Lecture 5 3_pascal_principle
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This document covers several key topics in fluid mechanics including Pascal's principle, Bernoulli's equation, Archimedes' principle, and their applications. It includes definitions of key terms like viscosity, diffusion, and osmosis. It also works through multiple examples applying concepts like pressure, buoyancy, and continuity to problems involving things like fluid flow through pipes and the height of towers or pools.
Lecture 5 3_pascal_principle
- 1. Fluids Mechanics • Pascal Principle • Bernoulli’s Equation
- 2. Pressure & Pascal’s Principle “Pressure applied to any part of an enclosed fluid is transmitted undimished to every point of the fluid and to the walls of the container” Each face feels same force P FA
- 3. Transmitting force An applied force F1 can be “amplified”: Hydraulic press Examples: hydraulic brakes, forklifts, car lifts, etc. P F1A1 F2A2F2F1A2A1
- 4. Pressure and Depth w is weight Sum forces to zero, Factor A wMgVgAhgPAP0Aw0PP0gh
- 5. Example 9.5 (skip) Find the pressure at 10,000 m of water. DATA: Atmospheric pressure = 1.015x105 Pa. 9.82x107 Pa
- 6. Example 9.6 Assume the ultimate strength of legos is 4.0x104 Pa. If the density of legos is 150 kg/m3, what is the maximum possible height for a lego tower? 27.2 m
- 7. Example 9.7 Estimate the mass of the Earth’s atmosphere given that atmospheric pressure is 1.015x105 Pa. Data: Rearth=6.36x106 m 5.26x1018 kg
- 8. Archimedes Principle Any object completely or partially submerged in a fluid is buoyed up by a force whose magnitude is equal to the weight of the fluid displaced by the object.
- 9. Example 9.8 A helicopter lowers a probe into Lake Michigan which is suspended on a cable. The probe has a mass of 500 kg and its average density is 1400 kg/m3. What is the tension in the cable? 1401 N
- 10. Example 9.9a A wooden ball of mass M and volume V floats on a swimming pool. The density of the wood is wood <H20. The buoyant force acting on the ball is: a) Mg upward b) H20gV upward c) (H20-wood)gV upward
- 11. Example 9.9b A steel ball of mass M and volume V rests on the bottom of a swimming pool. The density of the steel is steel >H20. The buoyant force acting on the ball is: a) Mg upward b) H20gV upward c) (steel-H20)gV upward
- 12. Example 9.10 A small swimming pool has an area of 10 square meters. A wooden 4000-kg statue of density 500 kg/m3 is then floated on top of the pool. How far does the water rise? Data: Density of water = 1000 kg/m3 40 cm
- 13. Paint Thinner Demo (SKIP) When I pour in the paint thinner, the cylinder will: a) Rise b) Fall
- 14. Equation of Continuity What goes in must come out! Mass that passes a point in pipe during time Dt mass density DMADxAvDtEq. of Continuity 1A1v12A2v2
- 15. Example 9.11 Water flows through a 4.0 cm diameter pipe at 5 cm/s. The pipe then narrows downstream and has a diameter of of 2.0 cm. What is the velocity of the water through the smaller pipe? 20 cm/s
- 16. Laminar or Streamline Flow •Fluid elements move along smooth paths •Friction in laminar flow is called viscosity
- 17. Turbulence •Fluid elements move along irregular paths •Sets in for high velocity gradients (small pipes)
- 18. Ideal Fluids • Laminar Flow -> No turbulence • Non-viscous -> No friction between fluid layers • Incompressible -> Density is same everywhere
- 19. Bernoulli’s Equation Sum of P, KE/V and PE/V is constant How can we derive this? P 12 v2gyconstant
- 20. Bernoulli’s Equation: derivation Consider a volume DV of mass DM of incompressible fluid, DKE 12Mv22 12Mv12 12 DVv22 12 DVv12DPEMgy2Mgy1 DVgy2DVgy1WF1Dx1F2Dx2 P1A1Dx1P2A2Dx2 P1DVP2DVP1gh1 12 v12P2gh2 12 v22
- 21. Example 9.12 A very large pipe carries water with a very slow velocity and empties into a small pipe with a high velocity. If P2 is 7000 Pa lower than P1, what is the velocity of the water in the small pipe? 3.74 m/s Venturi Meter
- 22. Applications of Bernoulli’s Equation •Venturi meter •Curve balls •Airplanes Beach Ball & Straws Demos
- 23. Example 9.13a Consider an ideal incompressible fluid, choose >, < or = 1 ____ 2 a) = b) < c) >
- 24. Example 9.13b Consider an ideal incompressible fluid, choose >, < or = Mass that passes “1” in one second _____ mass that passes “2” in one second a) = b) < c) >
- 25. Example 9.13c Consider an ideal incompressible fluid, choose >, < or = v1 ____ v2 a) = b) < c) >
- 26. Example 9.13d Consider an ideal incompressible fluid, choose >, < or = P1 ____ P2 a) = b) < c) >
- 27. Example 9.14 Water drains out of the bottom of a cooler at 3 m/s, what is the depth of the water above the valve? a b 45.9 cm
- 28. Three Vocabulary Words •Viscosity •Diffusion •Osmosis
- 29. Viscosity •Friction between the layers •Pressure drop required to force water through pipes (Poiselle’s Law) •At high enough v/d, turbulence sets in FAvd
- 30. Diffusion •Molecules move from region of high concentration to region of low concentration •Fick’s Law: •D = diffusion coefficient Diffusionrate Masstime DAC2C1L
- 31. Osmosis Movement of water through a boundary while denying passage to specific molecules, e.g. salts