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Transcript
AP Physics B: Ch.11 - Fluid Mechanics. Subdivisions of matter solids liquids gases rigid will flow will flow fluids dense and incompressible dense and incompressible condensed matter Q: what about thick liquids and soft solids? low density compressible Fluid mechanics Ordinary mechanics Mass and force identified with objects Fluid mechanics Mass and force “distributed” Density and Pressure Density r for element of fluid m r V mass M volume V m r V for uniform density mass M volume V units kg m-3 Density and Pressure Pressure p force per unit area F p A for uniform force F p A units N m-2 or pascals (Pa) Atmospheric pressure at sea level p0 on average 101.3 x103 Pa or 101.3 kPa Gauge pressure pg excess pressure above atmospheric p = pg + p0 Density and Pressure Gauge pressure pg pressure in excess of atmospheric gauge atmospheric total p = pg + p0 typical pressures total gauge atmospheric 1.0x105 Pa 0 car tire 3.5x105 Pa 2.5x105 Pa deepest ocean 1.1x108 Pa 1.1x108 Pa best vacuum 10-12 Pa - 100 kPa Example to pump 30 cms 15 cms The can shown has atmospheric pressure outside. The pump reduces the pressure inside to 1/4 atmospheric • What is the gauge pressure inside? • What is the net force on one side? Fluids at rest (hydrostatics) Hydrostatic equilibrium laws of mechanical equilibrium pressure just above surface is atmospheric, p0 hence, pressure just below surface must be same, i.e. p0 surface is in equilibrium Fluids at rest (hydrostatics) Hydrostatic equilibrium element of fluid surface area A height y laws of mechanical equilibrium S Fy =0 pA - (p+p)A - mg = 0 -p A - rAyg = 0 (p+p)A p =- rgy y Pressure at depth h at distance h below surface, pressure is larger by rgh p = p0+rgh pA mg = rAyg Question How far below surface of water must one dive for the pressure to increase by one atmosphere? What is total pressure and what is the gauge pressure, at this depth? ? Pascal’s principle The pressure at a point in a fluid in static equilibrium depends only on the depth of that point Pascal’s principle The pressure at a point in a fluid in static equilibrium depends only on the depth of that point Open tube manometer (i) If h=6 cm and the fluid is mercury (r=13600 kg m-3) find the gauge pressure in the tank (ii) Find the absolute pressure if p0 =101.3 kPa Pascal’s principle The pressure at a point in a fluid in static equilibrium depends only on the depth of that point Barometer Find p0 if h=758 mm Pascal’s principle The pressure at a point in a fluid in static equilibrium depends only on the depth of that point A change in the pressure applied to an enclosed incompressible fluid is transmitted to every point in the fluid Hydraulic press p Fi Fo Ai Ao Fo Fi Ao Ai alternative argument based on conservation of energy Fo do Fidi work out = work in Ao di Ai do Ao di = do Ai volume moved is same on each side Ao Ao Fo do Fi do Fo Fi Ai Ai Archimedes’s principle When a body is fully or partially submerged in a fluid, a buoyant force from the surrounding fluid acts on the body. The force is upward and has a magnitude equal to the weight of fluid displaced. Fg imagine a hole in the water-a buoyancy force exists fill it with fluid of mass mf and equilibrium will exist Fb=mfg stone more dense than water so sinks Fb Fg wood less dense than water so floats now the water displaced is less -just enough buoyancy force to balance the weight of the wood Fb=Fg Flotation volume immersed Vi Fb For object of uniform density r Fb=Fg rfluid Vi g= r V g Vi/V = r/rfluid Fg Example 1 What fraction of an iceberg is submerged? (rice for sea ice =917 kg m-3 and rsea for sea water = 1024 kg m-3) Example 2 A “gold” statue weighs 147 N in vacuum and 139 N when immersed in salt water of density 1024 kg m-3 . What is the density of the “gold”? total volume V Fluid Dynamics The study of fluids in motion. turbulent laminar Ideal Fluid 1. Steady flow Velocity of the fluid at any point fixed in space doesn’t change with time. This is called “ laminar flow”, and for such flow the fluid follows “streamlines”. 2. Incompressible We will assume the density is fixed. Accurate for liquids but not so likely for gases. 3. Inviscid “Viscosity” is the frictional resistance to flow. Honey has high viscosity, water has small viscosity. We will assume no viscous losses. Our approach will only be true for low viscosity fluids. Equation of continuity Streamlines tube of flow Conservation of mass in tube of flow means mass of fluid entering A1 in time t = mass of fluid leaving A2 in time t For incompressible fluid this means volume is also conserved. Volume entering and leaving in time t is V V = A1 v1 t =A2 v2 t Therefore A1 v1 = A2 v2 Equation of continuity (Streamline rule) Bernoulli’s equation (Daniel Bernoulli, 1700-1782) 1 1 p1 2 rv12 rgy1 p2 2 rv22 rgy2 1 p 2 rv 2 rgy constant For special case of fluid at rest (Hydrostatics!) p2 p1 rg(y2 - y1 ) For special case of height constant (y1=y2) 1 p1 2 r v12 1 p2 2 rv2 2 The pressure of a fluid decreases with increasing speed Proof of Bernoulli’s equation Note: same volume V with mass m enters A1 as leaves A2 in time t Work done at A1 in time t (p1A1)v1 t =p1 V Use work energy theorem work done by external force (pressure) =change in KE + change in PE W=K+ U Work done W p1V - p2 V -(p2 - p1 )V Change in KE 1 1 1 1 K 2 mv2 2 - 2 mv12 2 rVv22 - 2 rVv12 Change in PE U r Vgy2 - r Vgy1 Problem Titanic had a displacement of 43 000 tonnes. It sank in 2.5 hours after being holed 2 m below the waterline. Calculate the total area of the hole which sank Titanic. Examples of Bernoulli’s relation at work Venturi meter Aircraft lift Examples of Bernouilli’s relation at work “spin bowling”