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Pressure and Fluids

Pressure and Fluids. § 12.1–12.3. Density. Relating “how big” to “how much”. § 12.1. Density. Mass per volume r = dm / dV ; m =  r dV If r is constant, m = r V SI units kg/m 3 ; conventionally g/cm 3 Intensive quantity Depends on substance, not amount. Densities. Pressure.

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Pressure and Fluids

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  1. Pressure and Fluids § 12.1–12.3

  2. Density Relating “how big” to “how much” § 12.1

  3. Density • Mass per volume r = dm/dV;m = rdV • If r is constant, m = rV • SI units kg/m3; conventionally g/cm3 • Intensive quantity • Depends on substance, not amount

  4. Densities

  5. Pressure One type of stress § 12.2

  6. Pressure • Force per unit area p = F/A • F is component perpendicular to surface • Scalar quantity

  7. Units of Pressure • SI Units: N/m2 = pascal = Pa • Bar = 100,000 Pa; mbar = hPa = 100 Pa • Atmosphere = atm = mean sea level pressure = 101,325 Pa = 1.01 bar • Torr: Height in mm of a mercury barometer (760 torr = 1 atm) • lb/in2 = psi (14.696 psi = 1 atm)

  8. Gauge and Absolute Pressure • Gauge: increase above surrounding atmosphere • e.g. tire pressure 32 psig • Absolute: total

  9. Poll Question When a molecule bounces elastically off a surface, what is the direction of its momentum change? A. B. C. D.

  10. Poll Question When a molecule bounces elastically off a surface, what is the direction of the force on the surface? A. B. C. D.

  11. Force Collisions • Collision exerts a force into surface • Pressure from force averaged over time and area

  12. Pressure in a fluid • Pascal’s Principle: pressure in a fluid is exerted uniformly in all directions throughout the fluid.

  13. Hydraulics just another simple machine

  14. Quick Question If a force F1is applied to the small piston with area A1, what is the pressure increase in the fluid? Dp = F1A1. Dp = F1 + A1. Dp = A1/F1. Dp = F1/A1. 1 2

  15. Poll Question If a force F1is applied to the narrow piston with area A1, How does the magnitude of the force F2 exerted by the fluid on the wide piston compare to F1? F2 = F1. F2 < F1. F2 > F1. 1 2

  16. = = F1 F1 F2 A1 A1 F2 A2 A2 Hydraulics • Pistons have different areas • Pressure p=F/A for both pistons • rearranges to

  17. Dx2 ? Dx1 Poll Question If the narrow piston with area A1is moved a distance Dx1, How does the distance Dx2 that the wide piston moves compare to Dx1? Dx2 = Dx1. Dx2 < Dx1. Dx2 > Dx1. 1 2

  18. = F1 F2 = DV DV F1 F2 A1 A2 A1 A2 Hydraulics are Simple Machines • Volume changesDV are opposites • DV = A1Dx1, soDx1=DV/A1;Dx2=–DV/A2 F1Dx1 = –F2Dx2 • input, output work equal and opposite

  19. Example Problem In an auto shop a hydraulic jack is used to lift vehicles for service. If the radius of the cylinder below piston 1 is r1 = 0.02 m, and the radius of the cylinder below the piston holding the car is r2 = 0.1 m, what force F1 must be exerted on the small piston to lift a 1500-kg car?

  20. Pressure with Depth

  21. Supports weight above Pressure beneath a fluid • Force comes from weight of fluid above • Pressure increases with depth

  22. weight of fluid above cross-section = mass  g h A = density  volume  g A = rAhg A Pressure in a liquid p = F/A = = rgh p = pressure here

  23. h Depth Pressure Formula p = p0 + rgh p0 = pressure here • p0 = pressure at depth 0 • p = pressure at depth h • r = constant density of liquid • h = depth under top of liquid p = pressure here

  24. Buoyancy How do things float? § 12.3

  25. What forces are present?

  26. What forces are present?

  27. What forces are present?

  28. What forces are present?

  29. Pressure in a fluid Pressure increases with depth Greater pressure at bottom than top of an immersed object Results in upward buoyancy force that is the (vector) sum of all pA forces F =  p dA A

  30. Buoyancy from Ug Force = –dUg/dh Ug = Ug obj + Ug displaced fluid DUg = mg(Dh) + rVg(–Dh) ∑F = –dUg/dh = –mg + rVg –mg = gravity force rVg = buoyancy force Dh

  31. Quick example The density of fresh water r = 1000 kg/m3. What is the buoyancy force on a 1-m3 parcel of water? What is the buoyancy force on a 1-m3 rock submerged under water? What is the buoyancy force on a 1-m3 chunk of Styrofoam submerged under water?

  32. Quick Questions All blocks are cubes with 1-m sides. Which ones float? Which one makes the water level change the most? Which one makes the water level change the least?

  33. Buoyancy and Density If an object is more dense than the surrounding fluid, it sinks If an object is less dense than the surrounding fluid, it rises A floating object displaces exactly enough fluid to support its weight

  34. Quick Question Which boat (with cargo) has the greatest volume? The high boat (left). The low boat (right). Both have the same volume.

  35. Quick Question Which boat (with cargo) has the greatest volume under the water line? The high boat (left). The low boat (right). Both have the same volume under water.

  36. Quick Question Which boat (with cargo) experiences the greatest buoyancy force? The high boat (left). The low boat (right). Both experience the same buoyancy force.

  37. Quick Question Which boat (with cargo) has the greatest mass? The high boat (left). The low boat (right). Both have the same mass.

  38. Poll Question After a boat sinks, it displaces more waterthan when it floated. less water than when it floated. the same volume of water as when it floated.

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