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Physics 2 – April 18, 2017 Do Now – Sign up for Physics day activities. Must know today if you plan to go, or not. Permission slip and $30 by the time you board the bus. Objectives/Agenda/Assignment Objective: IB Opt B.3 Fluids Assignment: Read sections B.2 and B.3. Do Corresponding sample problems. Agenda: Ideal Fluids Fluid pressure Archimedes’ Buoyant force Pascal and Continuity Bernoulli’s Equation and effect Real fluids Fluids (liquids) Rapidly get very complicated, so we deal only with ideal fluids: Laminar Flow: Velocity does not change with time. Still, or moving with a constant speed. No acceleration, no net forces. Incompressible: Density constant throughout fluid. Non-viscous: No drag forces. (“frictionless” fluid) Streamlines track the path of a small unit of fluid. Streamlines do not cross. A collection of streamlines creates a flow tube. A flow tube will have a cross-sectional area that is perpendicular to the streamlines. Important Applications: Aerodynamics, Aeronautics, Hydraulics, Medicine. Liquid pressure Liquid pressure, p, defined the same as gas pressure: Force/area. p = F/A Pressure at a depth increases with the depth. (recall the dependence on depth in the ocean.) Pressure at a depth, p, is equal to the surface pressure, po, plus gravity, g, times the depth, h, times the density, . p = p0 + gh Ex: What is the change in pressure for 10 m of water? = 1000 kg/m3, g = 9.81 m/s2 h = 10 m. gh = 9.81 x 104 Pa / 101325 Pa = 0.97 atm Buoyancy / Archimedes’ Principle Why do things float? Density of object is less than density of fluid. In other words, it experiences a buoyant force that is equal to it weight. Buoyant force, B = weight of water displaced by the object. W = mg. For water, the mass of the displaced volume is V (from definition of density) Therefore: Wdisplaced = Vg B = Vg in Data packet with f subscripts to specify that it is the density and volume of the fluid displaced that is important. Known as Archimedes’ Principle Archimedes sample problem A basketball floats in a bathtub of water. The ball has a mass of 0.5 kg and a diameter of 22 cm. (a) What is the buoyant force? (b) What is the volume of water displaced by the ball? (c) What percentage of the ball is submerged? Pascal’s Principle Pascal’s Principle: When a pressure is applied to any point of an ideal fluid, the pressure will immediately be transmitted to all other parts of the liquid and the walls of the container of the fluid. Application to hydraulics: P1 = P2 F1/A1 = F2/A2 The work done by an external force on a fluid is pV = pAh F1A1 h1/A1 = F2A2h2/A2 So… F1h1 = F2h2 A small force pushed through a long distance can lift a large force through a short distance. Equation of continuity Consider a flow tube of a given cross-sectional area with laminar flow and constant density. If the cross-sectional area, A, changes, the speed of the fluid v, must also change so that the total volume rate stays the same. Otherwise, the density of the fluid would change. Analogous to a current flowing though a wire, then resistor, then wire. Current has to stay the same. Same charge per second. Here it’s the same mass per second. Equation of continuity: Av = constant A1v1 = A2v2 Ex: Plugging up part of a garden hose to get a fast water flow, provides more momentum and greater impact when used to wash a car, e.g. Bernoulli’s Equation Bernoulli’s equation describes how the pressure of a fluid changes when there is a change in flow tube cross-sectional area and/or a change in height for the fluid. p1 + gz1 + ½v12 = p2 + gz2 + ½v22 Can be derived from W = K.E. P and gz comes from finding the work Done by the fluid (PV work) Work done by gravity ½v2 is from kinetic energy Bernoulli’s Equation Problem A dam holds back the water in a lake. If the dam has a small hole 1.4 meters below the surface of the lake, at what speed does water exit the hole? A hose lying on the ground has water coming out of it at a speed of 5.4 meters per second. You lift the nozzle of the hose to a height of 1.3 meters above the ground. At what speed does the water now come out of the hose? The Bernoulli effect A faster flowing fluid has a lesser pressure. Applications: Airplane wing shape causes air to flow faster on top, decreasing the pressure there, creating a net lift force. Water aspirators in a laboratory use large volumes of water through a smaller faucet. A side port on the faucet provide a useful relative vacuum, a lesser pressure, due to the fast flowing water. Sail shape creates the same kinds of pressure differences as wing shape. Explains why a curve ball curves. Real fluids – Stoke’s Law Ideal fluids have no drag, but real fluids do. Stoke’s Law is about drag force. Fd = 6rv where is the “viscosity coefficient” (unit Pa s - to produce Newtons for drag force) As an object falls through a viscous fluid, it’s speed increases. As the speed increases, so does the drag, because it depends on velocity. When the drag increases to form a net force = 0, the speed will stop increasing and the object will fall with a terminal speed. Same mechanism works for air resistance and falling through air. Real fluids – Reynold’s number The second aspect of ideal fluids to get relaxed is the laminar flow. The as the flow speed increases, first the friction between the walls and the fluid slow the edges. (middle image). At higher speeds, the flow develops cross currents and becomes turbulent (lower image) The judgements of which mode of flow you have (turbulent vs laminar) is based on the value of the Reynold’s number. Above 1000 is turbulent. R = vr/ Same var. defs.
