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Lecture 2

Hot air balloon. Buoyancy (in the Dead Sea). Lecture 2. Buoyancy. Fluid dynamics. Cohesion (water bubble in space). Laminar flow. Vacuum gun. Sealed tube, air pumped out. Ping-pong ball. What happens if we punch a little hole on one side?.

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Lecture 2

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  1. Hot air balloon Buoyancy (in the Dead Sea) Lecture 2 Buoyancy. Fluid dynamics. Cohesion (water bubble in space) Laminar flow

  2. Vacuum gun Sealed tube, air pumped out Ping-pong ball What happens if we punch a little hole on one side? Atmospheric pressure pushes ball through tube and accelerates to high speed. Realistic calculation of ball speed is complicated and needs to take turbulent air and friction into account. DEMO: Vacuum gun

  3. Ftop Net force by liquid: Fbottom Buoyancy and the Archimedes’ principle A box of base A and height h is submerged in a liquid of density ρ. ytop A h ybottom Archimedes’s principle: The liquid exerts a net force upward called buoyant force whose magnitude is equal to the weight of the displaced liquid.

  4. FB mg The sphere sinks if In-class example: Hollow sphere A hollow sphere of iron (ρFe = 7800 kg/m3) has a mass of 5 kg. What is the maximum diameter for this sphere to be completely submerged in water? (ρwater = 1000 kg/m3) • It will always be submerged. • 0.11 m • 0.21 m • 0.42 m • It will always only float.

  5. Density rule DEMO: Frozen helium balloon A hollow sphere of iron (ρFe = 7800 kg/m3) has a mass of 5 kg. What is the maximum diameter necessary for this sphere to be fully submerged in water? (ρwater = 1000 kg/m3) Answer: R = 0.106 m. And what is the average density of this sphere? • An object of density ρobject placed in a fluid of density ρfluid • sinks if ρobject > ρfluid • is in equilibrium anywhere in the fluid if ρobject = ρfluid • floats if ρobjectρfluid This is why you cannot sink in the Dead Sea (ρDead Sea water = 1240 kg/m3 , ρhuman body = 1062 kg/m3 ) !

  6. On the surface: Net force on a molecule is inward. In the bulk: Net force on a molecule is zero. Attraction between molecules Wood floats on water because it is less dense than water. But a paper clip (metal, denser than water!) also floats in water… (?) . Molecules in liquid attract each other (cohesive forces that keep liquid as such!) Very small attraction by air molecules. …And this force is compensated by the incompressibility of the liquid.

  7. Liquid adopts the shape that minimizes the surface area. Any attempt to increase this area is opposed by a restoring force. The surface of a liquid behaves like an elastic membrane. The weight of the paper clip is small enough to be balanced by the elastic forces due to surface tension. Surface tension Overall, the liquid doesn’t “like” surface molecules because they try to compress it.

  8. Drops and bubbles Water drops are spherical (shape with minimum area for a given volume) Adding soap to water decreases surface tension. This is useful to: • Force water through the small spaces between cloth fibers • Make bubbles! (Large area and small bulk)

  9. How wet is water? Molecules in a liquid are also attracted to the medium it is in contact with, like the walls of the container (adhesive forces). Water in wax- or teflon-coated glass Water in a glass Fadhesive < Fcohesive Fadhesive > Fcohesive Or: surface tension in air-liquid interface is larger/smaller than surface tension in wall-liquid interface

  10. Fluid flow Laminar flow: no mixing between layers Turbulent flow: a mess…

  11. Dry water, wet water Within the case of laminar flow: Slower near the walls Real (wet) fluid: friction with walls and between layers (viscosity) Faster in the center Ideal (dry) fluid: no friction (no viscosity) Same speed everywhere

  12. dx = vdt A Mass flow rate Volume flow rate Flow rate Consider a laminar, steady flow of an ideal, incompressible fluid at speed v though a tube of cross-sectional area A

  13. A2 v2 A1 v1 Continuity equation The mass flow rate must be the same at any point along the tube (otherwise, fluid would be accumulating or disappearing somewhere) ρ1 ρ2 If fluid is incompressible (constant density):

  14. Thin tube, large speed Thick tube, small speed Incompressible fluid:

  15. Volume flow rate Example: Garden hose • When you use your garden faucet to fill your 3 gallon watering can, it takes 15 seconds. You then attach your 1.5 cm thick garden hose fitted with a nozzle with 10 holes at the end. You turn on the water, and 4 seconds later water spurts through the nozzle. When you hold the nozzle horizontally at waist level (1 m from the ground), you can water plants that are 5 m away. • How long is the hose? • How big are the openings in the nozzle?

  16. h We use kinematics to determine vnozzle: x • When you use your garden faucet to fill your 3 gallon watering can, it takes 15 seconds. You then attach your 1.5 cm thick garden hose fitted with a nozzle with 10 holes at the end. You turn on the water, and 4 seconds later water spurts through the nozzle. When you hold the nozzle horizontally at waist level (1 m from the ground), you can water plants that are 5 m away. • How long is the hose? • How big are the openings in the nozzle?

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