Transcript Physics 9
Physics Chapter 9: Fluid Mechanics Fluids Fluids Definition - Materials that Flow Liquids Definite Volume Non-Compressible Gasses No Definite Volume Compressible Fluids Mass Density Mass / Unit Volume Symbol r (rho) m = mass (kg) V = volume (m3) m r V Fluids Mass Density Density of Water (@ 40C) 1.0 g/cm3 = 1.0 g/cc = 1.0 g/mL 1.0L = 1x10-3 m3 3 1.0 g / mL 1kg / L 1kg / 10 m 3 1x10 kg / m 3 3 Fluids Mass Density Density of Other Materials Page 319 Table 9-1 Gasses Liquids Solids Fluids Archimedes’ Principle Any Fluid Applies a Buoyant Force to an Object that is partially or Completely Immersed in it; The Magnitude of the Buoyant Force Equals the Weight of the Fluid that the Object Displaces. Fbuoyant W fluid Fluids Archimedes’ Principle Apparent Weight The Weight of an Object You “Feel” The Net Force Acting on a Submerged or Partially Submerged Object Wapparent Fg Fbuoyant Fluids Archimedes’ Principle Fbuoyant Wdisplacedfluid m fluid g m fluid rV Fbuoyant r fluidVg Fluids Archimedes’ Principle If Fg > Fbuoyant the Object Will Sink If Fg < Fbuoyant the Object Will Float If Fg = Fbuoyant the Object Will be Neutrally Buoyant Fbuoyant rVg Fluids Archimedes’ Principle For a Floating Object the Buoyant Force is Equal to the Weight of the Object Fbuoyant mobject g Wobject Fbuoyant Wdisplacedfluid m fluid g rgV mobject g Fbuoyant r fluid gV r object gV Fg Fluids Archimedes’ Principle Fbuoyant r fluid gV r object gV Fg Object Volume and Gravity are Constant r object Fg r fluid Fbuoyant Fluids Pressure Force Acting Perpendicular to a Surface Divided by the Area of the Surface Units – N/m2 = Pascal (Pa) F P A Fluids Pressure Pascal 1Pa = 1N/m2 Very Small Quantity F P A ATM 1.013x105 Pa = 1.013 bar = 14.70 lb/in2 Fluids Pascal’s Principle Any Change in the Pressure Applied to a Completely Enclosed Fluid Is Transmitted Undiminished to All Parts of the Fluid and the Enclosing Walls. P1 P2 F1 F2 A1 A2 Fluids Pascal’s Principle F1 F2 A1 A2 A2 F2 F1 A1 Fluids Pressure and Depth Fluid, at a Given Depth, has Pressure Applied to It by the Fluid Above It F P A Fluids Pressure and Depth F mg rgV rgAh P rgh A A A A P2 P1 rgh Fluids Pressure and Depth F1 P1 A F2 P2 A Fluids Pressure and Depth F2 F1 F2 F1 mg Fluids Pressure and Depth Pnet P2 P1 rhg Fnet Pnet A rVg m fluid g Fluids Pressure and Depth Air is a Fluid Atmospheric Pressure Works by the Same Principles Fluids Problem Neutron stars consist only of neutrons and have very high densities. A typical mass and radius of a neutron star is 2.7x1028kg and 1.2x103m. What is the density of this star? Fluids Solution m = r = 1.2x103m 2.7x1028kg 4 3 V r 3 m m 3m 3(2.7 x10 28 kg) 18 3 r 3 . 73 x 10 kg / m V 4 r 3 4r 3 4 (1.2 x103 m)3 3 Fluids Problem A paperweight weighs 6.9N in air, but when immersed in water weighs 4.3N. What is the volume of the paperweight? Fluids Wapparent Fnet Fbuoyant Fg Solution W = 6.9N Wapparent = 4.3N Wapparent Fbuoyant Fg Fbuoyant Fg Wapparent rVg V Fg Wapparent rg 6.9 N 4.3N 4 3 2.7 x10 m 3 3 2 (1x10 kg / m )(9.81m / s ) Fluids Problem In a car lift the output plunger has a radius of 9.0cm. The weight of the plunger and the car is 21,600N. What is the gage pressure of the hydraulic oil used in operating the lift? Fluids Solution F P A F = r = 9.0cm = 0.09m 2.16x104N 4 F 2.16 x10 N 5 P 2 8 . 5 x 10 Pa 2 r (0.09m) Fluids Problem A dentist chair with a patient in it weighs 2100N. The output plunger of a hydraulic system begins to lift the chair when the dentist’s foot applies a force of 55N to the input piston. Neglect any height difference between the plunger and the piston. What is the ratio of the radius of the plunger to the radius of the piston? Fluids Solution F1 = 55N F2 = 2100N A2 F2 F1 A1 r22 F2 r22 F2 2 2 F1 r1 F1 r1 r F2 r2 2100 N r r F 55 N 1 1 2 2 2 1 6.2 Fluids Problem A spring (k=1600N/m) is attached to the input piston of a hydraulic chamber and a rock with a mass of 40.0kg rests on the output plunger. The piston and the plunger are nearly at the same height and each has negligible mass. By how much is the spring compressed from its unstrained position? Fluids Solution k = 1.6x103N/m A2 = 15cm2 A1 = 65cm2 A2 mr = 40.0kg F1 kx A1 A2 F2 F1 A1 F2 kx A2 mg kx A1 mg A2 (40.0kg)(9.8m / s 2 ) 15cm2 2 x 5 . 7 x 10 m 3 2 k A1 1.6 x10 N / m 65cm Fluids Homework Pages 343 – 344 Problems 8 (a, 6.3x103kg/m3 b, 9.2x102kg/m3) 16 (1.9x104N) 18 (14N Downward) 19 (a, 2.6x106Pa b, 1.8x105N) Fluids Fluids in Motion Fluid Flow Laminar Flow (Streamline Flow) Constant Flow of Fluid Particles All Particles have Same Velocity Turbulent Flow Velocity Changes with Time Due to Obstructions or Openings Fluids Fluids in Motion Fluid Flow Turbulent Flow Direction of Flow Eddy Currents Fluids Fluids in Motion Fluid Flow Compressible Gasses Incompressible Liquids Some Gasses Under Specific Circumstances Fluids Fluids in Motion Viscosity (Resistance to Flow) Viscous Inefficient Use of Energy Non-viscous Efficient Use of Energy Fluids Fluids in Motion Fluid Flow Ideal Fluid Incompressible Fluid with Zero Viscosity Fluids Fluids in Motion Equation of Continuity Initial Flow Rate = Final Flow Rate m1 m2 Fluids Fluids in Motion Equation of Continuity Mass Flow Rate m rV V Ax r1 A1v1t r2 A2v2t x vt Fluids Fluids in Motion Equation of Continuity Mass Flow Rate r1 A1v1t r2 A2v2t Assuming Same Density and Time A1v1 A2v2 Fluids Fluids in Motion A1v1 A2v2 Fluids Fluids in Motion Bernoulli’s Principle The Pressure in a Fluid Decreases as the Fluid’s Velocity Increases Fluids Bernoulli’s Equation Pressure Drops with Decreased Area Fluids Bernoulli’s Equation Pressure Drops with Increased Height Fluids Bernoulli’s Equation In the steady flow of a nonviscous, incompressible fluid of density r, the pressure P, the fluid speed v, and the elevation y at any two points are related. 1 2 P rv rvh constant 2 Fluids Bernoulli’s Equation 1 2 1 2 P1 rv1 rvh1 P2 rv2 rvh2 2 2 For Static Fluids P2 P1 rgh Fluids Bernoulli’s Equation For Moving Fluids with No Change in Elevation 1 2 1 2 P1 rv1 P2 rv2 2 2 Fluids Fluids in Motion 1 2 1 2 P1 rv1 rvh1 P2 rv2 rvh2 2 2 A1v1 A2v2 Fluids Gasses The Ideal Gas Law Pressure Volume Number of Particles Boltzmann’s Constant (kB) = 1.38x10-23J/K Temperature (in Kelvin) PV nkBT Fluids Gasses The Ideal Gas Law PV nkBT If Number of Particles is Constant P1V1 P2V2 T1 T2 Fluids Problem A dump truck traveling at 27m/s has its load covered by a tarp. By how much does the pressure inside the cargo area beneath the tarp exceed the outside pressure? Fluids Solution rair = 1.29kg/m3 v1 = 0m/s v2 = 27m/s 1 2 1 2 P1 rv1 P2 rv2 2 2 1 2 2 P1 P2 r v2 v1 2 1 2 3 P1 P2 (1.29kg / m )27m / s 470 Pa 2 Fluids Problem Laura can fill a bucket from a water hose in 30.0s. If she covers up part of the hose’s opening with her thumb the velocity of the water doubles. How long will it take Laura to fill the bucket now? Fluids Solution t1 = 30.0s v2 = 2v1 r1 A1v1t r2 A2v2t A1v1t1 A2v2t2 1 A1v1t1 A2 2v1t 2 2 t1 t2 Fluids Problem The water supply of a building is fed through a main pipe that is 6.0cm in diameter. A 2.0cm diameter faucet tap is positioned 2.00m above the main pipe and can fill a 2.5x10-2m3 container in 30.0s. What is the velocity that the water leaves the faucet? Fluids Solution r1 = 3.0cm r2 = 1.0cm h2-h1 = 2.00m r = 1000kg/m3 t = 30.0s V = 0.025m3 Flow Rate A2v2 V A2 v2 t V V v2 2 A2t r2 t 3 0.025m v2 2.7m / s 2 (0.01m) (30.0s) Fluids Problem The water supply of a building is fed through a main pipe that is 6.0cm in diameter. A 2.0cm diameter faucet tap is positioned 2.00m above the main pipe and can fill a 2.5x10-2m3 container in 30.0s. What is the gage pressure in the main pipe (difference in pressure)? Fluids Solution r1 = 3.0cm r2 = 1.0cm h2-h1 = 2.00m r = 1000kg/m3 t = 30.0s V = 0.025m3 v2 = 2.7m/s A1v1 A2v2 A2 v2 v1 0.3m / s A1 1 2 1 2 P1 rv1 rvh1 P2 rv2 rvh2 2 2 1 2 1 2 P1 P2 r v2 v1 g h2 h1 2 2 P1 P2 2.32 x10 Pa 4 Fluids Problem The pressure on an ideal gas is cut in half, resulting in a decrease in temperature to ¾ of the original value. What is the ratio of the final volume to the original volume of the gas? Fluids Solution P2 = ½ P1 T2 = ¾ T1 P1V1 P2V2 T1 T2 V2 P1T2 V1 T1 P2 3 P1 T1 V2 4 6 1.5 V1 1 4 T1 P1 2 Fluids Homework Page 344 - 345 Problems 24 (12.6m/s) 29 (474K) 30 (21.3K)