Lecture 2 Buoyancy. Fluid dynamics. Hot air balloon Buoyancy (in the Dead Sea) 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? DEMO: Vacuum gun 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.

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

In-class example: Hollow sphere A hollow sphere of iron (ρ Fe = 7800 kg/m 3 ) has a mass of 5 kg. What is the maximum diameter for this sphere to be completely submerged in water? (ρ water = 1000 kg/m 3 ) A.It will always be submerged. B.0.11 m C.0.21 m D.0.42 m E.It will always only float. FBFB mg The sphere sinks if

Density rule A hollow sphere of iron (ρ Fe = 7800 kg/m 3 ) has a mass of 5 kg. What is the maximum diameter necessary for this sphere to be fully submerged in water? (ρ water = 1000 kg/m 3 ) Answer: R = m. And what is the average density of this sphere? An object of density ρ object placed in a fluid of density ρ fluid sinks if ρ object > ρ fluid is in equilibrium anywhere in the fluid if ρ object = ρ fluid floats if ρ object ρ fluid This is why you cannot sink in the Dead Sea (ρ Dead Sea water = 1240 kg/m 3, ρ human body = 1062 kg/m 3 ) ! DEMO: Frozen helium balloon

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

Surface tension Overall, the liquid doesn’t “like” surface molecules because they try to compress it. 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.

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)

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 a glass Water in wax- or teflon-coated glass F adhesive > F cohesive F adhesive < F cohesive Or: surface tension in air-liquid interface is larger/smaller than surface tension in wall-liquid interface

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

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

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

Continuity equation A1A1 A2A2 v1 v1 v2v2 The mass flow rate must be the same at any point along the tube (otherwise, fluid would be accumulating or disappearing somewhere) If fluid is incompressible (constant density): ρ1 ρ1 ρ2 ρ2

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

Example: Garden hose When you use your garden faucet to fill your 3 gallon watering can, it takes 15 seconds. You then attach your 3 cm thick garden hose fitted with a nozzle with 40 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. a)How long is the hose? b)How big are the openings in the nozzle? Volume flow rate

When you use your garden faucet to fill your 3 gallon watering can, it takes 15 seconds. You then attach your 3 cm thick garden hose fitted with a nozzle with 40 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. a)How long is the hose? b)How big are the openings in the nozzle? We use kinematics to determine v nozzle : x h