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# Lecture Ch 10

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### Lecture Ch 10

1. 1. Chapter 10 Fluids
2. 2. Units of Chapter 10 <ul><li>Phases of Matter </li></ul><ul><li>Density and Specific Gravity </li></ul><ul><li>Pressure in Fluids </li></ul><ul><li>Atmospheric Pressure and Gauge Pressure </li></ul><ul><li>Pascal’s Principle </li></ul><ul><li>Measurement of Pressure; Gauges and the Barometer </li></ul><ul><li>Buoyancy and Archimedes’ Principle </li></ul>
3. 3. Units of Chapter 10 <ul><li>Fluids in Motion; Flow Rate and the Equation of Continuity </li></ul><ul><li>Bernoulli’s Equation </li></ul><ul><li>Applications of Bernoulli’s Principle: from Torricelli to Airplanes, Baseballs, and TIA </li></ul>
4. 4. 10-1 Phases of Matter The three common phases of matter are solid , liquid , and gas . A solid has a definite shape and size . A liquid has a fixed volume but can be any shape. A gas can be any shape and also can be easily compressed . Liquids and gases both flow , and are called fluids .
5. 5. 10-2 Density and Specific Gravity The density ρ ( lower case Greek rho) of an object is its mass per unit volume : The SI unit for density is kg/m 3 . Density is also sometimes given in g/cm 3 ; to convert g/cm 3 to kg/m 3 , multiply by 1000. Water at 4 ° C has a density of 1 g/cm 3 = 1000 kg/m 3 . The specific gravity of a substance is the ratio of its density to that of water. SG = ( ρ / ρ water ) = 10 -3 ρ (10-1)
6. 6. 10-3 Pressure in Fluids Pressure is defined as the force per unit area . Pressure is a scalar; the units of pressure in the SI system are pascals : 1 Pa = 1 N/m 2 Pressure is the same in every direction in a fluid at a given depth ; if it were not, the fluid would flow.
7. 7. 10-3 Pressure in Fluids Also for a fluid at rest, there is no component of force parallel to any solid surface – once again, if there were the fluid would flow.
8. 8. 10-3 Pressure in Fluids The pressure at a depth h below the surface of the liquid is due to the weight of the liquid above it. We can quickly calculate: This relation is valid for any liquid whose density does not change with depth. <ul><li>At rest  ∑ F y = 0 </li></ul><ul><li>F - mg = 0  F = mg </li></ul><ul><li>F = mg = ρ Vg, V =Ah </li></ul><ul><li>F = ρ Ahg </li></ul><ul><li> P  F/A = ρ gh </li></ul><ul><li>Pressure at depth h (fluid at rest) </li></ul><ul><li>P = ρ gh </li></ul>
9. 9. Section 10-4: Atmospheric Pressure <ul><li>Earth’s atmosphere: A fluid. </li></ul><ul><ul><li>But doesn’t have a fixed top “surface”! </li></ul></ul><ul><li>Change in height  h above Earth’s surface: </li></ul><ul><li> Change in pressure:  P = ρ g  h </li></ul><ul><li>Sea level: P A  1.013  10 5 N/m 2 </li></ul><ul><li> = 101.3 kPa  1 atm </li></ul><ul><ul><li>Old units: 1 bar = 1.00  10 5 N/m 2 </li></ul></ul><ul><li>Physics: Cause of pressure at any height: </li></ul><ul><li>Weight of air above that height! </li></ul><ul><li>This pressure does not crush us, as our cells maintain an internal pressure that balances it. </li></ul>
10. 10. <ul><li>Show Clip </li></ul>
11. 11. 10-4 Atmospheric Pressure and Gauge Pressure Most pressure gauges measure the pressure above the atmospheric pressure – this is called the gauge pressure . The absolute pressure is the sum of the atmospheric pressure and the gauge pressure.
12. 12. Conceptual Example 10-4 <ul><li>P = ? </li></ul><ul><li>Pressure on A : </li></ul><ul><li>P down = P + P mg </li></ul><ul><li>P up = P A </li></ul><ul><li>At rest  ∑ F y = 0 </li></ul><ul><li> P up = P down </li></ul><ul><li>or P A = P + P mg </li></ul><ul><li> P = P A - P mg < P A </li></ul><ul><li>So, air pressure holds </li></ul><ul><li>fluid in straw! </li></ul>
13. 13. 10-5 Pascal’s Principle If an external pressure is applied to a confined fluid, the pressure at every point within the fluid increases by that amount. This principle is used, for example, in hydraulic lifts and hydraulic brakes .
14. 14. 10-6 Measurement of Pressure; Gauges and the Barometer There are a number of different types of pressure gauges (Most use P - P A = ρ gh = P G = gauge pressure ) . This one is an open-tube manometer . The pressure in the open end is atmospheric pressure; the pressure being measured will cause the fluid to rise until the pressures on both sides at the same height are equal .
15. 15. 10-6 Measurement of Pressure; Gauges and the Barometer Here are two more devices for measuring pressure: the aneroid gauge and the tire pressure gauge .
16. 16. Various Pressure Units <ul><li>Gauge Pressure: P G = ρ gh </li></ul><ul><li> Alternate unit of pressure: Instead of calculating ρ gh , common to use standard liquid </li></ul><ul><li>(mercury, Hg or alcohol, where ρ is standard) & measure h </li></ul><ul><li> Quote pressure in length units ! For example: </li></ul><ul><li>“ millimeters of mercury”  mm Hg </li></ul><ul><li> For h = 1 mm Hg = 10 -3 m Hg </li></ul><ul><li>ρ mercury gh = (1.36  10 4 kg/m 3 ) (9.8 m/s 2 )(10 -3 m) </li></ul><ul><li>= 133 N/m 2 = 133 Pa  1 Torr </li></ul><ul><li>(another pressure unit!) </li></ul><ul><li>mm Hg & Torr are not proper SI pressure units! </li></ul>
17. 17. 10-6 Measurement of Pressure; Gauges and the Barometer This is a mercury barometer, developed by Torricelli to measure atmospheric pressure . The height of the column of mercury is such that the pressure in the tube at the surface level is 1 atm . Therefore, pressure is often quoted in millimeters (or inches) of mercury.
18. 18. 10-6 Measurement of Pressure; Gauges and the Barometer Any liquid can serve in a Torricelli-style barometer, but the most dense ones are the most convenient. This barometer uses water .
19. 19. Prob. 17: (A variation on Example 10-2) Tank depth = 5 m Pipe length = 110 m Hill slope = 58º Gauge Pressure P G = ? Height water H shoots from broken pipe at bottom? Height of water level in tank from house level: h = (5 + 110 sin58º) = 98.3 m P G = ρ water gh = (1  10 3 kg/m 3 )(9.8 m/s 2 )(98.3 m) = 9.6  10 5 N/m 2 Conservation of energy: H = h = 98.3 m (Neglects frictional effects, etc.)
20. 20. 10-7 Buoyancy and Archimedes’ Principle This is an object submerged in a fluid. There is a net force on the object because the pressures at the top and bottom of it are different. The buoyant force is found to be the upward force on the same volume of water:
21. 21. Archimedes Principle <ul><li> The total (upward) buoyant force F B on an object of volume V completely or partially submerged in a fluid with density ρ F : </li></ul><ul><li> F B = ρ F Vg (1) </li></ul><ul><li> ρ F V  m F  Mass of fluid which would take up same volume as object, if object were not there. (Mass of fluid that used to be where object is!) </li></ul><ul><li> Upward buoyant force </li></ul><ul><li>F B = m F g (2) </li></ul><ul><li>F B = weight of fluid displaced by the object! </li></ul><ul><li>(1) or (2)  Archimedes Principle </li></ul><ul><li>Proved for cylinder. Can show valid for any shape </li></ul>
22. 22. 10-7 Buoyancy and Archimedes’ Principle The net force on the object is then the difference between the buoyant force and the gravitational force.
23. 23. 10-7 Buoyancy and Archimedes’ Principle If the object’s density is less than that of water, there will be an upward net force on it, and it will rise until it is partially out of the water.
24. 24. 10-7 Buoyancy and Archimedes’ Principle For a floating object, the fraction that is submerged is given by the ratio of the object’s density to that of the fluid.
25. 25. 10-7 Buoyancy and Archimedes’ Principle This principle also works in the air; this is why hot-air and helium balloons rise.
26. 26. 10-8 Fluids in Motion; Flow Rate and the Equation of Continuity If the flow of a fluid is smooth ( neighboring layers of the fluid slide smoothly ), it is called streamline or laminar ( in layers ) flow (a). Above a certain speed, the flow becomes turbulent (b). Turbulent flow has eddies (or eddy currents : erratic, small, whirlpool-like circles) ; the viscosity ( internal friction ) of the fluid is much greater when eddies are present.
27. 27. We will deal with laminar flow. The mass flow rate is the mass that passes a given point per unit time. The flow rates at any two points must be equal , as long as no fluid is being added or taken away. 10-8 Fluids in Motion; Flow Rate and the Equation of Continuity This gives us the equation of continuity :
28. 28. 10-8 Fluids in Motion; Flow Rate and the Equation of Continuity If the density doesn’t change – typical for liquids – this simplifies to . Where the pipe is wider , the flow is slower .
29. 29. Example 10-11: Estimate Blood Flow <ul><li>r cap = 4  10 -4 cm, r aorta = 1.2 cm </li></ul><ul><li>v 1 = 40 cm/s, v 2 = 5  10 -4 cm/s </li></ul><ul><li>Number of capillaries N = ? </li></ul><ul><li>A 2 = N  (r cap ) 2 , A 1 =  (r aorta ) 2 </li></ul><ul><li>A 1 v 1 = A 2 v 2 </li></ul><ul><li> N = (v 1 /v 2 )[(r aorta ) 2 /(r cap ) 2 ] </li></ul><ul><li>N  7  10 9 </li></ul>
30. 30. 10-9 Bernoulli’s Equation A fluid can also change its height . By looking at the work done as it moves, we find: This is Bernoulli’s equation. One thing it tells us is that as the speed goes up, the pressure goes down.
31. 31. <ul><li>Work & energy in fluid moving from Fig. a to Fig. b : </li></ul><ul><li>a) Fluid to left of point 1 exerts pressure P 1 on fluid mass M = ρ V </li></ul><ul><li>V = A 1   1 = A 2   2 . Moves it   1 . Work done: </li></ul><ul><li>W 1 = F 1   1 = P 1 A 1   1 </li></ul><ul><li>b) Fluid to right of point 2 exerts pressure P 2 on fluid mass M = ρ V , </li></ul><ul><li>V = A 2   2 . Moves it   2 . Work done: </li></ul><ul><li>W 2 = -F 2   2 = -P 2 A 2   2 </li></ul><ul><li>a)  b) Mass M moves from height y 1 to height y 2 . Work done against </li></ul><ul><li>gravity: </li></ul><ul><li>W 3 = -Mg(y 2 – y 1 ) ; W G = -  PE </li></ul>
32. 32. Sect. 10-9: Bernoulli’s Eqtn <ul><li>Total work done from a)  b): </li></ul><ul><li>W net = W 1 + W 2 + W 3 </li></ul><ul><li> W net = P 1 A 1   1 - P 2 A 2   2 - Mg( y 2 – y 1 ) (1) </li></ul><ul><li>Recall the Work-Energy Principle: </li></ul><ul><li>W net =  KE = (½)M(v 2 ) 2 – (½)M(v 1 ) 2 (2) </li></ul><ul><li>Combining (1) & (2): </li></ul><ul><li>(½)M(v 2 ) 2 – (½)M(v 1 ) 2 </li></ul><ul><li>= P 1 A 1   1 - P 2 A 2   2 - Mg( y 2 – y 1 ) (3) </li></ul><ul><li>Note that M = ρ V = ρ A 1   1 = ρ A 2   2 </li></ul><ul><li>divide (3) by V = A 1   1 = A 2   2 </li></ul>
33. 33. <ul><li> (½) ρ (v 2 ) 2 - (½) ρ (v 1 ) 2 = P 1 - P 2 - ρ g( y 2 – y 1 ) (4) </li></ul><ul><li>Rewrite (4) as: </li></ul><ul><li>P 1 + (½) ρ (v 1 ) 2 + ρ gy 1 = P 2 + (½) ρ (v 2 ) 2 + ρ gy 2 </li></ul><ul><li> Bernoulli’s Equation </li></ul><ul><li>Another form: </li></ul><ul><li>P + (½) ρ v 2 + ρ gy = constant </li></ul><ul><li>Not a new law, just work & energy of system in fluid language. ( Note : P  ρ g(y 2 -y 1 ) since fluid is NOT at rest!) </li></ul><ul><li> Work Done by Pressure =  KE +  PE </li></ul>
34. 34. Flow on the level <ul><li>P 1 + (½) ρ (v 1 ) 2 + ρ gy 1 = P 2 + (½) ρ (v 2 ) 2 + ρ gy 2 </li></ul><ul><li>Flow on the level  y 1 = y 2  </li></ul><ul><li>P 1 + (½) ρ (v 1 ) 2 = P 2 + (½) ρ (v 2 ) 2 </li></ul><ul><li>Explains many fluid phenomena & is a quantitative statement of Bernoulli’s Principle: </li></ul><ul><li>“ Where the fluid velocity is high, the pressure is low, and where the velocity is low, the pressure is high.” </li></ul>
35. 35. 10-10 Applications of Bernoulli’s Principle: from Torricelli to Airplanes, Baseballs, and TIA Using Bernoulli’s principle, we find that the speed of fluid coming from a spigot on an open tank is: This is called Torricelli’s theorem. (10-6) P 1 + (½) ρ (v 1 ) 2 + ρ gy 1 = P 2 + (½) ρ (v 2 ) 2 + ρ gy 2 P 1 = P 2 P 2 P 1
36. 36. 10-10 Applications of Bernoulli’s Principle: Lift on an airplane wing is due to the different air speeds and pressures on the two surfaces of the wing. P TOP < P BOT  LIFT force! A 1  Area of wing top, A 2  Area of wing bottom F TOP = P TOP A 1 F BOT = P BOT A 2 Plane will fly if ∑ F = F BOT - F TOP - Mg > 0 !
37. 37. 10-10 Applications of Bernoulli’s Principle: from Torricelli to Airplanes, Baseballs, and TIA A ball’s path will curve due to its spin , which results in the air speeds on the two sides of the ball not being equal.
38. 38. 10-10 Applications of Bernoulli’s Principle: from Torricelli to Airplanes, Baseballs, and TIA A person with constricted arteries will find that they may experience a temporary lack of blood to the brain ( TIA; Transient Ischemic Attack ) as blood speeds up to get past the constriction , thereby reducing the pressure.
39. 39. 10-10 Applications of Bernoulli’s Principle: from Torricelli to Airplanes, Baseballs, and TIA A venturi meter can be used to measure fluid flow by measuring pressure differences.
40. 40. 10-10 Applications of Bernoulli’s Principle: from Torricelli to Airplanes, Baseballs, and TIA Air flow across the top helps smoke go up a chimney , and air flow over multiple openings can provide the needed circulation in underground burrows.
41. 41. Problem 46: Pumping water up Street level: y 1 = 0 v 1 = 0.6 m/s, P 1 = 3.8 atm Diameter d 1 = 5.0 cm ( r 1 = 2.5 cm ). A 1 = π (r 1 ) 2 18 m up : y 2 = 18 m, d 2 = 2.6 cm ( r 2 = 1.3 cm ). A 2 = π (r 2 ) 2 v 2 = ? P 2 = ? Continuity: A 1 v 1 = A 2 v 2  v 2 = (A 1 v 1 )/(A 2 ) = 2.22 m/s Bernoulli: P 1 + (½) ρ (v 1 ) 2 + ρ gy 1 = P 2 + (½) ρ (v 2 ) 2 + ρ gy 2  P 2 = 2.0 atm
42. 42. Summary of Chapter 10 <ul><li>Phases of matter: solid, liquid, gas. </li></ul><ul><li>Liquids and gases are called fluids. </li></ul><ul><li>Density is mass per unit volume. </li></ul><ul><li>Specific gravity is the ratio of the density of the material to that of water. </li></ul><ul><li>Pressure is force per unit area. </li></ul><ul><li>Pressure at a depth h is ρ gh . </li></ul><ul><li>External pressure applied to a confined fluid is transmitted throughout the fluid. </li></ul>
43. 43. Summary of Chapter 10 <ul><li>Atmospheric pressure is measured with a barometer. </li></ul><ul><li>Gauge pressure is the total pressure minus the atmospheric pressure. </li></ul><ul><li>An object submerged partly or wholly in a fluid is buoyed up by a force equal to the weight of the fluid it displaces. </li></ul><ul><li>Fluid flow can be laminar or turbulent. </li></ul><ul><li>The product of the cross-sectional area and the speed is constant for horizontal flow. </li></ul>
44. 44. Summary of Chapter 10 <ul><li>Where the velocity of a fluid is high, the pressure is low, and vice versa. </li></ul>