Hello!

• I’m Chris Blake, your lecturer for the rest of semester • We’ll cover: fluid motion, thermal physics, electricity, revision • MASH centre in AMDC 503 - 09.30-16.30 daily • My consultation hours: Tues 10.30-12.30 • Wayne’s consultation hours: Thurs 2.30-4.30 • [email protected] • 03 9214 8624 Fluid Motion

• Density and • Hydrostatic Equilibrium and Pascal’s Law • Archimedes' Principle and • Fluid Dynamics • Conservation of Mass: Continuity Equation • Conservation of Energy: Bernoulli’s Equation • Applications of Fluid Dynamics A microscopic view

Solid rigid body

Liquid Fluid Incompressible

Gas Fluid compressible What new physics is involved?

• Fluids can flow from place- to-place

• Their density can change if they are compressible (for example, gasses)

• Fluids are pushed around by pressure

• An object immersed in a fluid experiences buoyancy Density

• The density of a fluid is the of mass

= = 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚 𝑘𝑘𝑘𝑘 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝜌𝜌 𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈 𝑎𝑎𝑎𝑎𝑎𝑎 3 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 𝑉𝑉 𝑚𝑚 • Mass = 100 g = 0.1 kg

• Volume = 100 cm3 = 10-4 m3

• Density = 1 g/cm3 = 1000 kg m3 0% The shown cubic vessels contain the stated matter. Which fluid has the highest density ? . . 1 2 3 4 5 1 of 1 of 𝟏𝟏 water at 𝟐𝟐 water at 73𝑘𝑘𝑘𝑘° 273𝑘𝑘𝑘𝑘°

1 𝐶𝐶 1 𝐶𝐶

. 𝑚𝑚 . 𝑚𝑚 1 of 1 of 𝟑𝟑 water at 𝟒𝟒 water at 273𝑘𝑘𝑘𝑘° 373𝑘𝑘𝑘𝑘° 1 3 𝐾𝐾 𝐶𝐶 𝑚𝑚 𝑚𝑚 Pressure

• Pressure is the concentration of a – the force exerted per unit area

Exerts a pressure on the Greater pressure! sides and through the fluid (same force, less area) Pressure

= 𝐹𝐹 𝑝𝑝 𝐴𝐴

• Units of pressure are N/m2 or Pascals (Pa) – 1 N/m2 = 1 Pa

• = 1 atm = 101.3 kPa = 1 x 105 N/m2 A What is responsible for the force which holds 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣B𝑣𝑣 urban climber B in place when using suction cups.

1. The force of friction 0% 2. Vacuum pressure exerts a pulling force 3. Atmospheric pressure exerts a pushing force 4. The normal force of the glass. 1. 2. 3. 4. Hydrostatic Equilibrium

• Pressure differences drive fluid flow

• If a fluid is in equilibrium, pressure forces must balance

• Pascal’s law: pressure change is transmitted through a fluid Hydrostatic Equilibrium with Gravity

Derivation: + =

𝑃𝑃 𝑑𝑑𝑑𝑑 𝐴𝐴= − 𝑝𝑝𝑝𝑝 𝑚𝑚𝑚𝑚

𝑑𝑑𝑑𝑑 𝐴𝐴 =𝜌𝜌 𝐴𝐴 𝑑𝑑� 𝑔𝑔 𝑑𝑑𝑑𝑑 𝜌𝜌𝜌𝜌 =𝑑𝑑�0 + 𝑃𝑃 𝑃𝑃 𝜌𝜌𝜌𝜌휌

Pressure in a fluid is equal to the of the fluid per unit area above it: = +

𝑃𝑃 𝑃𝑃0 𝜌𝜌𝜌𝜌휌 Consider the three open containers filled with water. How do the at the bottoms compare ?

A. B. C.

1. = = 0% 2. < = 𝑷𝑷𝑨𝑨 𝑷𝑷𝑩𝑩 𝑷𝑷𝑪𝑪 3. < < 𝑷𝑷𝑨𝑨 𝑷𝑷𝑩𝑩 𝑷𝑷𝑪𝑪 4. < < 𝑷𝑷𝑨𝑨 𝑷𝑷𝑩𝑩 𝑷𝑷𝑪𝑪 5. Not enough information 1. 2. 3. 4. 5. 𝑷𝑷𝑩𝑩 𝑷𝑷𝑨𝑨 𝑷𝑷𝑪𝑪 The three open containers are now filled with oil, water and honey respectively. How do the pressures at the bottoms compare ?

honey A. B. C. oil water 1. = = 0% 2. < = 𝑷𝑷𝑨𝑨 𝑷𝑷𝑩𝑩 𝑷𝑷𝑪𝑪 3. < < 𝑷𝑷𝑨𝑨 𝑷𝑷𝑩𝑩 𝑷𝑷𝑪𝑪 4. < < 𝑷𝑷𝑨𝑨 𝑷𝑷𝑩𝑩 𝑷𝑷𝑪𝑪 5. Not enough information 1 2 3 4 5 𝑷𝑷𝑩𝑩 𝑷𝑷𝑨𝑨 𝑷𝑷𝑪𝑪 Calculating Crush Depth of a Submarine Q. A nuclear submarine is rated to withstand a pressure difference of 70 before catastrophic failure. If the internal air pressure is maintained at 1 , what is the maximum permissible depth ? 𝑎𝑎𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎𝑎𝑎

= 0 +

𝑃𝑃 𝑃𝑃 𝜌𝜌𝑔𝑔� 6 3 3 0 = 70 = 7.1 × 10 ; = 1 × 10 /

𝑃𝑃 − 𝑃𝑃 𝑎𝑎𝑎𝑎𝑎𝑎7.1 × 106 𝑃𝑃𝑃𝑃 𝜌𝜌 𝑘𝑘𝑘𝑘 𝑚𝑚 = 0 = = 720 1 × 103 × 9.8 𝑃𝑃 − 𝑃𝑃 ℎ 𝑚𝑚 𝜌𝜌𝜌𝜌 Measuring Pressure Q. What is height of mercury (Hg) at 1 ?

𝑎𝑎𝑎𝑎𝑎𝑎 = 13.6 / 3 𝐻𝐻𝐻𝐻 =𝜌𝜌 0 + 𝑔𝑔 𝑐𝑐𝑐𝑐= /

𝑃𝑃 𝑃𝑃 1 ×𝜌𝜌𝜌𝜌10휌5→ ℎ 𝑃𝑃 𝜌𝜌𝜌𝜌 = = 0.75 1.36 × 104 × 9.8 ℎ 𝑚𝑚

Atmospheric pressure can support a 10 meters high column of water. Moving to higher density fluids = + allows a table top barometer to be easily constructed. 𝑝𝑝 𝑝𝑝0 𝜌𝜌𝜌𝜌휌 Pascal’s Law • Pressure force is transmitted through a fluid

Q. A large piston supports a car. The total mass of the piston and car is 3200 . What force must A2 F2 A be applied to the smaller piston ? 1 𝑘𝑘𝑘𝑘

Pressure at the same height is the same! (Pascal’s Law) × 0.152 1 = 2 = 1 = × 3200 × 9.8 = 490 1 × 1.202 𝐹𝐹1 𝐹𝐹2 𝐴𝐴2 𝜋𝜋 𝐹𝐹 𝑚𝑚𝑚𝑚 𝑁𝑁 𝐴𝐴 𝐴𝐴 𝐴𝐴 𝜋𝜋 Gauge Pressure

Gauge Pressure is the pressure difference from atmosphere. (e.g. Tyres) = +

𝑃𝑃𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑃𝑃𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎푎𝑎𝑎𝑎𝑎𝑎𝑎 𝑃𝑃𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 Archimedes’ Principle and Buoyancy

Why do some things float and other things sink ? Archimedes’ Principle and Buoyancy

Objects immersed in a fluid experience a Buoyant Force!

= =

𝐹𝐹𝐵𝐵 𝑚𝑚𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑔𝑔 𝜌𝜌𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑉𝑉𝑉𝑉

= =

𝑊𝑊 𝑚𝑚𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑔𝑔 𝜌𝜌𝑠𝑠𝑠𝑠𝑙𝑙𝑙𝑙𝑙𝑙𝑉𝑉𝑉𝑉 The Buoyant Force is equal to the weight of the displaced fluid ! Archimedes’ Principle and Buoyancy

The hot-air balloon floats because the weight of air displaced (= the buoyancy force) is greater than the weight of the balloon

The Buoyant Force is equal to the weight of the displaced fluid ! Which of the three cubes of length shown below has the largest buoyant force ? water stone wood𝑙𝑙

water FB FB FB

m1g m2g m3g A. B. C. 1. 0% 2. 𝒘𝒘𝒘𝒘𝒘𝒘𝒘𝒘𝒘𝒘 3. 𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔 4. 𝒘𝒘𝒘𝒘𝒘𝒘𝒘𝒘 5. Not enough information 1 2 3 4 5 𝒕𝒕𝒕𝒕𝒕𝒕 𝒃𝒃𝒃𝒃𝒃𝒃𝒃𝒃𝒃𝒃𝒃𝒃𝒃𝒃 𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇 𝒊𝒊𝒊𝒊 𝒕𝒕𝒕𝒕𝒕𝒕 𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔 Example Archimedes’ Principle and Buoyancy Q. Find the apparent weight of a 60 concrete block when you lift it under water, = 2200 / 3 Evaluate𝑘𝑘𝑘𝑘 Interpret 𝜌𝜌𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑘𝑘𝑘𝑘 𝑚𝑚 Water provides a buoyancy force Fnet = mg − Fb = wapparent Apparent weight should be less F = m g = ρ Vg Develop b disp water water ρ mg w = mg − water app ρ Fb con m ρwater ρcon = wapp = mg(1− ) V ρcon m 1000 V = = 60×9.8×(1− ) = 321 N ρ 2200 w = mg con Assess The Buoyant Force is equal to the weight of the displaced fluid. Floating Objects

Q. If the density of an iceberg is 0.86 that of seawater, how much of an iceberg’s volume is below the sea?

= 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓= 𝐹𝐹𝐵𝐵 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤� 𝑜𝑜𝑜𝑜 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑

𝑠𝑠𝑠𝑠=𝑠𝑠 = 𝑉𝑉 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 𝐵𝐵 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠𝑠𝑠 𝐹𝐹 𝑚𝑚 , 𝑔𝑔= 𝜌𝜌 𝑉𝑉 𝑔𝑔

𝐼𝐼𝐼𝐼 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒= 𝐹𝐹𝐵𝐵 =𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤� 𝑜𝑜𝑜𝑜 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖

𝐹𝐹𝐵𝐵 𝑚𝑚𝑖𝑖𝑖𝑖𝑖𝑖 𝑔𝑔 𝜌𝜌𝑖𝑖𝑖𝑖𝑖𝑖 𝑉𝑉𝑖𝑖𝑖𝑖𝑖𝑖 𝑔𝑔 = = = 0.86 𝑉𝑉𝑠𝑠𝑠𝑠𝑠𝑠 𝜌𝜌𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝜌𝜌𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑉𝑉𝑠𝑠𝑠𝑠𝑠𝑠 𝑔𝑔 𝜌𝜌𝑖𝑖𝑖𝑖𝑖𝑖𝑉𝑉𝑖𝑖𝑖𝑖𝑖𝑖 𝑔𝑔 → 𝑉𝑉𝑖𝑖𝑖𝑖𝑖𝑖 𝜌𝜌𝑖𝑖𝑖𝑖𝑖𝑖 A beaker of water weighs . A block weighting is suspended in the water by 1 a spring balance reading 𝑤𝑤. Does the 2 0% scale read 𝑤𝑤 1. 𝑤𝑤3 2. + 𝒘𝒘𝟏𝟏 3. + 𝒘𝒘𝟏𝟏 𝒘𝒘𝟐𝟐 scale 4. 𝟏𝟏 + 𝟑𝟑 𝒘𝒘 𝒘𝒘 1 2 3 4 5 5. + 𝒘𝒘𝟏𝟏 𝒘𝒘𝟐𝟐 − 𝒘𝒘𝟑𝟑 𝒘𝒘𝟏𝟏 𝒘𝒘𝟑𝟑 − 𝒘𝒘𝟐𝟐 Centre of Buoyancy

The Centre of Buoyancy is given by the Centre of Mass of the displaced fluid. For objects to float with stability the Centre of Buoyancy must be above the Centre of Mass of the object. Otherwise Torque yield Tip ! Fluid Dynamics Laminar (steady) flow is where each particle in the fluid moves along a smooth path, and the paths do not cross.

Streamlines spacing measures velocity and the flow is always tangential, for steady flow don’t cross. A set of streamlines act as a pipe for an incompressible fluid

Non-viscous flow – no internal friction (water OK, honey not)

Turbulent flow above a critical speed, the paths become irregular, with whirlpools and paths crossing. Chaotic and not considered here. Conservation of Mass: The Continuity Eqn.

“The water all has to go somewhere”

The rate a fluid enters a pipe must equal the rate the fluid leaves the pipe. i.e. There can be no sources or sinks of fluid. Conservation of Mass: The Continuity Eqn. A A1 2

fluid in → fluid out →

v 1 v2 Q. How much fluid flows across each area in a time : v2∆t

v1∆t ∆𝑡𝑡

A2 A1

∆m = ρV1 = ρ A1 v1∆t ∆m = ρV2 = ρ A2v2∆t

∆m flow rate : = ρ Av continuity eqn : A v = A v ∆t 1 1 2 2 Conservation of Mass: The Continuity Eqn. Q. A river is 40m wide, 2.2m deep and flows at 4.5 m/s. It passes through a 3.7-m wide gorge, where the flow rate increases to 6.0 m/s. How deep is the gorge?

2 = 2 2

1 = 1 1 𝐴𝐴 𝑤𝑤 𝑑𝑑

𝐴𝐴 𝑤𝑤 𝑑𝑑 1 1 = 2 2 1 1 1 = 2 2 2 40 × 2.2 × 4.5 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑒𝑒=𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒1 1 1∶ =𝐴𝐴 𝑣𝑣 𝐴𝐴 𝑣𝑣 → 𝑤𝑤 =𝑑𝑑 18𝑣𝑣 𝑤𝑤 𝑑𝑑 𝑣𝑣 2 3.7 × 6.0 𝑤𝑤 𝑑𝑑2 𝑣𝑣2 𝑑𝑑 𝑚𝑚 𝑤𝑤 𝑣𝑣 Conservation of Energy: Bernoulli’s Eqn. What happens to the energy density of the fluid if I raise the ends ?

v1∆t

y2 ∆ y1 v2 t

1 2 1 2 Energy per unit p1 + 2 ρ v1 + ρ g y1= p2 + 2 ρ v2 + ρ g y2= const volume

Total energy per unit volume is constant + 1 ρ 2 + ρ = at any point in fluid. p 2 v g y const Conservation of Energy: Bernoulli’s Eqn. Q. Find the velocity of water leaving a tank through a hole in the side 1 metre below the water level.

+ 2 + = 1 𝑃𝑃 2𝜌𝜌:𝜌𝜌 =𝜌𝜌1𝜌𝜌𝜌𝜌 ,𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐= 0, = 1

𝐴𝐴𝐴𝐴 𝑡𝑡𝑡𝑡� 𝑡𝑡𝑡𝑡𝑡𝑡 𝑃𝑃: =𝑎𝑎𝑎𝑎𝑎𝑎1 𝑣𝑣, =?𝑦𝑦, = 0𝑚𝑚

𝐴𝐴𝐴𝐴 𝑡𝑡𝑡𝑡� 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏+ 𝑃𝑃 = 𝑎𝑎𝑎𝑎𝑎𝑎+ 𝑣𝑣2 𝑦𝑦 𝑚𝑚 1 = 𝑃𝑃 =𝜌𝜌𝜌𝜌𝜌𝜌2 ×𝑃𝑃9.8 ×2𝜌𝜌1𝜌𝜌 = 4.4 /

𝑣𝑣 2𝑔𝑔𝑔𝑔 𝑚𝑚 𝑠𝑠 Which of the following can be done to increase the flow rate out of the water tank ?

ℎ 1. Raise the tank ( ) 0% 2. Reduce the hole size 𝐻𝐻 ↑ 𝑯𝑯 3. Lower the water level ( ) 4. Raise the water level ( ) ↓ 𝒉𝒉 5. None of the above

↑ 𝒉𝒉 1 2 3 4 5 Summary: fluid dynamics

Continuity equation: mass is conserved! × × =

𝜌𝜌 𝑣𝑣For𝐴𝐴 liquids:𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 = × =

𝜌𝜌 ( 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐, → 𝑣𝑣, 𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐) 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝜌𝜌 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 𝑣𝑣 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝐴𝐴 Bernoulli’s equation: energy is conserved! + 2 + = 1 ( 𝑃𝑃 2𝜌𝜌, 𝜌𝜌 𝜌𝜌𝜌𝜌𝜌𝜌, 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐, ) 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝜌𝜌 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 𝑣𝑣 ℎ𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒� 𝑦𝑦 Bernoulli’s Effect and Lift

Newton’s 3rd law + 2 + = (air pushed downwards) 1 𝑃𝑃 2𝜌𝜌𝜌𝜌 𝜌𝜌𝜌𝜌𝜌𝜌 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 Lift on a wing is often explained in textbooks by Bernoulli’s Principle: the air over the top of the wing moves faster than air over the bottom of the wing because it has further to move (?) so the pressure upwards on the bottom of the wing is smaller than the downwards pressure on the top of the wing.

Is that convincing? So why can a plane fly upside down? Chapter 15 Fluid Motion Summary

• Density and Pressure describe bulk fluid behaviour • Pressure in a fluid is the same for points at the same height • In hydrostatic equilibrium, pressure increases with depth due to gravity • The buoyant force is the weight of the displaced fluid • Fluid flow conserves mass (continuity eq.) and energy (Bernoulli’s equation) • A constriction in flow is accompanied by a velocity and pressure change. • Reread, Review and Reinforce concepts and techniques of Chapter 15

Examples 15.1 , 15.2 Calculating Pressure and Pascals Law Examples 15.3 , 15.4 Buoyancy Forces: Working Underwater + Tip of Iceberg Examples 15.5 Continuity Equation: Ausable Chasm Examples 15.6, 15.7 Bernoulli's Equation – Draining a Tank and Venturi Flow