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Liquids and Gasses Matter that “Flows” Density and Specific Gravity Density is mass per unit volume kg SI unit is m3 m V g But this is also used cm 3 Specific Gravity Ratio of density to the density of water at 4oC. Density of water at 4oC=1 g/cm3 The 4oC is important since water’s density changes with temperature It has nothing to do with gravity Relative density would be a better term Pressure A static fluid will exert a force perpendicular to any surface it is in contact with Pressure is defined as Force per unit Area F p A Note this is scalar SI unit of N/m2=pascal=pa Atmospheric Pressure The big volume of air above us exerts pressure downwards. Pressure at sea level pa 1atm 1.013x10 pa 5 Changes depending on elevation, temperature, weather condition etc. Pressure at depth Assume that is constant all throughout the liquid. Fluid is incompressible Force is from weight of liquid F m g Vg Ahg p A A A A p hg Pressure at depth If fluid is compressible (gasses) density changes with height and equation is only valid for very thin slices. Pascal’s Law Equations independent of Area, only height matters. Pressure is the same at any two points at the same level of the fluid. If pressure is applied to one end it is distributed throughout the fluid and to the walls of the container Pascal’s Law Application Hydraulic Lift F1 F2 p A1 A2 However, volume of the liquid remains the same V1 V2 A1x1 A2 x2 Hydraulic Lift F1 A1 x2 F2 A2 x1 F1x1 F2 x2 Work done by F1 = Work done by F2 Example A car lift used in a service station uses compressed air to exert a force on a small piston with a circular cross section of radius 5.00cm. The pressure is transmitted by a liquid to a second piston of radius 15.0 cm. What force must the compressed air exert to lift a car weighing 13300N? What air pressure produces this force? Example F1 F2 p A1 A2 A1 r12 F1 F2 2 F2 A2 r1 52 F1 2 13300 15 F1 1480N Note: no need to convert to meters since units cancel F1 F2 p A1 A2 1480N p 188000Pa 2 (0.05) Absolute Pressure and Gauge Pressure Absolute Pressure Total Pressure within a vessel Gauge Pressure Excess Pressure above atmospheric pressure Gauge Pressure – Ex. A tire shows 220kPa Absolute Pressure of tire is 220kPa+1atm=320kPA Flat tire actually has 1 atm of pressure inside Pressure Gauges - Manometer Many different devices used to measure pressure Manometer – consists of a U-shaped tube filled with a liquid of known density. It is open in one end and the other end is connected to a container filled with gas whose pressure to be measured. Pressure Gauges - Manometer Bottom of tube has same pressure p gy1 patm gy2 p patm g ( y2 y1 ) p patm gh Gauge pressure is proportional to height difference Pressure Gauges - Barometer Mercury barometer – consists of filled tube of mercury closed on one end and the other end is inverted into a dish of mercury. Pressure in the closed end of the tube can be approximated to be zero. Pressure Gauges - Barometer patm gh p0 patm gh mmHg or Torr, is another common unit of measure of pressure. But it is affected by temperature (density of Hg changes) as well as gravity. Buoyancy Bodies immersed in water will weigh less Bouyant Force is exerting an upward force Assume a submerged object with uniform cross section area A Bouyancy B FB FT B PB A PT A A( PB PT ) B A( ghB ghT ) B Ag (h) B Vg B mg Bouyancy Archimedes’ Principle- When a body is immersed in a fluid, the fluid exerts an upward force on the body equal to the weight of the fluid displaced by the body. Example Archimedes was supposedly tasked to determine if the crown of the king was pure gold. He solved this by first weighing the crown in air, and then weighing the crown when submerged in water. Suppose the scale read 7.84N in air and 6.84N in water, what is the crown made of? Example F 0 0 T1 m g T1 m g 1 wVg 1 1 V w g 1000(9.8) T1 T2 B wVg V 1.02x10 4 m3 mc 7.84 c 4 V (1.02x10 )9.8 7.84 6.84 wVg c 7840kg m 1 wVg gold 19300kg m 0 T2 B m g T2 m g B 3 3 Bouyancy An object will float if B Wobject lVobject g mobject g lVobject g objectVobject g l object Fluid Flow – Fluid Dynamics Complicated, but can be simplified with certain assumptions Ideal Fluid Incompresible- same density throughout Not viscous - has no internal friction Minimal turbulence Fluid Flow Assume the flow is steady m flow t V Al flow t t flow Av Flow rate is constant 1 A1v1 2 A2v2 Fluid Flow 1 A1v1 2 A2v2 Since incompresible A1v1 A2v2 Continuity Equation Example A water hose of 2.50cm diameter is used by a gardener to fill a 30.0L bucket. The gardener notes that it takes 1.00min to fill the bucket. A nozzle with an opening of cross-sectional area 0.500cm2 is attached and held so water is projected horizontally from a point 1.00m above ground. Over what horizontal distance can the water be projected? Example Flow A1v1 3 30L 0.03m 3 m 0.0005 s min 60s 3 m 0.0005 s Av m3 0.0005 s m v 10 . 0 s 2 0.00005m Example v 10.0 m s 1 2 y gt 2 t 2 / g 2 /(9.8)(1) t 0.45s R vxt 10(0.45) R 4.5m Bernoulli’s Equation Pressure in a fluid depends on velocity and height Assuming ideal fluid, fluid at certain cross section will do work on other parts of the fluid. W1 F1d1 p1 A1d1 W2 F2 d 2 p2 A2 d 2 Bernoulli’s Equation Wg m gh m g( y2 y1 ) W W1 W2 Wg W p1 A1d1 p2 A2 d 2 m g( y2 y1 ) K p1 A1d1 p2 A2 d 2 m g( y2 y1 ) 1 2 1 2 m v1 p1 A1d1 m gy1 m v2 p2 A2 d 2 m gy2 2 2 m V Ad 1 2 1 2 v1 p1 gy1 v2 p2 gy2 2 2 Example Water enters a house through a pipe with an inside diameter of 2.0 cm at an absolute pressure of 4.0x105 Pa. A 1.0cm diameter pipe leads to the second floor bathroom 5.0 m above. When the flow speed at the inlet pipe is 1.5 m/s, find the flow speed, pressure and volume flow in the bathroom. Example A1v1 A2 v2 A1 v2 v1 A2 (1) 2 m v2 1 . 5 6 s 2 (0.5) 1 2 1 2 v1 p1 gy1 v2 p2 gy2 2 2 1 1 2 5 (1000)(1.5) 4.0 x10 0 (1000)6 2 p2 (1000)(9.8)(5) 2 2 p2 3.34x105 Pa Example Flow A1v1 Flow (0.5 x10 2 ) 2 6 4 m3 Flow 4.7 x10 s Problems Giancoli 10-20 In working out Pascal’s principle, Pascal showed dramatically how force can be multiplied with fluid pressure. He placed a long thin tube of radius r=0.30cm, vertically into a wine barrel of radius R=21cm. He found that when the barrel was filled with water and the tube filled to a height of 12m, the barrel burst. Calculate (a) the mass of water in the tube? (b) the net force exerted on the barrel lid just before rupture. Serway 14-20 A U tube of uniform cross-secitonal area, open to the athmosphere, is partially filled with mercury. Water is then poured into both arms. If the equilibrium configuration fo the tube is as shown with h2=1.00cm, find h1. Young and Friedman 14.31 A cubical block of wood, 10 cm on a side, floats in the interface of oil and water with its lower surface, 1.50cm below the interface. The density of oil is 790kg/m3. (a) what is the gauge pressure at the upper face of the block? (b) What is the gauge pressure at the lower face of the block? (c) What are the mass and density of the block? Young and Friedman 14.46 A golf course sprinkler system discharges water from a horizontal pipe at the rate of 7200cm3/s. At one point in the pipe, where the radius is 4.00cm, the absolute pressure is 2.40x105 Pa. At a second point in the pipe, the water passes through a connection where the radius is 2.00cm. What is the absolute pressure as it flows through the constriction?