PHYS 1443 – Section 501Lecture #24AnnouncementsPascal’s Law and HydraulicsBuoyant Forces and Archimedes’ PrincipleFlow Rate and the Equation of ContinuityExample for Equation of ContinuityBernoulli’s EquationBernoulli’s Equation cont’dBernoulli’s Equation cont’dExample for Bernoulli’s EquationSimple Harmonic MotionEquation of Simple Harmonic MotionMore on Equation of Simple Harmonic MotionSimple Harmonic Motion continuedExample for Simple Harmonic MotionPHYS 1443 – Section 501Lecture #24Monday, April 26, 2004Dr. Andrew Brandt1. Fluid Dymanics : Flow rate and Continuity Equation2. Bernoulli’s Equation3. Simple Harmonic Motion4. Simple Block-Spring SystemMonday April 26, 2004 1PHYS 1443-501, Spring 2004Dr. Andrew BrandtAnnouncements• The last HW on Ch. 12+13 is due Weds. 4/28 (decided not to have HW during last week)• An optional take-home quiz to replace your lowestquiz is due tonight• No optional sections (*) in Ch 11, 12, 13Explicitly, covered 11.1-5,7 12.1-6 13.1-8 14.1-5• We’ll stop class a little early for teaching evaluationsMonday April 26, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt2Pascal’s Law and HydraulicsA change in the pressure applied to a fluid is transmitted undiminished to every point of the fluid and to the walls of the container.The resultant pressure P at any given depth h increases as much as the change in P0. Monday April 26, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt3This is the principle behind hydraulic pressure. Therefore, the resultant force F2isWhat happens if P0is changed?AFP ≡2/11 mNPa ≡VM≡ρghPPρ+=0PSince the pressure change caused by the the force F1applied on to the area A1is transmitted to the F2on an area A2.d1d2A2F22FIn other words, the initial force multiplied by the ratio of the areas A2/A1is transmitted to F2on the surface.Note the actual displaced volume of the fluid is the same and the work done by the forces are still the same.2F11AF=22AF=121Fdd=112FAA=F1A1Buoyant Forces and Archimedes’ PrincipleWhy is it so hard to put a beach ball under water while a piece of small steel sinks in the water?The water exerts a force on an object immersed in the water. This force is called Buoyant force.The magnitude of the buoyant force always equals the weight of the volume of fluid displaced by the submerged object.How does the Buoyant force work?This is called Archimedes’ principle.Let’s consider a cube whose height is h and is filled with fluid and at its equilibrium. Then the weight Mg is balanced by the buoyant force B.BMghgF=Mg=And the pressure at the bottom of the cube is larger than the top by ρgh.BTherefore,P∆AB /=ghρ=BPA∆=ghAρ=Vgρ=BgF=Vgρ=Mg=Where Mg is the weight of the fluid.Monday April 26, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt4Flow Rate and the Equation of ContinuityStudy of fluid in motion: Fluid DynamicsHydro-dynamics If the fluid is water: Monday April 26, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt5•Streamline or Laminar flow: Each particle of the fluid follows a smooth path, a streamline•Turbulent flow: Erratic, small, whirlpool-like circles called eddy current or eddies which absorb a lot of energyTwo main types of flowFlow rate: the mass of fluid that passes a given point per unit time/mt∆∆since the total flow must be conserved1mt∆=∆11Vtρ∆=∆11 1Altρ∆=∆111Avρ111 2 22AvAvρρ=12mmtt∆∆=∆∆Equation of ContinuityExample for Equation of ContinuityHow large must a heating duct be if air moving at 3.0m/s along it can replenish the air every 15 minutes in a room of 300m3volume? Assume the air’s density remains constant.111 2 22Av AvUsing equation of continuityρρ=Since the air density is constant11 2 2Av Av=Now let’s view the room as a large section of the duct2211AvAv==221/Altv=21Vvt=⋅23000.113.0 900m=×Monday April 26, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt6Bernoulli’s EquationMonday April 26, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt7Bernoulli’s Principle: Where the velocity of fluid is high, the pressure is low, and where the velocity is low, the pressure is high. Amount of work done by the force, F1, that exerts pressure, P1, at point 11W=Work done by the gravitational force to move the fluid mass, m, from y1to y2is11Fl∆=11 1PAl∆Amount of work done on the other section of the fluid is2222WPAl=−∆()321Wmgyy=− −Bernoulli’s Equation cont’dThe net work done on the fluid is123WWWW=++11 1 2 2 2 2 1PAl PAl mgymgy=∆− ∆− +From the work-energy principle22211122mv mv−=11 1 2 2 2 2 1PAl PAl mgy mgy∆−∆− +Since mass, m, is contained in the volume that flowed in the motion11 2 2mAl Alρρ=∆= ∆11 2 2AlAl∆= ∆and222 112121122vvAlAlρρ−∆∆Thus,11 2 2 2 2 1 1112 2Al Al APP gyl gyAlρρ=− − +∆∆∆ ∆Monday April 26, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt8Bernoulli’s Equation cont’dSince22 11 11 22 22 1222112 2111122Al Al Al Al AlvvPP gAlygyρρ ρρ−=−−+∆∆∆∆∆∆211112Pvgyρρ++=Bernoulli’s Equation21111.2Pvgycon s tρρ++=222212Pvgyρρ++222112211122v v P P gy gyρρ ρρ−=−−+We obtainRe-organizeThus, for any two points in the flow()21 12 1PP gyy P ghρρ=+ − =+Pascal’s LawFor static fluid()2221 1212PP vvρ=+ −For the same heightsMonday April 26, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt9The pressure at the faster section of the fluid is smaller than slower section.Example for Bernoulli’s EquationWater circulates throughout a house in a hot-water heating system. If the water is pumped at a speed of 0.5m/s through a 4.0cm diameter pipe in the basement under a pressure of 3.0atm, what will be the flow speed and pressure in a 2.6cm diameter pipe on the second floor 5.0m above? Assume the pipes do not divide into branches.Using the equation of continuity, flow speed on the second floor is112AvA=21122rvrππ=20.0200.5 1.2 /0.013ms⎛⎞×=⎜⎟⎝⎠2v =Using Bernoulli’s equation, the pressure in the pipe on the second floor isMonday April 26, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt10()()2221 12 1212PP vv gyyρρ=+ − + −()()5322313.0 10 1 10 0.5 1.2 1 10 9.8 52=×+× − +×××−522.5 10 /Nm=×Simple Harmonic MotionHarmonic Motion is motion that occurs due to a force that depends on displacement, with the force always directed toward the system’sequilibrium position.kx−=FWhen a spring is stretched from its equilibrium position by a length x, the force acting on the mass is What is a
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