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FLUIDS What is a fluid? A substance that can flow. A fluid is a substance that cannot maintain its own shape but takes the shape of its container. Gases and liquids are called fluids because neither have an orderly arrangement. In rigid bodies we expressed Newton's laws in terms of mass and force. We describe fluids in terms of density and pressure Pressure force per surface area; symbol is P; SI unit is Pa (Pascal), or N/m2. P=F/A where F is force or weight in Newtons A is cross-sectional area in m2 The pressure at any point in a fluid acts equally in all directions. Also, the force due to the fluid pressure always acts perpendicularly to any surface the fluid is in contact with. Atmospheric pressure= 1.013 x 105 Pa (measured at sea level). This can be approximated as 101 kPa. Gauge Pressure Pressure gauges measure the pressure over and above atmospheric pressure. This is called gauge pressure. Absolute Pressure To get the absolute pressure at a point, one must add the atmospheric pressure to the gauge pressure. For example, if gauge pressure is 100 kPa, the absolute pressure at that point is the sum of 100 kPa and 101 kPa, or 201 kPa. Manometer A U-shaped tube partially filled with liquid used to measure pressure. The pressure is equal to the difference in height of the two levels of the liquid according to P = Patm + gh. Density the ratio of mass to volume; density is a characteristic property of a any pure substance. Its SI unit is kg/m3 =m/V where is density in kg/m3 m is mass in kilograms V is volume in m3 (H2O) = 1000 kg/m3 Sometimes densities are given in g/cm3. To convert to the SI unit of kg/m3 simply multiply by 1000. Density Lab - Float or Sink: You find out!" http://www.explorelearning.com/index.cfm?method=cResource.dspDetail&Resou rceID=17 Specific Gravity The ratio of the density of that substance to the density of water at 4.0. It has no units. substance density, kg/m3 substance density, kg/m3 aluminum 2700 iron and steel 7800 copper 8900 lead 11300 mercury 13600 ethyl alcohol 790 Hydrostatics the study of fluids at rest 1. Pascal’s principle - any change in pressure at any point in a fluid is transmitted unchanged throughout the fluid. Or, pressure applied to a confined fluid increases the pressure throughout the fluid by the same amount. F1/A1 = F2/A2 This is the basis for squeezing a tube of toothpaste, hydraulic brakes, and for the Heimlich maneuver. 2. Hydrostatic pressure - pressure due to a fluid’s depth P=gh where is density of fluid in kg/m3 h is the height (depth) of fluid Pressure increases with depth. The pressure at any depth depends only upon that depth and not upon any horizontal dimension. For example, Hoover Dam holds back Lake Mead which is 700 ft deep. The bottom of Hoover Dam must withstand the same pressure if it were only holding back a few thousand gallons of water 700 ft deep. Also, the pressure at equal depths within a fluid is the same. 3. Archimede’s principle - an object immersed in a fluid is buoyed up by a force equal to the weight of the displaced fluid. A fluid provides some support for any object placed in it. The upward force on an object placed in a fluid is called the buoyant force. FB = g V where is the density of the fluid in kg/m3 V is the volume of the displace fluid FB is the buoyant force (The buoyant force occurs because the pressure in a fluid increases with depth. The upward force on the bottom surface of a submerged object is greater than the downward pressure on its top surface.) o o o If an solid floats partially submerged in a liquid, the volume of liquid displaced is less than the volume of the object. According to Archimedes Principle, the weights of the object and its displaced fluid are the same. The fractional part of an object that is submerged is equal to the ratio of the density of the solid to the density of the liquid in which is floats (for example, about 90% of an iceberg is submerged because the density of the ice is about 90% that of sea water). An object floats when the buoyant force is equal to its weight. Objects submerged in a fluid appear to weigh less than they do when outside the fluid. Applet demonstrating buoyant force http://www.sciencejoywagon.com/physicszone/lesson/otherpub/wfendt/buoyforce.htm Fluid Dynamics study of fluids in motion Hydrodynamics study of water in motion Aerodynamics study of air in motion Lift on an airplane wing http://www.explorelearning.com/index.cfm?method=cResource.dspDetail&Resou rceID=24 Equation of Continuity the volume of fluid passing two points per second is equal A1 V1 = A2 V2 In a narrow tube, the velocity of the liquid is high; in a wide tube, the velocity of the liquid is low. Bernoulli’s principle as the velocity of a fluid increases, the pressure exerted by that fluid decreases on top of an airfoil there is low pressure due to high velocity airflow on the bottom of an airfoil there is high pressure due to low velocity airflow Fluids in Motion There are two type of fluid flow, streamline (laminar) and turbulant flow. If the slow is smooth (layers of fluid slide by each other smoothly), the flow is said to be steamlined, or laminar. Above a certain speed (which depends upon many factors), a flow becomes turbulent. Turbulent flow is characterized by the formation of eddies. Viscosity Internal friction in a fluid. Mass Flow Rate the ratio of the mass of a fluid that passes a certain point in a certain interval of time (or, m/t) Volume Rate of Flow the ratio of the volume of a fluid that passes a certain point in a certain interval of time (or, V/t). In SI units, this is m3/sec (or the same thing as the product of area, A, and velocity, v.) Bernoulli's Equation Where the velocity of a fluid is high, the pressure is low; where the velocity is low, the pressure is high. Bernoulli's equation is an expression of the law of conservation of energy. P1 + 1/2 v12 + gh1 = P2 + 1/2 v22 + gh2 Torricelli's theorem A liquid leaves a spigot at the bottom os a reservoir with the same speed that a freely falling object falling through the same height. v1 = (2g(h2-h1))1/2 AP Multiple Choice Questions Fluid questions were added to the AP B test beginning in 2002. Be able to determine the buoyant force acting on an object when given the weight in air and the weight in the fluid. Be able to recognize that the pressure acting on the bottom of a container is due to the weight of the fluid above it. Recognize that high velocity fluid has low pressure. Predict the consequences of low pressure due to high velocity fluid. AP Free Response Questions Fluid questions were added to the AP B test beginning in 2002. All the ones that have occurred since 2002 have dealt with either a fluidbased laboratory exercise or with fluids at rest. Be able to calculate gauge pressure. Be able to calculate absolute pressure. Be able to calculate pressure. Kinematics is frequently integrated into fluid problems. AP Physics B - Fluids Objectives Students should understand that fluids exert pressure in all directions. Students should understand that a fluid at rest exerts pressure perpendicular to any surface that it contacts. Students should understand and be able to use the relationship between pressure and depth in a fluid, P = gh. Students should understand that the difference in pressure on the upper and lower surfaces of an object immersed in liquid results in an upward force on the object. Students should understand and be able to apply Archimedes Principle: the buoyant force on a submersed object is equal to the weight of the liquid it displaces. Students should understand that for laminar flow, the flow rate of a liquid through its cross section is the same at any point along its path. Students should understand and be able to apply the equation of continuity, 1A1v1 = 2A2v2. Students should understand that the pressure of a flowing liquid is low when the velocity is high, and vice versa. Students should understand and be able to apply Bernoulli's equation: P + gy + 1/2 v2 = constant AP Fluids Sample Problems-Hydrostatics 1. Estimate the pressure exerted on a floor by a 50 kg model standing momentarily on a single spiked heel (area = 0.05 cm2) and compare it to the pressure exerted by a 1500 kg elephant standing on one foot (area = 800 cm2). Ans: 9.8 x 107 Pa; 1.8375 x 105 Pa 2. A tire gauge reads 220 kPa. What is the absolute pressure within the tire? Ans: 321 kPa 3. What is the pressure due to a column of water 100 m high? Ans: 980,000 Pa 4. The area of the output piston in a hydraulic lift is 20 times that of the input cylinder. What force would it take to lift a 4000 lb car? Ans: 200 lb 5. A geologist finds that a moon rock whose mass is 8.20 kg has an apparent mass of 6.18 kg when submerged in water. What is the density of the rock? Ans: 4059.41 kg/m3 6. Water and then oil are poured into a U-shaped tube, open at both ends, and do not mix. They come to equilibrium as shown. What is the density of the oil? The height of the oil column is 27.2 cm; the distance from the top of the water column to the top of its tube is 9.41 cm; the top of its tube is even with the top of the oil column. (Hint: pressures at a and b are equal.) Ans: 654.04 kg/m3 AP Fluids Sample Problems-Hydrodynamics 1. A stream of water emerges from a pipe. The cross-sectional area of the water stream at the tap is 1.2 cm2 and 43 mm lower than the tap is 0.35 cm2. At what volume rate of flow does the water flow from the tap? Ans: 3.21 x 10-5 m3/s 2. Water is pumped from a pipeline 2 m above the ground to a water tower 15 m above the ground. If the pipeline velocity is 8 m/s, its pressure is 3.103 x 105 Pa, and water enters the tank at a pressure of one atmosphere, with what velocity does the water enter the tank? Ans: 15.09 m/s 3. A gas is flowing through a pipe whose cross-sectional area is 0.07 m2. The gas has a density of 1.30 kg/m3. A Venturi meter is used to measure the speed of the gas. It has a cross-sectional area of 0.05 m2. The pressure difference between the pipe and the Venturi meter is found to be 120 Pa. Find the speed of the gas in the pipe and the volume rate of flow of the gas. Ans: 13.87 m/s; 0.97 m3/s 4. An aneurysm is an abnormal enlargement of a blood vessel such as the aorta. Suppose that, because of an aneurysm, the cross-sectional area of the aorta increases to a value 1.7 times greater than the original. The speed of the blood (density of blood is =1060 kg/m3) through a normal portion of the aorta is 0.40 m/s. Assuming the aorta is horizontal (the person is lying down), determine the amount by which the pressure in the enlarged region exceeds that in the normal region. Ans: 55.53 Pa