# Archimede's and Pascal's Principles

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```					                                  Thermal Physics                                                      Archimede’      Pascal’
Archimede’s and Pascal’s Principles
     Pascal’s Principle: A change in pressure             F1
applied to an enclosed fluid is transmitted
undiminished to all portions of the fluid and to
the walls of its container.
A
     This principle is used in hydraulic system       A1
1
   Bernoulli’s Equation Cont.                                                P1 = P2                                                              F2
   Temperature                                                               (F1 / A1) = (F2 / A2)                                       A
2
   Thermal Expansion                                                   Archimede’s Principle: Buoyant Force
   Atoms, Molecules and Moles                                               weight of fluid displaced
   Ideal Gas Law                                                              » B = ρ fluid g Vdisplaced, W = ρobject g Vobject
» object sinks if ρobject > ρ fluid
» object floats if ρ object < ρfluid
If object float B=W
» Therefore ρfluid g Vdisplaced = ρ object g Vobject
» Therefore Vdisplaced/Vobject = ρ object / ρ fluid
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Bernoulli’
Review: Bernoulli’s Equation                                                                    Sailing Against the Wind
1                                                                                       1
P + "v 2 + "gy = const                                                                 P + "v 2 + "gy = const
2                                                                                       2
1 2 1                                                                        1
#P = P1 \$ P2 = "v 2 \$ "v12 + "gy 2 \$ "gy1                                              P1 + "v12 = const
2      2                                                                     2
!       1 2
    Based on conservation of energy                                                       P2 + "v 2 = const
    Pressure drops in a rapidly moving fluid or as the fluid rises                            2
!           whether or not the fluid is confined to a tube                                 !         v2>v1
    For incompressible, frictionless fluid (ideal), liquid or gas                                                                                 v1
   Therefore:
    Applications: Wings and sails                                                                P1>P2
P1
!      
 Higher velocity on one side of sail or wing versus the                                   Pressure                                    v2
other results in a pressure difference that can even                                      difference                                              P2
allow the boat to sail into the wind                                                      causes a force                 !
    Water pressure in your house                                                                                                         !
    Velocity measurement
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Physics 103, Spring 2007, U. Wisconsin                 4
!

Temperature :
Thermodynamics
Zeroth Law of Thermodynamics
      Concerned with the concepts of energy transfers between a system and its                   Two objects in thermal equilibrium with each other are at the same
environment and the resulting temperature variations                                        temperature
      Concerns matter in all of its forms: solid, liquid, and gas                                Temperature is the property that determines whether or not an object is
in thermal equilibrium with other objects
      The process by which energy is exchanged between objects                                   Measured by many methods: volume of liquid, pressure of gas
because of temperature differences is called heat
      Objects are in thermal contact if energy can be exchanged                                         Celsius Scale
      Thermal equilibrium exists when two objects in thermal contact -                         Temperature of an ice-water mixture is
may take some time to reach thermal equilibrium                                           defined as 0º C
 This is the freezing point of water
If objects A and B are in                                                                        Temperature of a water-steam mixture is
thermal equilibrium with a
third object, C, then A and                                                                       defined as 100º C
B are in thermal equilibrium                                                                        This is the boiling point of water
with each other.
     Distance between these points is divided
into 100 segments

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Pressure-Temperature Graph                                                                       Modern Definition of Kelvin Scale
      All gases extrapolate to the same temperature at 0 pressure                                        Defined in terms of two points
      This temperature is absolute zero                                                                     Agreed upon by International
Committee on Weights and Measures
in 1954
Kelvin Scale                              Pressure varies with temperature
    First point is absolute zero
when maintaining a constant volume                  Second point is the triple point of water
   When the pressure of a gas
goes to zero, its temperature is                                                                         Triple point is the single point where
–273.15º C                                                                                                 water can exist as solid, liquid, and gas
   This temperature is called                                                                                   » Same temperature and pressure
absolute zero                                                                                                » Occurs at 0.01º C, 273.16 K and P
   This is the zero point of the                                                                                  = 4.58 mm Hg
Kelvin scale


 –273.15º C = 0 K
To convert: TC = TK – 273.15
TC = TK " 273.15
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!

Thermal Expansion                                                                                   Question: Expansion
      The thermal expansion of an object is a consequence of the change in                               Two spheres are made of the same metal and have the same radius,
the average separation between its constituent atoms or molecules                                   but one is hollow and the other is solid at room temperature. The
      At ordinary temperatures, molecules vibrate with a small amplitude                                  temperature of both spheres is increased by the same amount. Which
sphere expands more?
      As temperature increases, the amplitude increases
 Solid sphere
 This causes the overall object as a whole to expand
 Hollow sphere
      Linear, area, and volume expansion for small temperature changes
 They expand by the same amount
 Coefficient of linear expansion, alpha, depends on material

"L = # Lo "t                                                                                            All dimensions increase
The increase is proportional to
"A = # Ao "t, # = 2\$                                                                                      original size of that dimension
For the sphere the relevant
"V = # Vo "t, # = 3\$
!   4/5/08                     Physics 103, Spring 2007, U. Wisconsin                            9      4/5/08                          Physics 103, Spring 2007, U. Wisconsin        10

!
!
Lecture 19, Preflight 2                                                                              Lecture 19, Preflight 4
Sometimes small gaps are left in the roadways on bridges                                              Liquids always expand when heated.
A. Intentionally to help slow down the traffic.                                                        A. yes.
B. Intentionally to avoid buckling the roadways when seasons                                           B. no.       CORRECT
change.         CORRECT
C. Unintentionally. This usually is a result of poor civil
engineering.                                                                                                                              48%

52%

0%      20%   40%         60%

4%
Unusual behavior of water.

94%                             As the temperature of water increases from 0ºC to 4
1%                                                               ºC, it contracts and its density increases
   Above 4 ºC, water exhibits the expected expansion
0%        20%   40%   60%   80%   100%
with increasing temperature
   Maximum density of water is 1000 kg/m3 at 4 ºC
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Applications of Thermal Expansion
Thermometers
– Bimetallic Strip
    Make use of physical properties that change with
temperature
    Many physical properties can be used
 volume of a liquid
 length of a solid
 pressure of a gas held at constant volume
 volume of a gas held at constant pressure
 electric resistance of a conductor
   Thermostats                                                color of a very hot object
 Use a bimetallic strip
 Two metals expand differently

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The Periodic Table and Moles
The Ideal Gas Law

 Quantify    how pressure or volume change with
increasing temperature

proton
neutron                                        PV      = NkBT
electron
   1 mole = 6.022 x 1023 molecules                                                      P = pressure in N/m2 (or Pascals)
   V = volume in m3
   1 µ = 1 atomic mass unit = (mass of 12C atom)/12                                     N = number of molecules
» approximately #neutrons + #protons                                             T = absolute temperature (K)
» atomic weight W                                                                kB = Boltzmann’s constant
» kB = 1.38 x 10-23 J/K
   Mass of 1 mole of “stuff” in grams = molecular
mass in µ                                                                              » note: PV has units of N-m or J (Work or energy!)
• e.g., 1 mole of N2 has mass of 2x14=
28 grams
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The Ideal Gas Law                                                             Hot Air Balloon

    PV = NkBT                                                                   How does a hot air balloon work?
    Alternate way to write this
 N = number of moles (n) x NA molecules/mole                           The flame heats the air, which pushes the balloon up.
 PV= NkBT                                                               Is this correct?
» nNAk B T                                                                  Yes
» n(NAk B)T                                                                 No
» nRT
    PV = nRT
 R = ideal gas constant = NAk B = 8.31 J/(mol-K)

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Hot Air Balloon                                                             Hot Air Balloon
How does a hot air balloon work?
Where is the pressure higher?                                                     How does a hot air balloon work?
Inside the balloon where air is warm
Outside the balloon where air is cooler
Pressure is the same both inside and outside                               As the air is heated it expands, increasing the volume of the
balloon. Therefore the density decreases and the balloon floats.

The pressure has not changed - the volume of the balloon has                                           PV = nRT
increased.
The kinetic energy of the air molecules inside the balloon is on              Note: this is not a pressure effect..it is a volume effect. As T
increases, volume increase, so that density decreases…balloon then
average higher. Therefore, the energy transferred to the                    floats due to Archimedes principle.
balloon membrane when molecules collide with it is higher -
that energy results in higher potential energy of the elastic                                 B = W, ρ air > ρballoon
membrane - it expands - the volume of the balloon with hot air
increases.
B = ρ air g Vdisplaced
The air in the balloon is now less dense than regular air
W = ρballoon g V ballon
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