Fluids: a problem. g (L 2 d) each of the two different fluids. To find the total buoyant force, g (L 2 (L-d)) imagine that the wood block is

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1 Fluids, elasticity, matter Newtonian mechanics of deformable media Fluids: a roblem A beaker contains a thick layer of oil (shown in green) of density ρ 2 floating on water (shown in blue), which has density ρ 3. A cubical block of wood of density ρ with side length L is gently lowered into the beaker, so as not to disturb the layers of liquid, until it floats eacefully between the layers, as shown in the figure. What is the distance d between the to of the wood cube (after it has come to rest) and the interface between oil and water? Hint: After the wood block has come to rest, it is in static equilibrium. Thus, the magnitude of the buoyant force (directed uward) must exactly equal the magnitude of the gravitational force (directed downward). The buoyant force will deend on the quantity d that you are trying to find. The total buoyant force has two contributions, one from F oil = ρ 2 g (L 2 d) each of the two different fluids. To find the total buoyant force, F water = ρ 3 g (L 2 (L-d)) imagine that the wood block is W = ρ divided into two ieces, one in g L 3 oil and one in water. Aly W = F Archimedes' rincile to each, oil + F water and add the two buoyant forces to find the total force. DC

2 Fluids: how are airlanes ket aloft? xx higher velocity lower ressure lower velocity higher ressure Note: density of flow lines reflects velocity, not density. We are assuming an incomressible fluid. but. this is not enough! Bernoulli only works in closed systems, and air in the sky really is not a closed system Simle Bernoulli calculation Boeing Dimensions: Length: 23 ft 0 inches Wingsan: 2 ft 5 in Height: 63 ft 8 in Weight: Emty: 4 x 0 5 lbs Cargo: 2.5 x 0 5 lbs Passengers+fuel:.5 x 0 5 lbs Full at takeoff: 8 x 0 5 lbs Performance: Cruising Seed: 583 mh Range: 7,230 miles r (v 22 - v 2 ) / 2 = P P 2 = DP Using Bernoulli s equation and a surface area of 200 ft x 5 ft you only roduce an uward lift of 2 x 0 4 lbs too low by a factor of 40! Let v 2 = m/s v 2 = 20 m/s Airlanes are ket aloft because the wings scatter So DP = 3 x 0 3 Pa = 0.03 atm off air molecules, and scatter more downward than or 0.5 lbs/in 2 uward htt://

3 Elasticity Elasticity, 92, by Umberto Boccioni, futurist ainter Elasticity Describes the deformation of solids and liquids under stress. Linear stretch and comression F A = Y ΔL L tensile stress Young s modulus strain olume comression F A Bulk modulus Δ = = B ressure volume strain

4 thermodynamics: a macroscoic descrition of matter 3 Phases of matter solid: rigid, definite shae. Nearly incomressible. liquid: molecules held together by bonds, but able to flow. Nearly incomressible. gas: molecules move freely. Comressible. All 3 hases exist at different,t conditions Phase diagram Trile oint of water: = atm T = 0.0 C Trile oint of CO 2 : = 5 atm T = -56 C demo: geyser & collasing tank ideal gas Atoms and molecules are small, hard sheres traveling freely through sace. Occasionally they collide with each other or with the walls. The molecules have a distribution of seeds The model is valid when the density is low and the temerature is high, well above the condensation oint = nrt or = Nk B T R = 8.3 J/mol K universal gas constant n = number of moles k B =.38 x 0-23 J/K Boltzmann s constant N = 6.02 x 0 23 molecules/mol Avogadro number,, T must be in SI units: Pa, m 3, K

5 ideal gas Isochoric rocess: = const Isobaric rocess: = const 2 T = 2 T 2 2 T = 2 T 2 Isothermal rocess: T = const 2 = 2 2 T = const air bubble rising A diver roduces an air bubble underwater, where the absolute ressure is = 3.5 atm. The bubble rises to the surface, where the ressure is 2 = atm. The water temeratures at the bottom and the surface are, resectively, T = 4 C, T 2 = 23 C What is the ratio 2 / of the volume of the bubble as it reaches the surface ( 2 ) to its volume at the bottom ( )? 2 / = 3.74 Is it safe for the diver to ascend while holding his breath? No. Air in the lungs would exand, and he could have a lung ruture. This is addition to the bends, or decomression sickness, which is due to the ressuredeendent solubility of gas (in air, mostly nitrogen). At deth and at higher ressure N 2 is more soluble in blood. As divers ascend, N 2 dissolved in their blood stream becomes gaseous again and forms N 2 bubbles in blood vessels, which in turn can obstruct blood flow, and therefore rovoke ain and in some cases even strokes or deaths. Fortunately, this only haens when diving deeer than 30 m (00 feet). The diver in this question only went down 25 meters. How do we know that? DC