 N

exerts 100,000Non 1 m2

Answer: D. The force on the bottom squaremeter is 100,000 N + (4 x 10,000 N) = 140,000 N.The pressure is 140,000 N / 1 m2 = 140,000 Pa.

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Why doesn’t water sink?

1m1m

1m

1m

1m

1m

Atmosphericpressure:100,000 Pa

10,000 N

10,000 N

10,000 N

10,000 N

exerts 100,000Non 1 m2

pressure100,000 Pa

120,000 Pa

130,000 Pa

110,000 Pa

140,000 Pa

120,000 N

110,000 N

mg=10,000 N

Note: pressures from sides areleft out of drawing

No net force!Pressure differenceexactly balancesweight!

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Force of water on other stuff: buoyancy

1m1m

1m

1m

1m

1m

Atmosphericpressure:100,000 Pa

pressure100,000 Pa

120,000 Pa

130,000 Pa

110,000 Pa

140,000 Pa

mg=80,000 N

Remove one cubic meter of water;replace with one cubic meter of steel.Steel density: 8 g/cc = 8000 kg/m3

Fnet = 70,000 N downward--steel sinks in water

120,000 N

110,000 N

1m1m

displacedwater

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Force of water on other stuff: buoyancy

1m1m

1m

1m

1m

1m

Atmosphericpressure:100,000 Pa

pressure100,000 Pa

120,000 Pa

130,000 Pa

110,000 Pa

140,000 Pa

Remove one cubic meter of water;replace with one cubic meter of pine woodPine density: 0.5 g/cc = 500 kg/m3

1m1m

mg=

120,000 N

130,000 N

5,000 N

What is the net forceon the cube of wood?

Fnet = 5,000 N upwards--the wood floats

displacedwater

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Calculating buoyancy forces

• To calculate force of a fluid on an immersed object:1. Find volume V of object (or of the part of the object

below the surface of the fluid).2. Find the weight of a volume V of the fluid ( mg = ρVg,

where ρ is the mass density of the fluid).3. The buoyant force of the fluid on the immersed object

is equal and opposite to the weight of the displacedfluid.

• Why? The force of the fluid on the immersed objectwill be the same as the force of the fluid on the fluiddisplaced by the object; that force is exactly equal tothe weight of the displaced fluid (because when thefluid was where the object is, it was not accelerating).

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V is the volume of what?

• A the object• B the fluid desplaced by the object• C A and B

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Calculating buoyancy forces• To calculate force of a fluid on an immersed object:

1. Find volume V of object.2. Find the weight of the same volume V of the fluid ( mg =

ρVg, where ρ is the mass density of the fluid).3. The buoyant force of the fluid on the immersed object

is equal and opposite to the weight of the displacedfluid.

Clicker Question :A brick has a density of about 2000 kg/m3.What is the net force on a brick of volume 0.001 m3 immersed in water? (for convenience, use g~10m/s2)

A. 20 N upwardB. 10 N upwardC. 0D. 10 N downwardE. 20 N downward

Answer: D. The weight of the brick is 20 N(downward); the weight of 0.001 m3

displaced water is 10 N. The buoyant forceon the brick is therefore 10 N upward. Thenet force on the brick is 10 N downward.The brick sinks.

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Calculating buoyancy forces in air• To calculate force of a fluid on an immersed object:

1. Find volume V of object.2. Find the weight of the same volume V of the fluid ( mg =

ρVg, where ρ is the mass density of the fluid).3. The buoyant force of the fluid on the immersed object

is equal and opposite to the weight of the displacedfluid.

Clicker Question :A brick has a density of about 2000 kg/m3.Air (at room temp/sea-level pressure) has a density around 1 kg/m3.What is the buoyant force on a brick of volume 0.001 m3 immersed in air?(for convenience, use g~10m/s2)

A. 0B. 0.0001 N upwardC. 0.001 N upwardD. 0.01 N upwardE. 0.1 N upward

Answer: D. The mass of 0.001 m3 of displacedair is 0.001 kg; its weight is 0.01 N. Thebuoyant force on the brick is therefore 0.01 Nupward. The brick, which would weigh 20 N invacuum, appears to weigh only 19.99 N in air.

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Buoyancy depends on mass density ofobject relative to mass density of fluid

• If object is less dense than fluid it floatsbecause the buoyancy is greater than thegravitational force

• Examples:Wood in water: floatsOil in water: floatsLead in water: sinks

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Swimmers: you can float or sink dependingon the air in your lungs

• Humans are approximately neutrallybuoyant because they are made ofwater

• Jump in, at surface take in as much airas possible - float

• Now empty your lungs all the way - sink• Caveat: must not have clothing that

traps air bubbles

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For Stationary Water,Pressure Changes with Height

1m1m

Δh

weight = mg = ρV g = ρ (1 m2) Δh gThe pressure increase is theincreased weightdivided by the area:p2 = p1 + mg = p1 + ρg Δh

Water density: ρ = 1000 kg/m3

To support the weight of the above water, the water pressuremust increase by 10,000 Pa per meter depth(about 1/10 of an atmosphere per meter).

p1

p2

Gravity

p2 = p1 + ρg Δhor

p2 + ρgh2 = p1 + ρgh1

(increasing h meanslowering p)

or

p + ρgh = constant

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p + ρgh = constant

Important: the difference in pressures at two points within a staticwater column depends only on the differences in height of the twopoints. For example, water pressure increases by about 10,000 Pa(actually 9800 Pa) for every meter increase in depth (on the surface ofthe Earth), regardless of its container.Remember work done on body to move up a hight h.

Also important: this is true for water and most liquids; it is not true forair and other gases. Why?

The density ρ is not constant for gases.

For Stationary Water,Pressure Changes with Height

Kind of like “conservation of energy” for non-moving water:Pressure + gravitational potential energy = constant

Pressure is energy density (energy per unit volume)

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p + ρgh + (1/2)ρv2 = constant

Kind of like “conservation of energy” for moving water:Pressure + gravitational potential energy + kinetic energy = constant

For Moving Water,Bernoulli’s Principle

Pressure is energy density (energy per unit volume)

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P? P?12

Clicker question :a. Pressure in case 1 is larger than in case 2.b. Pressure in case 2 is larger than in case 1.c. Pressure in case 1 is same as in case 2.d. Pressures can’t be determined because thehoses are contorted into complicated shapes.

Answer: c. In both cases, the end of the hose is the same distancebelow the water level. The path and shape of the hose make nodifference (in the absence of viscosity, or fluid friction). GRAVITY IS ACONSERVATIVE FIELD; REMEMBER THE RAMP PROBLEMS

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Clicker Question

Water streams out from three spigots in an open bucket. From which hole will the water emergetraveling fastest?

A

B

C

Answer: CThe water pressure increases with depth;a greater water pressure exerts a greaterforce on the water emerging from thespigot, causing greater acceleration,giving the water a greater exit velocity.

D. A, B, and C will spout water at the same speed

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Note to Last Clicker QuestionThe acceleration of water as it emerges from aspigot is governed not merely by the pressureinside the bucket, but by the pressure differenceacross the spigot.

p1=100,000 Pa + ρgh p2 = 100,000 Paatmospheric pressureplus water pressure atmospheric pressure

100,000 Pa

resulting flow

100,000 Pa + ρgh

h

100,000 Pa

Δp = ρgh

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Speed of Water Emerging Under Pressure

Water leaves spout with kinetic energy; where does the energy come from?

ABCD

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Speed of Water Emerging Under Pressure

100,000 Pa

Water leaves spout with kinetic energy; where does the energy come from?

Δh

h

Δp = ρgh

AreaA

Grav. Potential Energy lost: mghKinetic Energy Gained: (1/2)mv2

Energy Conservation: mgh = (1/2)mv2

v

Because of Δp, water exits with speed v

v

!

v = 2gh

Does this look familiar?Water exits with same samevelocity that an object wouldgain falling (from rest) adistance equal to the heightof the water column.

Note:Volume of water leaving tank: A ΔhMass of water leaving tank: m = ρA Δh

m

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Review

• Pressure = Force / Area• Pressure of gases increases with density,

temperature (average kinetic energy)• Pressure increases with depth (both for

water and for air)• Boyancy force = weight of displaced fluid