Lecture 3Archimedes principle.

Fluid dynamics.

ACT: Side tube

A sort of barometer is set up with a tube that has a side tube with a tight fitting stopper. What happens when the stopper is removed?

vacuum

stopper

A. Water spurts out of the side tube.

B. Air flows in through the side tube.

C. Nothing, the system was in equilibrium and remains in equilibrium.

DEMO: Side tube

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 minimum diameter necessary for this sphere not to sink in water? (ρwater = 1000 kg/m3)

A. It will always sink.

B. 0.11 m

C. 0.21 m

D. 0.42 m

E. It will always float.

FB

mg

The sphere sinks if Bmg F

3water

43

mg R g

3

water

30.106 m

4m

R Minimum 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 minimum diameter necessary for this sphere not to sink 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 ρfluidThis is why you float on the sea (1025 kg/m3) but not on a pool (1000 kg/m3) …

DEMO: Frozen helium

balloon

ACT: Styrofoam and lead

A piece of lead is glued to a slab of Styrofoam. When placed in water, they float as shown.

What happens if you turn the system upside down?

A

The displaced volume in both cases must be the same (volume of water whose weight is equal to the weight of the lead+Styrofoam system)

Pb

styrofoam

Pb

styrofoam

Pb

styrofoam

B

C. It sinks.

1 2

ACT: Floating wood

Two cups are filled to the same level with water. One of the two cups has a wooden block floating in it. Which cup weighs more?

A. Cup 1

B. Cup 2

C. They weigh the same.

The weight of the wood is equal to the weight of the missing liquid (= “displaced liquid”) in 2.

Cup 2 has less water than cup 1.

DEMO: Bucket of water

with wooden block

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

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

dVAv

dtVolume flow

rate

A

v dt

dV Avdt

dmAv

dtMass flow rate

Continuity equation

A1

A2

v1

v2

1 1 1 2 2 2Av Av

The mass flow rate must be the same at any point along the tube (otherwise, fluid would be accumulating or disappearing somewhere)

1 1 2 2Av AvIf 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 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.

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

4 3

hose 22

hose

7.6 10 m / s1.1 m/ s

1.5 10 m

dVdtv

A

hoseLength of hose 1.1 m/ s 4 s 4.3 mv t

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.

a) How long is the hose?

b) How big are the openings in the nozzle?

22 22

nozzlenozzle

9.8 m/ s 5 m0 4.5 m/ s

2 2 2 6 m

g gxxh x v

v h x

We use kinematics to determine vnozzle:

nozzlenozzle

0 x

x v t tv

x

h

2nozzle0

2g

h v t t

hose hose nozzle nozzleA v A v

2 2hose hose nozzle nozzle10v v

hose hosenozzle

nozzle

1.5 cm 1.1 m/ s0.073 cm 0.73 mm

10 10 4.5 m/ s

v

v