Lecture 2 Buoyancy. Fluid dynamics. Hot air balloon Buoyancy (in the Dead Sea) Cohesion (water...
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Transcript of Lecture 2 Buoyancy. Fluid dynamics. Hot air balloon Buoyancy (in the Dead Sea) Cohesion (water...
Lecture 2Buoyancy.
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 ballspeed is complicated and needs to take turbulent air and friction into account.
Buoyancy and the Archimedes’ principle
ybottom
ytop
hA
A box of base A and height h is submerged in a liquid of density ρ.
bottom topAp Ap
atm bottom atm topA p gy A p gy
A hg
Archimedes’s principle: The liquid exerts a net force upward called buoyant force whose magnitude is equal to the weight of the displaced liquid.
direction upVg
topbottomF F F
Ftop
Fbottom
Net force by liquid:
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)
A. It will always be submerged.
B. 0.11 m
C. 0.21 m
D. 0.42 m
E. It will always only float.
FB
mg
The sphere sinks if BF mg
3water
43
R g mg
3
water
30.106 m
4m
R maxMaximum diameter 2 0.21 mR
Density rule
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?
3
sphere water33
5 kg1000 kg/ m
4 40.106 m
3 3
m
R
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 ρobject ρfluid
This is why you cannot sink in the Dead Sea (ρDead Sea water = 1240 kg/m3 , ρhuman body = 1062 kg/m3 ) !
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 glassWater in wax- or
teflon-coated glass
Fadhesive > Fcohesive
Fadhesive < Fcohesive
Or: surface tension in air-liquid interface is larger/smaller than surface tension in wall-liquid interface
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
dVAv
dtVolume flow
rate
A
dx = v
dt
dmAv
dtMass flow rate
Continuity equation
A1
A2
v1
v2
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 ρ2
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?3
4 33 gallons 3.785 liters 1 m7.6 10 m / s
15 s 1 gallon 1000 literdVdt
Volume flow rate
hose hose
dVA v
dt
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 vnozzle:
nozzlenozzle
0 x
x v t tv
x
h
202g
h t
hose hose nozzle nozzleA v A v
r
hose
2 vhose
40rnozzle
2 vnozzle
rnozzle
rhose
vhose
40vnozzle
1.5 cm 4.3 m / s
40 ×11 m / s1.4 m m