Subdivisions of matter

solids liquids gases

rigid will flow will flow

dense dense low density and incompressible and incompressible compressible

fluids

condensed matter

Q: what about thick liquids and soft solids?

AP Physics B: Ch.11 - Fluid Mechanics.

Fluid mechanics

Ordinary mechanics

Mass and force identified with objects

Fluid mechanics

Mass and force “distributed”

Density and Pressure

Density

mV

mV

for element of fluid

mass Mvolume V

for uniform density

mass Mvolume V

units kg m-3

Density and Pressure

Pressure p

p FA

p FA

force per unit area

for uniform force

units N m-2 or pascals (Pa)

Atmospheric pressure at sea level p0

on average 101.3 x103 Pa or 101.3 kPa

Gauge pressure pg

excess pressure above atmospheric p = pg + p0

Density and Pressure

Gauge pressure pg

p = pg + p0pressure in excess of atmospheric

typical pressures total gauge

atmospheric 1.0x105 Pa 0

car tire 3.5x105 Pa 2.5x105 Pa

deepest ocean 1.1x108 Pa 1.1x108 Pa

best vacuum 10-12 Pa - 100 kPa

atmosphericgauge

total

Example

to pump

30 cms

15 cms

The can shown has atmospheric pressure outside. The pump reduces the pressure inside to 1/4 atmospheric

• What is the gauge pressure inside?

• What is the net force on one side?

Fluids at rest (hydrostatics)

Hydrostatic equilibrium

laws of mechanical equilibrium

pressure just above surface is atmospheric, p0

hence, pressure just below surface

must be same, i.e. p0

surface is inequilibrium

Fluids at rest (hydrostatics)

Hydrostatic equilibrium

laws of mechanical equilibrium

(p+p)A

pA

y

element of fluidsurface area A

height y

pA - (p+p)A - mg = 0

Fy =0

p A - Ayg = 0

mg = Ayg

p =- gy

p = p0+gh

at distance h below surface,

pressure is larger by gh

Pressure at depth h

Question

How far below surface of water must one dive for the pressure to increase by one atmosphere?

What is total pressure and what is the gauge pressure, at this depth?

?

Pascal’s principle

The pressure at a point in a fluid in static equilibrium depends only on the depth of that point

Pascal’s principle

The pressure at a point in a fluid in static equilibrium depends only on the depth of that point

Open tube manometer

(i) If h=6 cm and the fluid is mercury (=13600 kg m-3) find the gauge pressure in the tank

(ii) Find the absolute pressureif p0 =101.3 kPa

Pascal’s principle

The pressure at a point in a fluid in static equilibrium depends only on the depth of that point

Barometer

Find p0 ifh=758 mm

Fodo Fidi

diAi do Ao di =Ao

Ai

do

Fodo Fi

Ao

Ai

do Fo Fi

Ao

Ai

Pascal’s principle

The pressure at a point in a fluid in static equilibrium depends only on the depth of that point

A change in the pressure applied to an enclosed incompressible fluidis transmitted to every point in the fluid

Hydraulic press

p Fi

Ai

Fo

Ao

Fo Fi

Ao

Ai

alternative argument based on conservation of energy

work out = work in

volume moved is same on each side

Archimedes’s principle

When a body is fully or partially submerged in a fluid, a buoyant force from the surrounding fluid acts on the body. The force is upward and has a magnitude equal to the weight of fluid displaced.

imagine a hole in the water-a buoyancy force exists

fill it with fluid of mass mf and equilibrium will exist

Fb=mfg

stone more dense than water so sinks

wood less dense than water so floats

now the water displaced is less-just enough buoyancy forceto balance the weight of the wood

Fb=Fg

Fb

Fg

Fg

Example 1

What fraction of an iceberg is submerged?(ice for sea ice =917 kg m-3 and sea for sea water = 1024 kg m-3)

Fb

Fg

Fb=Fg

fluid Vi g= V g

Vi/V = /fluid

volume immersed Vi totalvolume V

Example 2

A “gold” statue weighs 147 N in vacuum and 139 N when immersed in saltwater of density 1024 kg m-3 . What is the density of the “gold”?

Flotation

For object of uniform density

Fluid DynamicsThe study of fluids in motion.

Ideal Fluid

1. Steady flow Velocity of the fluid at any point fixed in space doesn’t change with time. This is called

“ laminar flow”, and for such flow the fluid follows “streamlines”.

2. Incompressible We will assume the density is fixed. Accurate for liquids but not so likely for gases.

3. Inviscid “Viscosity” is the frictional resistance to flow. Honey has high viscosity, water has small viscosity. We will assume no viscous losses. Our approach will only be true for low viscosity fluids.

laminar

turbulent

Equation of continuityStreamlines

Conservation of mass in tube of flow means mass of fluid entering A1 in time t = mass of fluid leaving A2 in time t

For incompressible fluid this means volume is also conserved.

Volume entering and leaving in time t is V

V = A1 v1 t =A2 v2 t

Therefore A1 v1 = A2 v2 Equation of continuity(Streamline rule)

tube of flow

Bernoulli’s equation (Daniel Bernoulli, 1700-1782)

For special case of fluid at rest (Hydrostatics!)

For special case of height constant (y1=y2)

The pressure of a fluid decreases with increasing speed

p1 1

2v1

2 gy1 p2 1

2v2

2 gy2

p 1

2v2 gy constant

p2 p1 g(y2 y1)

p1 1

2v1

2 p2 1

2v2

2

Proof of Bernoulli’s equation

Use work energy theoremwork done by external force (pressure)

=change in KE + change in PE

W=K+ UWork done

Change in KE

Change in PEU Vgy2 Vgy1

K 1

2mv2

2 1

2mv1

2 1

2Vv2

2 1

2Vv1

2

Note: same volume V with mass m enters A1 as leaves A2 in time t

Work done at A1 in time t

(p1A1)v1 t

=p1 V

W p1V p2V (p2 p1)V

ProblemTitanic had a displacement of 43 000 tonnes. It sank in 2.5 hours after being holed 2 m below the waterline.

Calculate the total area of the hole which sank Titanic.

Examples of Bernoulli’s relation at work

Venturi meter

Aircraft lift

Examples of Bernouilli’s relation at work

“spin bowling”