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# HARVARD EPS 5 - Lecture 6

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Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8What will happen when I add the oil on top of the water?Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Lecture 6. EPS 5: 23 Sep 20101. Review the concept of the barometric law (hydrostatic balance: each layer of atmosphere must support the weight of the overlying column mass of atmosphere). Discuss the distribution of pressure with altitude in the atmosphere, or depth in the ocean.2. Introduce buoyancy. Pressure force "upwards" on an object immersed in a fluid.3. Archimedes principle: the buoyancy force on an object is equal to the weight of the fluid displaced by the object. Role of gravity.4. The buoyancy of warm air.5. A brief look at global weather patterns—sea surface temperature and buoyancy.6. Introducing the properties of water.Z2Z1P1P2By how much is P1 > P2? The weight of the slab of fluid between Z1 and Z2 is given by the density, ρ, multiplied by volume of the slab) and g weight of slab = ρ(area  height) g. Set the area of the column to 1 m2, the weight isρ g  (Z2 -Z1): If the atmosphere is not being accelerated, there must be a difference in pressure (P2 - P1) across the slab that exactly balances the force of gravity (weight of the slab). Relationship between density, pressure and altitudeocean atmosphere1 bar = 105 N/m2BuoyancyBuoyancy is the tendency for less dense fluids to be forced upwards by more dense fluids under the influence of gravity. Buoyancy arises when the pressure forces on an object are not perfectly balanced. Buoyancy is extremely significant as a driving force for motions in the atmosphere and oceans, and hence we will examine the concept very carefully here. The mass density of air ρ is given by mn, where m is the mean mass of an air molecule (4.8110-26 kg molecule-1 for dry air), and n is the number density of air (n =2.69 1025 molecules m-3 at T=0o C, or 273.15 K). Therefore the density of dry air at 0 C is ρ = 1.29 kg m-3. If we raise the temperature to 10 C (283.15 K), the density is about 4% less, or 1.24 kg m-3. This seemingly small difference in density would cause air to move in the atmosphere, i.e. to cause winds.Buoyancy force: Forces on a solid body immersed in a tank of water. The solid is assumed less dense than water and to have area A (e.g. 1m2 ) on all sides. P1 is the fluid pressure at level 1, and P1x is the downward pressure exerted by the weight of overlying atmosphere, plus fluid between the top of the tank and level 2, plus the object. The buoyancy force is P1 – P1x (up ↑) per unit area of the submerged block. P1xNet Force (Net pressure forces – Gravity)The buoyancy force and Archimedes principle.1. Force on the top of the block: P2  A = ρwater D2 A g (A = area of top)weight of the water in the volume above the block2. Upward force on the bottom of the block = P1  A = ρwater D1 A g3. Downward force on the bottom of the block = weight of the water in the volume above block + weight of block = ρwater D2 A g + ρblock (D1 - D2) A gUnbalanced, Upward force on the block (  –  ):Fb = ρwater D1 A g – [ ρwater D2 A + ρblock (D1 - D2) A ] g= ρwater g Vblock – ρblock g Vblock = (ρ water – ρblock) (D1 – D2) A g weight of blockBUOYANCY FORCE = weight of the water (fluid) displaced by the blockVolume of the block = (D1 – D2) AArchimedes principle: the buoyancy force on an object is equal to the weight of the fluid displaced by the object•object immersed in a fluid•weight of fluid displaced•for the fluid itself, there will be a net upward force (buoyancy force exceeds object weight) on parcels less dense than the surrounding fluid, a net downward force on a parcel that is more dense.•buoyancy can accelerate parcels in the vertical direction (unbalanced force).•the derivation of the barometric law assumed that every air parcel experienced completely balanced forces, thus didn't accelerate. Buoyancy exactly balanced the weight of the parcel (“neutrally buoyant”) – this is approximately true even if the acceleration due to unbalanced forces is quite noticeable, because the total forces on an air parcel are really huge (100,000 N/m2), and thus only small imbalances are needed to produce significant accelerations.This experiment, done in this class, shows Archimedes principle. In frame A, the block is displacing water, and air. When we add oil, it displaces oil and water. Since oil has a higher density than air the buoyancy force increases, forcing the block upwards. It stops moving upwards when the weight of (oil + water) that it displaces equals the weight of the block.Density Data: water = 1000; HDPE=941; Veg oil = .894What will happen when I add the oil on top of the water?•1. Block will move down; •2. Block will sink to the bottom; •3. Block will rise, but will remain submerged in the oil; •4 block will float to the top of the oil..9A closer look at the U-tube experiment…compute the density of the paint thinner :Uh1h2h3ρw h1 =ρw h3 + ρp h2 ρw (h1 – h3) = ρp h2 buoyancy force: ρwhi g – ρphi g = (ρw – ρp )hi gLooks a lot like Archimedes' principle2 hi = h1 + h2 + h3Lecture 6. EPS 51. Review the concept of the barometric law (hydrostatic balance: each layer of atmosphere must support the weight of the overlying column mass of atmosphere). Discuss the distribution of pressure with altitude in the atmosphere, or depth in the ocean.2. Introduce buoyancy. Pressure force "upwards" on an object immersed in a fluid.3. Archimedes principle: the buoyancy force on an object is equal to the weight of the fluid displaced by the object. Role of gravity.4. The buoyancy of warm air.Cold, relatively dense air hashigher density than adjacent warmair, the warm air is buoyant (the coldair is "negatively buoyant"). The"warm air rises" (is buoyant!) .Buoyancy and air temperature.Consider two air parcels at the same pressure, but different temperatures.P = ρ1 (k/m) T1 = ρ2 (k/m) T2Then ρ1/ρ2 = T2/T1 ; if T1 > T2, ρ1 < ρ2 . Warmer air, lower density!Lecture 6. EPS 51. Review the concept of the barometric law (hydrostatic balance: each layer of atmosphere must support the weight of the overlying column mass of atmosphere). Discuss the distribution of pressure with altitude in the atmosphere, or depth in the ocean.2. Introduce buoyancy.

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