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Fluid Mechanics

Fluid Mechanics . Chapter 9 . Fluids and Buoyant Force . Defining a Fluid . A fluid is a nonsolid state of matter in which the atoms or molecules are free to move past each other, as in a gas or a liquid. . Both liquids and gases are considered fluids because they can flow and change shape. .

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Fluid Mechanics

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  1. Fluid Mechanics Chapter 9

  2. Fluids and Buoyant Force

  3. Defining a Fluid Afluidisanonsolidstateofmatterinwhichthe atoms or molecules are free to move past each other, as in a gas or a liquid. Both liquids and gases are considered fluids because they can flow and change shape. Liquids have a definite volume; gases do not.

  4. Density and Buoyant Force The concentration of matter of an object is calledthe massdensity. Mass density is measured as the mass per unit volume of a substance.  m /V V = Ah p = m/Ah density  mass volume

  5. Densities of Common Substances

  6. Density and Buoyant Force The buoyantforceistheupwardforce exerted by a liquid on an object immersed in or floating on the liquid. Buoyant forces can keep objects afloat.

  7. Buoyant Force and Archimedes’ Principle The Brick, when added will cause the water to be displaced and fill the smaller container. What will the volume be inside the smaller container? The same volume as the brick!

  8. Hydrostatic Pressure Pressure is a measure of how much force is distributed over an area. p= F┴/ A The unit of pressure is N/m2, called a pascal (Pa) A simple example N = mg p = N/A = mg/A

  9. Incompressible Fluid Water is incompressible. Gases are highly compressible. Fnet = 0 Fsurface = mg - Fdepth= 0 The equilibrium condition is p0 A + (pAh)g – pA = 0 → p = p0 + pgh Pascal’s principle: if p0 changes, the fluid changes by the same amount

  10. Buoyant Force and Archimedes’ Principle Archimedes’ principle describes the magnitude of a buoyant force. Archimedes’principle: Anyobjectcompletelyor partially submerged in a fluid experiences an upward buoyant force equal in magnitude to the weight of the fluid displaced by the object. FB= Fg(displacedfluid)= mfg magnitude of buoyant force = weight of fluid displaced

  11. Buoyant Force The raft and cargo are floating because their weight and buoyant force are balanced.

  12. Buoyant Force Now imagine a small hole is put in the raft. The raft and cargo sink because their density is greater than the density of the water. As the volume of the raft decreases, the volume of the water displaced by the raft and cargo also decreases, as does the magnitude of the buoyant force.

  13. Buoyant Force For a floating object, the buoyant force equals the object’s weight. The apparent weight of a submerged object depends on the density of the object. Foranobjectwithdensity Osubmergedinafluid ofdensity f,thebuoyantforce FBobeysthe following ratio: Fg (object )  O FB f

  14. Example A bargain hunter purchases a “gold” crown at a flea market. After she gets home, she hangs the crown from a scale and finds its weight to be 7.84 N. She then weighs the crown while it is immersed in water, and the scale reads 6.86 N. Is the crown made of pure gold? Explain.

  15. Solution Choose your equations: F - F  apparent weight g B F  g  O F  B f Rearrange your equations: F  F - apparent weight   B g F g    O f F B

  16. Solution Plug and Chug: F  7.84 N - 6.86 N = 0.98 N B F 7.84 N g 3 3     1.00  10 kg/m   O f F 0.98 N B 3 3   8.0  10 kg/m O From the table in your book, the density ofgoldis19.3 103kg/m3. Because8.0 103kg/m3<19.3 103 kg/m3,thecrowncannotbepuregold.

  17. Your Turn I A piece of metal weighs 50.0 N in air and 36.0 N in water and 41.0 N in an unknown liquid. Find the densities of the following: The metal The unknown liquid A 2.8 kg rectangular air mattress is 2.00 m long and 0.500 m wide and 0.100 m thick. What mass can it support in water before sinking? A ferry boat is 4.0 m wide and 6.0 m long. When a truck pulls onto it, the boat sinks 4.00 cm in the water. What is the weight of the truck?

  18. Fluid Pressure

  19. Pressure Deep sea divers wear atmospheric diving suits to resist the forces exerted by the water in the depths of the ocean. You experience this pressure when you dive to the bottom of a pool, drive up a mountain, or fly in a plane.

  20. Pressure Pressure isthemagnitudeoftheforceona surface per unit area. P  F A pressure = force area Pascal’s principle states that pressure applied to a fluid in a closed container is transmitted equally to every point of the fluid and to the walls of the container.

  21. Pressure The SI unit for pressure is the pascal, Pa. Itisequalto1N/m2. The pressure at sea level is about 1.01 x 105Pa. This gives us another unit for pressure, the atmosphere,where1 atm=1.01x105Pa

  22. Pascal’s Principle When you pump a bike tire, you apply force on the pump that in turn exerts a force on the air inside the tire. The air responds by pushing not only on the pump but also against the walls of the tire. As a result, the pressure increases by an equal amount throughout the tire.

  23. Pascal’s Principle A hydraulic lift uses Pascals principle. A small force is applied F1 (F1)toasmallpistonof area(A1)andcausea A2 A1 pressure increase on the F2 fluid. F F This increase in pressure P   1 2 inc A A (Pinc)istransmittedtothe 1 2 largerpistonofarea(A2) and the fluid exerts a A F  F 2 force(F2)onthispiston. 2 1 A 1

  24. Example The small piston of a hydraulic lift has an areaof0.20m2.Acarweighing1.20x104 N sits on a rack mounted on the large piston. The large piston has an area of 0.90m2.Howmuchforcemustbeapplied to the small piston to support the car?

  25. Solution F F A  1 2 F  F 1 A A 1 2 A 1 2 2 Plug and Chug: F1=(1.20x104N)(0.20m2/0.90m2) F1 =2.7x103N

  26. Your Turn II In a car lift, compressed air exerts a force on a piston with a radius of 5.00 cm. This pressure is transmitted to a second piston with a radius of 15.0 cm. How large of a force must the air exert to lift a 1.33 x 104Ncar? A person rides up a lift to a mountain top, but the person’s ears fail to “pop”. The radius of each ear drum is 0.40 cm. The pressure of the atmospheredropsfrom10.10x105Paatthe bottomto0.998x105Paatthetop. What is the pressure difference between the inner and outer ear at the top of the mountain? What is the magnitude of the net force on each eardrum?

  27. Pressure Pressure varies with depth in a fluid. The pressure in a fluid increases with depth. P  P   gh 0 absolute pressure = atmospheric pressure +   densityfree-fall accelerationdepth

  28. Fluids in Motion

  29. Fluid Flow Movingfluidscanexhibit laminar(smooth) flowor turbulent(irregular)flow. Laminar Flow Turbulent Flow

  30. Fluid Flow An ideal fluidisafluidthathasnointernal friction or viscosity and is incompressible. The ideal fluid model simplifies fluid-flow analysis

  31. Fluid Flow No real fluid has all the properties of an ideal fluid, it helps to explain the properties of real fluids. Viscosity refers to the amount of internal friction within a fluid. High viscosity equals a slow flow. Steadyflow is when the pressure, viscosity, and density at each point in the fluid are constant.

  32. Principles of Fluid Flow The continuity equation results from conservation of mass. ∆m/∆t = p A1x1/∆t= p A1v1 ∆m/∆t = p A2x2/∆t= p A2v2 p A1v1= p A2v2 Continuity equation: A1v1 = A2v2 Area speedinregion1=area speedinregion2

  33. Principles of Fluid Flow The speed of fluid flow depends on cross- sectional area. Bernoulli’s principle states that the pressure in a fluid decreases as the fluid’s velocity increases.

  34. Bernoulli’s Equation The continuity equation tells you when a horizontal pipe is constricted, the fluid speeds up. ½ p v2+ pgh + p = constant

  35. Key Equations Pressure p= F┴/ A Hydrostatic pressure p = p0 + pgh Pascal’s Principle F’ = F ( A’/A) Archimede’s Principle FB= Fg(displacedfluid)= mfg Continuity Principle A1v1 = A2v2 Bernoulli’s Principle ½ p v2+ pgh + p = constant

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