Phys I - KSIU Fall 2023 Lecture 6: Fluid Mechanics - PDF

Field of Basic Sciences All Programs Lecture 6 : (Physics I (PHY111): Fluid Mechanics) Essential books: Raymond A. Serway, John W. Jewett, Jr. Physics for Scientists and Engineers with Modern Physics, eighth edition, 2010. Halliday, Resnick, Walker Fundamentals of Physics Dr. Sh...

Field of Basic Sciences All Programs Lecture 6 : (Physics I (PHY111): Fluid Mechanics) Essential books: Raymond A. Serway, John W. Jewett, Jr. Physics for Scientists and Engineers with Modern Physics, eighth edition, 2010. Halliday, Resnick, Walker Fundamentals of Physics Dr. Shehab E. Ali Date : 21/ 11 / 2023 1 Chapter 14 Fluid Mechanics 14.1 Pressure 14.2 Variation of Pressure with Depth 14.3 Pressure Measurements 14.4 Buoyant Forces and Archimedes’s Principle States of Matter Solid Has a definite volume and shape Liquid Has a definite volume but not a definite shape Gas – unconfined Has neither a definite volume nor shape States of Matter, cont All of the previous definitions are somewhat artificial More generally, the time it takes a particular substance to change its shape in response to an external force determines whether the substance is treated as a solid, liquid or gas Fluids A fluid is a collection of molecules that are randomly arranged and held together by weak cohesive forces and by forces exerted by the walls of a container Both liquids and gases are fluids Statics and Dynamics with Fluids Fluid Statics Describes fluids at rest Fluid Dynamics Describes fluids in motion The same physical principles that have applied to statics and dynamics up to this point will also apply to fluids Forces in Fluids Fluids do not sustain shearing stresses or tensile stresses 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 Pressure The pressure P of the fluid at the level to which the device has been submerged is the ratio of the force to the area F P A Pressure, cont Pressure is a scalar quantity Because it is proportional to the magnitude of the force If the pressure varies over an area, evaluate dF on a surface of area dA as dF = P dA Unit of pressure is pascal (Pa) 1Pa = 1 N/m2 Pressure vs. Force Pressure is a scalar and force is a vector The direction of the force producing a pressure is perpendicular to the area of interest Measuring Pressure The spring is calibrated by a known force The force due to the fluid presses on the top of the piston and compresses the spring The force the fluid exerts on the piston is then measured Density Notes Density is defined as the mass per unit volume of the substance The values of density for a substance vary slightly with temperature since volume is temperature dependent The various densities indicate the average molecular spacing in a gas is much greater than that in a solid or liquid Density Table Variation of Pressure with Depth Fluids have pressure that varies with depth If a fluid is at rest in a container, all portions of the fluid must be in static equilibrium All points at the same depth must be at the same pressure Otherwise, the fluid would not be in equilibrium Pressure and Depth Examine the darker region, a sample of liquid within a cylinder It has a cross-sectional area A Extends from depth d to d + h below the surface Three external forces act on the region Pressure and Depth, cont The liquid has a density of r Assume the density is the same throughout the fluid This means it is an incompressible liquid The three forces are: Downward force on the top, P0A Upward on the bottom, PA Gravity acting downward, Mg The mass can be found from the density: M = rV = r Ah Pressure and Depth, final Since the net force must be zero:  F = PAˆj − Po Aˆj − Mgˆj This chooses upward as positive Solving for the pressure gives P = P0 + rgh The pressure P at a depth h below a point in the liquid at which the pressure is P0 is greater by an amount rgh Atmospheric Pressure If the liquid is open to the atmosphere, and P0 is the pressure at the surface of the liquid, then P0 is atmospheric pressure P0 = 1.00 atm = 1.013 x 105 Pa Pascal’s Law The pressure in a fluid depends on depth and on the value of P0 An increase in pressure at the surface must be transmitted to every other point in the fluid This is the basis of Pascal’s law Pascal’s Law, cont Named for French scientist Blaise Pascal A change in the pressure applied to a fluid is transmitted undiminished to every point of the fluid and to the walls of the container P1 = P2 F1 F2 = A1 A2 Pascal’s Law, Example Diagram of a hydraulic press (right) A large output force can be applied by means of a small input force The volume of liquid pushed down on the left must equal the volume pushed up on the right Pascal’s Law, Example cont. Since the volumes are equal, A1x1 = A2 x2 Combining the equations, which means Work1 = Work2 This is a consequence of Conservation of Energy F1x1 = F2 x2 Pascal’s Law, Other Applications Hydraulic brakes Car lifts Hydraulic jacks Forklifts Pressure Measurements: Barometer Invented by Torricelli A long closed tube is filled with mercury and inverted in a dish of mercury The closed end is nearly a vacuum Measures atmospheric pressure as Po = rHggh One 1 atm = 0.760 m (of Hg) Pressure Measurements: Manometer A device for measuring the pressure of a gas contained in a vessel One end of the U-shaped tube is open to the atmosphere The other end is connected to the pressure to be measured Pressure at B is P = P0+ρgh Absolute vs. Gauge Pressure P = P0 + rgh P is the absolute pressure The gauge pressure is P – P0 This is also rgh This is what you measure in your tires Buoyant Force The buoyant force is the upward force exerted by a fluid on any immersed object The parcel is in equilibrium There must be an upward force to balance the downward gravitational force Buoyant Force, cont The magnitude of the upward (buoyant) force must equal (in magnitude) the downward gravitational force The buoyant force is the resultant force due to all forces applied by the fluid surrounding the parcel Archimedes C. 287 – 212 BC Greek mathematician, physicist and engineer Computed ratio of circle’s circumference to diameter Calculated volumes of various shapes Discovered nature of buoyant force Inventor Catapults, levers, screws, etc. Archimedes’s Principle The magnitude of the buoyant force always equals the weight of the fluid displaced by the object This is called Archimedes’s Principle Archimedes’s Principle does not refer to the makeup of the object experiencing the buoyant force The object’s composition is not a factor since the buoyant force is exerted by the fluid Archimedes’s Principle, cont The pressure at the top of the cube causes a downward force of Ptop A The pressure at the bottom of the cube causes an upward force of Pbot A B = (Pbot – Ptop) A = rfluid g V = Mg Archimedes's Principle: Totally Submerged Object An object is totally submerged in a fluid of density rfluid The upward buoyant force is B = rfluid g V = rfluid g Vobject The downward gravitational force is Fg = Mg = = robj g Vobj The net force is B - Fg = (rfluid – robj) g Vobj Archimedes’s Principle: Totally Submerged Object, cont If the density of the object is less than the density of the fluid, the unsupported object accelerates upward If the density of the object is more than the density of the fluid, the unsupported object sinks The direction of the motion of an object in a fluid is determined only by the densities of the fluid and the object  Archimedes’s Principle: Floating Object The object is in static equilibrium The upward buoyant force is balanced by the downward force of gravity Volume of the fluid displaced corresponds to the volume of the object beneath the fluid level Vfluid robj = Vobj r fluid Archimedes’s Principle: Floating Object, cont The fraction of the volume of a floating object that is below the fluid surface is equal to the ratio of the density of the object to that of the fluid Use the active figure to vary the densities Archimedes’s Principle, Crown Example Archimedes was (supposedly) asked, “Is the crown made of pure gold?” Crown’s weight in air = 7.84 N Weight in water (submerged) = 6.84 N Buoyant force will equal the apparent weight loss Difference in scale readings will be the buoyant force Archimedes’s Principle, Crown Example, cont. SF = B + T2 – Fg = 0 B = Fg – T2 (Weight in air – “weight” in water) Archimedes’s principle says B = rgV Find V Then to find the material of the crown, rcrown = mcrown in air / V Archimedes’s Principle, Iceberg Example What fraction of the iceberg is below water? The iceberg is only partially submerged and so Vseawater / Vice = rice / rseawater applies The fraction below the water will be the ratio of the volumes (Vseawater / Vice) Archimedes’s Principle, Iceberg Example, cont Vice is the total volume of the iceberg Vwater is the volume of the water displaced This will be equal to the volume of the iceberg submerged About 89% of the ice is below the water’s surface 27