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An Opportunity You can switch to the calculus stream and study more physics (if you want!) – 16.107 and beyond – if you get a B (B+ for honours physics) in 16.102 31 Fluids, the story so far... Density = Mass/Volume Density of substance Specific gravity = Density of water at 4◦C Pressure = Force/Area Increase of pressure with depth: P2 = P1 + !gh Pascal’s Principle: pressure changes are communicated uniformly throughout the fluid Archimedes’ Principle: the buoyant force is equal to the weight of displaced fluid 32 11.38 A duck is floating on a lake with 25% of its volume beneath the water. What is the average density of the duck? Fb= buoyant force 0.25V mg = weight of duck The duck floats, so the weight of the duck is equal to the weight of water with a volume of 25% of the volume of the duck. That is, mg = !water × 0.25 ×V g But, m = !duck ×V Therefore, !duck = 0.25!water = 250 kg/m3 33 11.46 A solid object has an apparent weight of 15.2 N when T completely immersed in ethyl alcohol. When completely submerged in water, its apparent weight is 13.7 N. What is the volume of the object? Density of ethyl alcohol = 806 kg/m3. Fb Weigh the object by supporting it by a string and measuring the tension in the string. mg In ethyl alcohol: T1 = mg − Fb1 = 15.2 N In water: T2 = mg − Fb2 = 13.7 N So, mg = 15.2 + Fb1 = 13.7 + Fb2 Fb2 − Fb1 = 1.5 N and, Fb2 − Fb1 = (!H20 − !ethyl )V g Therefore, 1.5 = (1000 − 806)gV 1.5 V= = 7.9 × 10−4 m3 194g 34 11.44 What is the smallest number of logs (ρ = 725 kg/m3), radius 0.08 m, length = 3 m, that can be used to carry four people, each of mass 80 kg? The volume of each log is πr2L = 0.0603 m3, and its mass is m = (0.0603 m3) × (725 kg/m3) = 43.72 kg. When fully immersed in water, the buoyant force on a log is: Fb = (1000 kg/m3) × (0.0603 m3)g = 590.9 N With n logs, the total buoyant force is 590.9n and the total weight, including the people, is W = (4×80 + 43.72n)g. Set these forces to be equal: 590.9n = (320 + 43.72n)g n = 19.3, so need at least 20 logs. 35 Fluids in Motion Streamline (steady) flow – the velocity of a fluid at some point does not change with time. Streamlines show the path of fluid particles. Streamlines do not cross. 36 Fluids in Motion Unsteady flow – the velocity at a point in the fluid changes with time. Turbulent flow is an extreme case of unsteady flow for a fast-moving fluid – the velocity changes erratically from moment to moment, as at sharp obstacles or bends. Most liquids are incompressible, or nearly so – the density is almost independent of pressure. Gases are highly compressible, but their density is sometimes constant enough in a flowing gas for them to be considered incompressible. Viscous flow – a type of friction impeding the relative motion of layers of a fluid. 37 The Equation of Continuity What goes in must come out. !m2 !m1 Mass flowing in = mass flowing out Mass flowing in during time !t is, !m2 = "2 A2 v2!t Mass flowing out during time !t is, !m1 = "1 A1 v1!t !m1 = !m2 and so, "1 A1 v1 = "2 A2 v2 Equation of continuity 38 The Equation of Continuity !m = " A v = mass flow rate !t If the fluid is incompressible, ρ1 = ρ2 and v1 A1 = v2 A2 Q = v A = volume flow rate 39