Chapter 9

Solids and Fluids (c)

EXAMPLEEXAMPLEA small swimming pool has an area of 10 square meters. A wooden 4000-kg statue of density 500 kg/m3 is then floated on top of the pool. How far does the water rise?

Note: Density of water = 1000 kg/m3

SolutionSolutionGiven: wood/H20 = 0.5, A = 10 m2, M = 4000 kgFind: h

h

Level is the same as if 4000 kg of water were added = 4 m3

Consider problem: A volume V = 4 m3 of water is added to a swimming pool. What is h?

AVh / = 40 cm

Quiz

1. What is your section number?

2. Three objects rest on bathroom scales at a lake bottom.Object 1 is a lead brick of volume 0.2 m3

Object 2 is a gold brick of volume 0.2 m3

Object 3 is a lead brick of volume 0.1 m3

DATA: specific gravity of lead = 11.3 specific gravity of gold = 19.3

specific gravity of mercury = 13.6Which statement is true?a) #1 and #2 have the same buoyant forceb) #1 and #2 register the same weights on the scalesc) #1 and #3 have the same buoyant forced) #1 and #3 register the same weights on the scalese) If the lake were filled with mercury, the scales would not change.

Equation of ContinuityEquation of Continuity

What goes in must come out!

tAvxAM

Mass that passes a pointin pipe during time t

222111

Continuity of Eq.

vAvA

mass density

ExampleExample

222111

Continuity of Eq.

vAvA

Water flows through a 4.0 cm diameter pipe at 5 cm/s. The pipe then narrows downstream and has a diameter of of 2.0 cm. What is the velocity of the water through the smaller pipe?

1122

21

2

2211

4vvr

rv

vAvA

= 20 cm/s

SolutionSolution

Laminar Flow and TurbulenceLaminar Flow and TurbulenceLaminar or Streamline Flow: Fluid elements move along smooth paths that don’t

cross Friction in laminar flow is called viscosity

Turbulent flow Irregular paths Sets in for high gradients (large velocities or small

pipes)

Ideal FluidsIdeal Fluids

Laminar Flow No turbulence

Non-viscous No friction between fluid layers

Incompressible Density is same everywhere

Bernoulli’s EquationBernoulli’s Equation

constant gyvP 2

2

1

Physical content: the sum of the pressure, kinetic energy per unit volume, and the potential energy per unit volume has the same value at all points along a streamline.

How can we derive this?

Bernoulli’s Equation: derivationPhysical basis: Work-energy relation

All together now:

WithWe get:

Example: Venturi MeterExample: Venturi Meter

A very large pipe carries water with a very slow velocity and empties into a small pipe with a high velocity. If P2 is 7000 Pa lower than P1, what is the velocity of the water in the small pipe?

SolutionSolutionGiven: P = 7000 Pa, = 1000 kg/m3

Find: v

constant2

1 2

formula Basic

vghP

221 2

1vPP

P

v

22

v = 3.74 m/s

Applications of Bernoulli’s Applications of Bernoulli’s EquationEquation

•Venturi meter•Curve balls•Airplanes

Beach Ball DemoBeach Ball Demo

ExampleExampleWater drains out of the bottom of a cooler at 3 m/s, what is the depth of the water above the valve?

SolutionSolution

constant2

1 2

formula Basic

vghP Compare water at top(a) of cooler with water leaving valve(b).

22

2

1

2

1bbbaaa vghPvghP

g

vh

2

2

= 45.9 cm

a

b

Three Vocabulary WordsThree Vocabulary Words

•Viscosity•Diffusion•Osmosis

ViscosityViscosity

Viscosity refers to friction between the layersPressure drop required to force water through pipes(Poiselle’s Law)At high enough velocity, turbulence sets in

d

AvF

DiffusionDiffusion

Molecules move from region of high concentration to region of low concentrationFick’s Law:

D = diffusion coefficient

L

CCDA

time

MassrateDiffusion 12

OsmosisOsmosis

Osmosis is the movement of water through a boundary while denying passage to specific molecules, e.g. salts