Density (r), pressure (p), Summary Fluids Pressure transmitted uniformly and isotropically (all directions): Paschal s Principle Pressure vs depth for static liquid Bouyancy: upward force = weight of displaced liquid (Archimedes) Fluids in motion for ideal fluids (laminar, incompressible, nonviscous, ) Equation of continuity (rva=constant) Work-energy requirement (p + rv 2 /2 + rgy = constant): Bernouli s equation Implications (more today) And then Chapt. 19 Temperature, Heat, and the beginnings of Thermodynamics slide 1
review Hydrodynamic Relations Av = Av 1 1 2 2 Equation of Continuity 1 2 1 2 p + rv + rgy = p + rv + rgy 1 2 1 1 2 2 2 2 Bernouli's Equation A v p < A 2 1 > v 2 1 < p 2 1 Venturi tube demo Air flow demo slide 2
Sample Problem 15-9 p + rv + rgy = p + rv + rgy 1 2 1 2 1 2 1 1 2 2 2 2 Find velocity of water when it exits tank Top of tank <=> Hole in tank p gh p v 1 2 0 + 0+ r = 0 + 2 r + 0 v 2gh Note same v as if it were dropped from height h = Same pressure - Cons Energy slide 3
Viscosity and geometry limit utility of Bernouli s equation 1 2 1 2 p + rv + rgy = p + rv + rgy 1 2 1 1 2 2 2 2 Bernouli's Equation v? 2 gh Viscosity measures friction between fluid and pipe, nozzle, Vertical nozzle: doesn t quite rise to ideal height Clearly oil (more viscosity) rises even less But Bernouli explains a lot: Pix shows real case for water and nozzle slide 4
b a How airplane wings lift p + rv + rgy = p + rv + rgy 1 2 1 2 1 2 1 1 2 2 2 2 1 2 2 p - p = r( v -v ) b a F = ( p - p ) A b 2 a a a b b Potential energies ( y ) difference negligible Streamline flow requires time to travel a-> a be the same as time for b -> b For a the distance traveled is larger, so velocity must be higher (v a >v b ) From Bernouli s equation: p a < p b slide 5
House Plumbing Why have elbow or trap? needs water in trap to work properly (acceptably) Having only air between B and A produces smelly bathrooms due to the main sewer line Plumbing in left figure doesn t work: water in trap gets sucked out when large volume of water flows at A to main sewer (eg, someone upstream showers) so that pressure at A lowered Plumbing in right figure works: the way real house plumbing is done slide 6
1 Blood Flow in Arteries 2 A v p < A 2 1 > v 2 1 < p 2 1 Av = 1 1 2 2 1 2 1 2 1 2 1 1 2 2 2 2 Note opposite happens when artery inflates = embolism Av p + rv + rgy = p + rv + rgy Plaque buildup in artery (arteriosclerosis) means (eqn of continuity) speed in blockage higher than in unblocked artery Speed higher in blockage means pressure is lower there (Bernouli eqn) Lower pressure makes artery more likely to collapse slide 7
Throwing Curves p + rv + rgy = p + rv + rgy w 1 2 1 2 1 2 1 1 2 2 2 2 1 Left: no spin, thrown ball flies straight Right: ball spins (cc-wise) Viscosity makes air flow faster around side 2 than side 1 2 v p > v 2 1 < p 2 1 Bernouli s eqn: p 2 < p 1 Net force (side 2 side 1) produces curve ball Also works for drop ball slide 8
Heat and Temperature topics (begins in Chapter 19) Temperature vs Heat Measuring temperature Temperature scales and absolute zero Thermal expansion today Heat capacity (absorption) Specific heat phase transitions Heat and Work 1st law of thermodynamics heat transfer slide 9
Heat and Temperature Rely initially on your intuitive sense + temperature is a property of a body reflecting our sense of hot and cold heat is a form of energy Intuition: a body with high temperature has more heat than it would have at low temperature Temperature initially measured empirically to conform to this intutition Bodies at the same temperature are in thermal equilibrium Thermal equilibrium bodies in contact ultimately come to the same temperature 19 th century brought understanding that temperature is a measure of the average random speed of molecules slide 10
Joule s Experiment 1843 James Prescott Joule showed that a specific amount of mechanical energy reproducibly raised the temperature of a material Demonstrated that lost mechanical energy (in closed system) can be accounted as Heat: the Conservation of Total Energy = First Law of Thermodynamics 1 calorie = 4.186 J raises 1 gram of water from 14.5 o C to 15.5 o C Q Q CT D cmt D C = heat capacity c = specific heat Will return to this slide 11
Heat energy moves Heat energy (Q ) always flows from hot body to cold body aside: we will consider this later as a statement, or consequence, of the Second Law of Thermodynamics Like other forms of energy (kinetic, potential, ) Q is measured in Joules also calories, BTU,... Read text! In upper and lower illustration to left, the system and environment Will initially exchange heat energy Then rate of exchange will slow down as temperatures get closer Both systems will ultimately arrive at the same temperature, and then Q=0 (as in middle figure) Now we have two laws of thermodynamics First discuss ways of measuring temperature and effects of temperature on materials slide 12
Familiar Temperature Measures Rely on (for now) empirical fact Most materials expand when temperature increases Old fashioned glass thermometer, with bulb and capillary containing fluid If temperature increases, fluid expands into capillary Scale provides measure Capillary where fluid expands to Large volume bulb filled with fluid slide 13
Gas Thermometer Fixed amount of gas and fixed volume of bulb ~1800: find empirically pressure and temperature proportional to each other T p Gay-Lussac Law Will return to this and other ideal gas laws later slide 14
Temperature Scales set with Water (freeze and boil) Define temperature scales Gas thermometer can measure temperature of bath by extrapolating p (or V) of gas to zero freezing water o C o F K T T F C 9 = TC + 5 32 = T - K 273. 15 slide 15
Gas Thermometer and Absolute Zero Measure temperature and pressure over range where it remains a gas Most cases, a phase change will occur find empirically pressure and temperature proportional to each other (Gay-Lussac) T = Cp extrapolate to lower temperatures all gases: extrapolation intercepts p=0 at T 0 =-273.15 o C = absolute zero slide 16
Real Temperatures and Life Temperature (we now know and will soon demonstrate) is a measure of the average random motion (kinetic energy) of molecules No temperature can be lower than absolute zero Absolute zero is where all molecular motion stops life life slide 17
Temperature Scales and Expansion Three temperature scales Mass of any system stays constant (until Relativitistic Quant Mech) Empirically as temperature increases, most materials increase their volume Coefficient of linear expansion, a (small) Typically ~10-6 to 10-3 ( 0 C) -1 D L = La DT slide 18
Area and Volume Changes How much do circle, hole, and ruler change in area? D A Ag DT A +D A = ( L +D L )( L +DL ) = L ( 1 + ad TL ) ( 1 + adt) x» A( 1 + 2aDT) D A = A( 2a ) DT g = 2a x x x x y Temperature increase causes most material volumes to increase = density decrease How much do solids change in volume? D V V b DT b = 3a slide 19
Use Material Expansion for Thermometers See table for typical expansions Note: fluid in mercury thermometer expands more than the glass envelope Liquids typically expand more than solids Gases (fixed pressure) typically expand more than liquids Different gases have similar values see later slide 20
Insight into Expansion with Temperature Molecular explanation: as temperature of solid increases, molecules move faster around average locations, but further from each other (expansion) High enough temp (kinetic energy), bonds holding atoms in lattice break (liquid), but forces still hold atoms nearby (not fixed average location like solid) and more temp makes for greater separations (expansion) Raise temp even higher, bonds completely break --- gas --- more temp, more energy (expansion) PHASE CHANGES Solid Liquid Gas slide 21
Heat and Temperature Remember to leave HW8 in box ------------------ So Far Temperature vs Heat Measuring temperature Temperature scales and absolute zero Thermal expansion To Come Heat capacity (absorption) Specific heat phase transitions Heat and Work Develop 1st law of thermodynamics heat transfer conduction convection radiation Kinetic Theory of Gases slide 22