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Physics 106 Final Exam When? May. 10 Tuesday, 2:30 — 5:00 pm Duration: 2.5 hours Where? ECEC-100 (SEC-008, 10) What? Phys-106 (75%), Phys-105 (25%) How? Review sessions (today’s lecture and next week’s recitation) Equation sheet Prof. Janow review session on Monday May 9, 3:00-5:00 pm in THL-2 Sample exams on my website: web.njit.edu/~cao/106 What if? 28 multiple choice problems (2.5hr/28 ~ 5 min/prob) 24 correct answers yields a score of 100% Today follows Thursday schedule: Chap. 14.1-14.5 HW13 due by 11:00 pm on May 10 Copyright R. Janow Spring 2010 Physics 106 Lecture 13 Fluid Mechanics SJ 8th Ed.: Chap 14.1 to 14.5 • • • • • • • • • • • What is a fluid? Pressure Pressure varies with depth Pascal’s principle Methods for measuring pressure Buoyant forces Archimedes principle Fluid dynamics assumptions An ideal fluid Continuity Equation Bernoulli’s Equation Copyright R. Janow Spring 2010 What is a fluid? Solids: strong intermolecular forces definite volume and shape rigid crystal lattices, as if atoms on stiff springs deforms elastically (strain) due to moderate stress (pressure) in any direction Fluids: substances that can “flow” no definite shape molecules are randomly arranged, held by weak cohesive intermolecular forces and by the walls of a container liquids and gases are both fluids Liquids: definite volume but no definite shape often almost incompressible under pressure (from all sides) can not resist tension or shearing (crosswise) stress no long range ordering but near neighbor molecules can be held weakly together Gases: neither volume nor shape are fixed molecules move independently of each other comparatively easy to compress: density depends on temperature and pressure Fluid Statics - fluids at rest (mechanical equilibrium) R. Janow Spring 2010 Fluid Dynamics – fluid flow (continuity, energy Copyright conservation) Mass and Density • Density is mass per unit volume at a point: m V • • or m V • scalar • units are kg/m3, gm/cm3.. • water= 1000 kg/m3= 1.0 gm/cm3 Volume and density vary with temperature - slightly in liquids The average molecular spacing in gases is much greater than in liquids. Copyright R. Janow Spring 2010 Force & Pressure • The pressure P on a “small” area A is the ratio of the magnitude of the net force to the area F P or P F / A A F PA PAn̂ • • PA n̂ Pressure is a scalar while force is a vector The direction of the force producing a pressure is perpendicular to some area of interest • h At a point in a fluid (in mechanical equilibrium) the pressure is the same in any direction Pressure units: 1 Pascal (Pa) = 1 Newton/m2 (SI) 1 PSI (Pound/sq. in) = 6894 Pa. 1 milli-bar = 100 Pa. Copyright R. Janow Spring 2010 Forces/Stresses in Fluids • Fluids do not allow shearing stresses or tensile stresses. Tension • • Shear Compression The only stress that can be exerted on an object submerged in a static fluid is one that tends to compress the object from all sides The force exerted by a static fluid on an object is always perpendicular to the surfaces of the object Question: Why can you push a pin easily into a potato, say, using very little force, but your finger alone can not push into the skin even if you push very hard? Copyright R. Janow Spring 2010 Pressure in a fluid varies with depth Fluid is in static equilibrium The net force on the shaded volume = 0 y=0 Incompressible liquid - constant density • • • Horizontal surface areas = A Forces on the shaded region: – Weight of shaded fluid: – – Mg Downward force on top: F1 =P1A Upward force on bottom: F2 = P2A y1 F1 P1 h F2 y2 P2 Mg Fy 0 P2A P1A Mg • In terms of density, the mass of the shaded fluid is: The extra pressure at extra depth h is: M V Ah P P2 P1 gh P2 A P1A ghA h y1 y2 Copyright R. Janow Spring 2010 Pressure relative to the surface of a liquid air Example: The pressure at depth h is: P0 Ph P0 gh P0 is the local atmospheric (or ambient) pressure Ph is the absolute pressure at depth h The difference is called the gauge pressure All points at the same depth are at the same pressure; otherwise, the fluid could not be in equilibrium The pressure at depth h does not depend on the shape of the container holding the fluid P0 h Ph liquid Preceding equations also hold approximately for gases such as air if the density does not vary much across h Ph Copyright R. Janow Spring 2010 Atmospheric pressure and units conversions • P0 is the atmospheric pressure if the liquid is open to the atmosphere. • Atmospheric pressure varies locally due to altitude, temperature, motion of air masses, other factors. • Sea level atmospheric pressure P0 = 1.00 atm = 1.01325 x 105 Pa = 101.325 kPa = 1013.25 mb (millibars) = 29.9213” Hg = 760.00 mmHg ~ 760.00 Torr = 14.696 psi (pounds per square inch) Pascal (Pa) 1 Pa ≡ 1 N/m bar (bar) 2 10 −5 atmosphere (atm) 9.8692×10 −6 torr (Torr) pound-force per square inch (psi) − 7.5006×10 3 −6 145.04×10 1 bar 100,000 ≡ 106 dyn/cm2 0.98692 750.06 14.5037744 1 atm 101,325 1.01325 ≡ 1 atm 760 14.696 1 torr 133.322 1.3332×10−3 1.3158×10− 3 ≡ 1 Torr; ≈ 1 mmHg 19.337×10−3 1 psi 6.894×103 68.948×10−3 68.046×10−3 51.715 ≡ 1 lbf/in2 Copyright R. Janow Spring 2010 Pressure Measurement: Barometer • • near-vacuum • • Invented by Torricelli (1608-47) Measures atmospheric pressure P0 as it varies with the weather The closed end is nearly a vacuum (P = 0) One standard atm = 1.013 x 105 Pa. P0 Hggh Mercury (Hg) How high is the Mercury column? P0 1.013 105 Pa h 0.760 m Hgg (13.6 103 kg / m3 )(9.80 m/s 2 ) One 1 atm = 760 mm of Hg = 29.92 inches of Hg How high would a water column be? h P0 waterg 1.013 105 Pa (1.0 10 kg / m )(9.80 m/s ) 3 3 2 10.34 m Height limit for a suction pump Copyright R. Janow Spring 2010 Pascal’s Principle A change in the pressure applied to an enclosed incompressible fluid is transmitted undiminished to every point of the fluid and to the walls of the container. F P Example: open container • p0 The pressure in a fluid depends on depth h and on the value of P0 at the surface All points at the same depth have the same pressure. • ph Ph P0 gh • Add piston of area A with lead balls on it & weight W. Pressure at surface increases by P = W/A Pext P0 P • Ph Pressure at every other point in the fluid (Pascal’s law), increases by the same amount, including all locations at depth h. Ph Pext ghCopyright R. Janow Spring 2010 Pascal’s Law Device - Hydraulic press A small input force generates a large output force • • Assume the working fluid is incompressible Neglect the (small here) effect of height on pressure • The volume of liquid pushed down on the left equals the volume pushed up on the right, so: A1x1 A2x2 x 2 A1 x 1 A2 • Other hydraulic lever devices using Pascal’s Law: • Squeezing a toothpaste tube • Hydraulic brakes • Hydraulic jacks • Forklifts, backhoes Assume no loss of energy in the fluid, no friction, etc. Work1 F1x1 Work 2 F2x2 mechanical advantage F2 x1 A2 F1 Copyright x 2 R. A 1 Spring 2010 Janow Archimedes Principle • • • • C. 287 – 212 BC Greek mathematician, physicist and engineer Computed p and volumes of solids Inventor of catapults, levers, screws, etc. Discovered nature of buoyant force – Eureka! Why do ships float and sometimes sink? Why do objects weigh less when submerged in a fluid? identical pressures at every point ball of liquid in equilibrium • hollow ball same upward force An object immersed in a fluid feels an upward buoyant force that equals the weight of the fluid displaced by the object. Archimedes’s Principle – The fluid pressure increases with depth and exerts forces that are the same whether the submerged object is there or not. – Buoyant forces do not depend on the composition of submerged objects. – Buoyant forces depend on the density of the liquid and g. Copyright R. Janow Spring 2010 Archimedes’s principle - submerged cube A cube that may be hollow or made of some material is submerged in a fluid Does it float up or sink down in the liquid? • The extra pressure at the lower surface compared to the top is: P Pbot Ptop flgh • • • The pressure at the top of the cube causes a downward force of Ptop A The larger pressure at the bottom of the cube causes an upward force of Pbot A The upward buoyant force B is the weight of the fluid displaced by the cube: B PA flghA flgV Mflg • The weight of the actual cube is: Fg Mcube g cube gV The cube rises if B > Fg (fl > cube) The cube sinks if Fg> B (cube > fl) Similarly for irregularly shaped objects cubeCopyright is the R. average density Janow Spring 2010 Archimedes's Principle: totally submerged object Object – any shape - is totally submerged in a fluid of density fluid The upward buoyant force is the weight of displaced fluid: B fluid gVfluid The downward gravitational force on the object is: Fg Mobject g object gVobject The volume of fluid displaced and the object’s volume are equal for a totally submerged object. The net force is: Fnet B Fg [ fluid object ] g Vobject The direction of the motion of an object in a fluid is determined only by the densities of the fluid and the object – – If the density of the object the density of the fluid, the object accelerates upward If the density of the object the density of the fluid, the object sinks is less than unsupported The apparent weight is the external force needed to restore equilibrium, i.e. is more than unsupported Wapparent Fg B Fnet Copyright R. Janow Spring 2010 Archimedes’s Principle: floating object An object sinks or rises in the fluid until it reaches equilibrium. The fluid displaced is a fraction of the object’s volume. At equilibrium the upward buoyant force is balanced by the downward weight of the object: Fnet B Fg 0 B Fg The volume of fluid displaced Vfluid corresponds to the portion of the object’s volume below the fluid level and is always less than the object’s volume. Equate: B fluid gVfluid Fg object gVobject Solving: fluid Vfluid object Vobject object Vfluid Vobject fluid Objects float when their average density is less than the density of the fluid they are in. The ratio of densities equals the fraction of the object’s volume that is below the surface Copyright R. Janow Spring 2010 Example: What fraction of an iceberg is underwater? Apply Archimedes’ Principle displaced seawater object Vunderwater Vfluid Vtotal Vobject fluid glacial fresh water ice From table: object ice 0.917 103 kg/m3 fluid seawater 1.03 103 kg/m3 Water expands when it freezes. If not.... ...ponds, lakes, seas freeze to the bottom in winter Vunderwater 89.03% Vtotal What if iceberg is in a freshwater lake? fluid freshwater 1.00 103 kg/m3 Vunderwater 91.7% Vtotal Floating objects are more buoyant in saltwater Freshwater tends to float on top of seawater... Copyright R. Janow Spring 2010 Fluids’ Flow is affected by their viscosity • Viscosity measures the internal friction in a fluid. Low viscosity • gases • • Medium viscosity • water • other fluids that pour and flow easily High viscosity • honey • oil and grease • glass Viscous forces depend on the resistance that two adjacent layers of fluid have to relative motion. Part of the kinetic energy of a fluid is converted to internal energy, analogous to friction for sliding surfaces Ideal Fluids – four approximations to simplify the analysis of fluid flow: • • • • The fluid is nonviscous – internal friction is neglected The flow is laminar (steady, streamline flow) – all particles passing through a point have the same velocity at any time. The fluid is incompressible – the density remains constant The flow is irrotational – the fluid has no angular momentum about any point. A small paddle wheel placed anywhere does not feel a torque and rotate Copyright R. Janow Spring 2010 Flow of an ideal fluid through a short section of pipe Constant density and velocity within volume element dV Incompressible fluid means d/dt = 0 Mass flow rate = amount of mass crossing area A per unit time = a “current”sometimes called a “mass flux” cross-section area A velocity v volume of fluid in cylinder Imass mass flow rate A length dx dx Av dt dV Adx dM d (V ) dt dt Imass mass flow rate Av Ivol volume flow rate Av Jmass mass flow/unit area v Copyright R. Janow Spring 2010 Equation of Continuity: conservation of mass • • • • An ideal fluid is moving through a pipe of nonuniform diameter The particles move along streamlines in steady-state flow The mass entering at point 1 cannot disappear or collect in the pipe The mass that crosses A1 in some time interval is the same as the mass that crosses A2 in the same time interval. mass flow in 1A1v1 • mass flow out 2 A2v2 The fluid is incompressible so: 2 1 2 a cons tan t A1v1 A2v2 • • This is called the equation of continuity for an incompressible fluid The product of the area and the fluid speed (volume flux) at all points along a pipe is constant. 1 The rate of fluid volume entering one end equals the volume leaving at the other end Where the pipe narrows (constriction), the fluid moves faster, and vice versa Copyright R. Janow Spring 2010