REUPLOAD - PHSI191 Personal Cheat Sheet 2020 PDF

Preview

Summary

MATHSProportionality P ∝ Q c is the constant of proportionalityc=QPUnit Conversion Example #1: convert 100 miles per hour into speed in metres per second (1 mile = 1 km)100푚푖푙푒푠ℎ표푢푟×1. 61푘푚1푚푖푙푒×1000푚1푘푚×1 ℎ표푢푟60푚푖푛×1푚푖푛60푠푒푐= 44푚푠1Conversion factor: 61푘푚 1푚푖푙푒 ×1000푚1푘푚×1ℎ표푢푟60푚푖푛×1푚푖푛60푠푒푐= 0. 447...

Description

MATHS Proportionality P∝Q c is the constant of proportionality Q c= P

‘Centre Of Gravity’ Or ‘Centre Of Mass’ The centre of mass is a point through w hich an object’s entire mass appears to act. e.g. if you have a circle, there is a line of symmetry down the middle so you are able to balance it along that line of symmetry.

Unit Conversion Example #1: convert 100 miles per hour into speed in metres per second (1 mile = 1.61 km) 𝑚𝑖𝑙𝑒𝑠 1.61𝑘𝑚 1000𝑚 1 ℎ𝑜𝑢𝑟 1𝑚𝑖𝑛 = 44.7𝑚𝑠1 100 × × × × ℎ𝑜𝑢𝑟 1𝑚𝑖𝑙𝑒 1𝑘𝑚 60𝑚𝑖𝑛 60𝑠𝑒𝑐 Conversion factor: 𝑚𝑠−1 1.61𝑘𝑚 1000𝑚 1ℎ𝑜𝑢𝑟 1𝑚𝑖𝑛 × × × = 0.447 1𝑚𝑖𝑙𝑒 1𝑘𝑚 60𝑚𝑖𝑛 60𝑠𝑒𝑐 𝑚𝑝ℎ Therefore, 80𝑚𝑝ℎ × 0.447

𝑚𝑠 −1

𝑚𝑝ℎ

= 35.76𝑚𝑠 −1

Example #2: C onvert a volume of 20 ml into a volume in m3. Find a conversion factor which may be used to convert ml into m3. Note: 1 ml is 1cm3 1𝑐𝑚 3 1𝑚 3 20𝑚𝑙 × ×( ) = 20 × 10 −6~2 × 10−5 𝑚3 100𝑐𝑚 1𝑚𝑙 Conversion Factor: 𝑚3 −6 𝑐 = 10 𝑚𝑙 MECHANICS Change in velocity: 𝜟𝒗 = 𝒗 𝒇 − 𝒗 𝒊 Time, Distance, Speed 𝛥𝑥 𝑣= 𝛥𝑡 Average Speed ∆𝑥 𝑡𝑜𝑡𝑎𝑙 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑟𝑎𝑣𝑒𝑙𝑙𝑒𝑑 𝑣𝑎𝑣 = = ∆𝑡 𝑡𝑜𝑡𝑎𝑙 𝑡𝑖𝑚𝑒 𝑖𝑡 𝑡𝑎𝑘𝑒𝑠 At constant acceleration, the average speed is also given 𝑣𝑓+𝑣𝑖 by, 𝑣𝑎𝑣 = 2

Instantaneous Speed The average speed gets closer to the instantaneous speed if we reduce ∆𝑥 and ∆𝑡 Acceleration ∆𝑣 𝑎= ∆𝑡 Uniform Acceleration 1 If an object has a uniform acceleration, ∆𝑥 = 𝑣𝑖∆ 𝑡 + 𝑎∆𝑡 2

If the object starts from rest, ∆𝑥 = Note: 𝑣𝑓 = 𝑣𝑖 + 𝑎∆𝑡

1

2

2

𝑎∆𝑡 2

Vector Quantities  Displacement is the name for the vector which describes both the distance and the direction moved. It’s the final displacement minus the initial displacement gives the change in displacement  Velocity is the rate of change of the displacement  Acceleration is the rate of change of the velocity Acceleration Due To Gravity 𝑎 = 𝑔 = 9.8𝑚𝑠−2~ 10𝑚𝑠−2 Weight And Mass 𝐹 = 𝑚𝑎 ⇒ 𝐹 = 𝑚𝑔 𝑊 = 𝑚𝑔 Newton’s Laws Of Motion  Newton’s First Law: Any object continues at rest, or at constant velocity (i.e. at constant speed in a straight line), unless an external force acts on it  Newton’s Second Law: An external force gives the object an acceleration which is proportional to the force. o 𝐹 = 𝑚𝑎 o 𝑎 = 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 (𝑚𝑠 −2 ) o 𝑚 = 𝑚𝑎𝑠𝑠 (𝑘𝑔) o 𝐹 = 𝑓𝑜𝑟𝑐𝑒 (𝑁)

Friction  Friction is the force between two surfaces, parallel to the surfaces.  Friction always opposes relative motion (between the object and the surface).  There are two different kinds of friction: static and kinetic  Maximum friction force (between two surfaces), 𝒇𝒎𝒂𝒙 = 𝝁𝑵  𝝁 is the coefficient of friction  Friction does not depend on area Atmospheric Friction (‘Air Resistance’)  When an object is dropped, it accelerates at g = 9.8ms-2. As its velocity increases, atmospheric friction increases.  When atmospheric friction = mg  Net force is now zero; it will go no faster  Terminal velocity, VT, has been reached.

Centripetal acceleration, 𝑎 =

𝑣2 𝑟

Centripetal acceleration around a circle, 𝑎 =

4𝜋2𝑟 𝑇2

Centripetal Force  Because 𝐹 = 𝑚𝑎, this centripetal acceleration means there must be a centripetal force: 𝐹 =

𝑚𝑣 2

, whenever an 𝑟 object moves in a circle at constant speed.

Hooke’s Law  𝐹 = −𝑘𝑥  k is called the spring constant Energy In Hooke’s Law Deformations  𝑊=

1

𝑘𝑥 2

2 1

 𝑃𝐸 = 𝑘𝑥 2 2

Simple Harmonic Motion  If F = - kx, it will oscillate with simple harmonic motion  The time for each cycle of oscillation is called the period, T  The frequency, f, is the number of cycles per second: 1 1 ,𝑇= 𝑓= 𝑇 𝑓 Energy Conservation 1 1 1 1 2 2 2 2 = 2 𝑘𝑥 + 2 𝑚𝑣 Total energy: 𝐸 = 2 𝑘𝐴 = 2 𝑚𝑣𝑚𝑎𝑥

 (d1) Effort arm - distance from the fulcrum to which the

Period & Frequency Of SHM √𝑘 𝐴 𝑣𝑚𝑎𝑥 = 𝑚 𝑚 ∴ 𝑇 = 2𝜋√ 𝑘

point the arm is applied (F1)

 (d2) Load arm - distance from the fulcrum to which the

point the arm holds the load (F2)

 At balance, in equilibrium: effort x effort arm = load x load

arm  𝐹1 𝑑1 = 𝐹2 𝑑2

∴

1

𝑇

=𝑓=

1 𝑘 √ 2𝜋 𝑚

The (Simple) Pendulum 𝑚𝑔 The restoring force: 𝑇𝐻 = −𝑘𝑥 where 𝑘 =

Classes Of Levers

𝐿

And since, 𝑇 = 2𝜋√𝑘 for SHM. 𝑚

𝑇 = 2𝜋√ , for a simple pendulum 𝑙

Energy Energy changes from one form to another, BUT energy is always conserved Work It involves force and displacement 𝑊 = 𝐹𝑑 (Unit: Joules (J)) when force and displacement are not in the same direction Kinetic Energy The kinetic energy of an object is a measure of the work it can do because of its motion 1 𝑊 = 𝐾𝐸 = 𝑚𝑣 2 2

𝑔

Waves – Frequency, Wavelength, Velocity 1 𝜆 𝑣𝑤𝑎𝑣𝑒 = = 𝑓𝜆 (𝑠𝑖𝑛𝑐𝑒 = 𝑓) 𝑇 𝑇 Generally, 𝑣 = 𝑓𝜆 for any wave Transverse Waves The oscillation is transverse (perpendicular) to propagation direction. Longitudinal Waves Oscillation is in direction of propagation.

Bulk Stress And Strain As the pressure gets bigger, the volume strain decreases and ∆𝑉 gets smaller  There is a negative sign on -ΔP, because we want the bulk modulus to be a positive number, and as the pressure gets bigger, the volume (ΔV) is going to be smaller. So we can make sure B is a positive number by putting in the negative sign  Units of B: Pascals (Pa) or N ewtons per metre squared (𝑁𝑚𝑠 2 ) ∆𝑉 𝑣𝑜𝑙𝑢𝑚𝑒 𝑠𝑡𝑟𝑎𝑖𝑛 = 𝑉0 𝑠𝑡𝑟𝑒𝑠𝑠 −∆ 𝑃 𝐵= =− 𝑠𝑡𝑟𝑎𝑖𝑛 ∆𝑉/𝑉0

Density 𝑚 𝑝= 𝑣 𝑝 = 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 (𝑖𝑛 𝑘𝑔𝑚−3) 𝑚 = 𝑚𝑎𝑠𝑠 (𝑖𝑛 𝑘𝑔) 𝑣 = 𝑣𝑜𝑙𝑢𝑚𝑒 (𝑖𝑛 𝑚3 )

Pressure 𝐹 𝑃= 𝐴  A fluid exerts pressure in all directions  At a given depth, pressure is equal in all directions  Fluid pressure always acts perpendicular to the surface in contact with it  Pressure applied to an enclosed fluid is transmitted undiminished to all parts of the fluid. This is Pascal’s Law  Pressure can produce a change in volume (in the same way that stress can produce a change in length)  Pressure is the same at equal depths in a stationary fluid of uniform density, independent of the shape of the container  For a fluid with constant density, 𝑝 the difference in pressure, ∆𝑃 = 𝑃2 − 𝑃1 , between the fluid at height 𝑦1 and that at 𝑦2 is given by:  𝑃2 − 𝑃1 = −𝑝𝑔(𝑦2 − 𝑦1 )  𝑔 = 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑑𝑢𝑒 𝑡𝑜 𝑔𝑟𝑎𝑣𝑖𝑡𝑦  𝑃1 = 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑎𝑡 𝑝𝑜𝑖𝑛𝑡 1  𝑃2 = 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑎𝑡 𝑝𝑜𝑖𝑛𝑡 2  𝑃2 = 𝑃𝑠𝑢𝑟𝑓 = 𝑃𝑎𝑡𝑚 = 𝑎𝑡𝑚𝑜𝑠𝑝ℎ𝑒𝑟𝑖𝑐 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒  𝑃1 = 𝑃 so that, 𝑃𝑎𝑡𝑚 − 𝑃 = − 𝑝𝑔(ℎ − 0)  𝑃𝑎𝑏𝑠𝑜𝑙𝑢𝑡𝑒 = 𝑃𝑠𝑢𝑟𝑓 + 𝑝𝑔ℎ  𝑃𝑎𝑏𝑠𝑜𝑙𝑢𝑡𝑒 = 𝑃𝑎𝑡𝑚 + 𝑝𝑔ℎ  Absolute pressure is measured relative to a vacuum  Gauge pressure is measured relative to atmospheric pressure 𝑨𝒃𝒔𝒐𝒍𝒖𝒕𝒆 𝒑𝒓𝒆𝒔𝒔𝒖𝒓𝒆 = 𝒂𝒕𝒎𝒐𝒔𝒑𝒉𝒆𝒓𝒊𝒄 𝒑𝒓𝒆𝒔𝒔𝒖𝒓𝒆 + 𝒈𝒂𝒖𝒈𝒆 𝒑𝒓𝒆𝒔𝒔𝒖𝒓𝒆

Superposition Of Waves

Potential Energy An object may store energy because of its position e.g. Ball next to a compressed spring 𝑃𝐸 = 𝑚𝑔ℎ Conservative Forces  Conservative forces result in conservation of mechanical energy  Non-conservative forces are dissipative - e.g. friction  Work done by friction, fd, causes mechanical energy to ‘disappear’ (dissipate) but this energy is not lost - it reappears as an equivalent amount of heat energy (molecules shaking)  Other examples: electrical, spring forces Conservation Of Total Energy  Energy can be changed into different forms, but it is never

created or destroyed  This is the principle of conservation of energy  RECALL Mechanical energy conservation occurs only in

special circumstances  BUT total energy is always conserved. Power 𝑃𝑜𝑤𝑒𝑟 =

or 𝑃 = 𝑡𝑖𝑚𝑒 𝑡𝑎𝑘𝑒𝑛 𝑡 i.e. it is the rate of doing work Units: 1 Watt (W) = 1 joule 𝑤𝑜𝑟𝑘 𝑑𝑜𝑛𝑒

𝜂 = 𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 =

Centripetal Acceleration

equals the negative of their relative speed after the collision”

Law Of The Lever (Principle Of Moments)

Efficiency

Motion In a Circle 2𝜋𝑟 𝑣= 𝑇  Assuming constant speed, the magnitude of the velocity is constant, but the velocity is changing because its direction is changing.  Even when the speed is constant, if an object is moving around in a circle its velocity is changing because its direction is changing.  Acceleration towards the centre of the circle is called centripetal acceleration

 Combining these equations: 𝑣1𝑖 − 𝑣2𝑖 = −(𝑣1𝑓 − 𝑣2𝑓 )  “The relative speed of the objects before the collision

𝑊

𝑤𝑜𝑟𝑘 𝑜𝑢𝑡 𝑤𝑜𝑟𝑘 𝑖𝑛

=

per second

(Js-1 )

𝑤𝑜𝑟𝑘 𝑜𝑢𝑡𝑝𝑢𝑡 𝑒𝑛𝑒𝑟𝑔𝑦 𝑢𝑠𝑒𝑑

Linear Momentum Momentum is a vector quantity 𝑝 = 𝑚𝑣 Newton’s Second Law: Net sum of external forces – ∆𝑣 ∆(𝑚𝑣) ∆𝑝 𝐹 = 𝑚𝑎 = 𝑚 = = ∆𝑡

∆𝑡

∆𝑡

“The (time) rate of change of momentum is proportional to the net external force” Impulse, 𝐹∆𝑡 = ∆𝑝 Types Of Collisions Inelastic:  Momentum is conserved (if no net external forces).  Kinetic energy is not conserved. (Some of it's converted: heat energy, sound…) Sticky Inelastic C ollisions: A collision is often called “totally” inelastic if the objects stick together after they have collided Elastic:  Momentum is conserved (if no net external forces)  Kinetic energy is conserved  Momentum Conservation: 𝑚1 𝑣1𝑖 + 𝑚 2 𝑣2𝑖 = 𝑚1 𝑣1𝑓 + 𝑚2 𝑣2𝑓  Kinetic Energy Conservation: 1 𝑚1 𝑣1𝑖2 + 1 𝑚2 𝑣22𝑖 = 1 𝑚1 𝑣1𝑓 + 𝑚2 𝑣2𝑓 2

2

2

SOLIDS & FLUID S Solids, Liquids & Gases  Solids – Molecules oscillate about more or less fixed centres. The amplitude of the vibration is small and the fixed centres frequently form a repeated spatial pattern. This is called long-range order  Liquids – Intermolecular distances and amplitudes of vibration are slightly larger than for solids. The centres are now free to move, and the material takes the shape of its container. Liquids show regularity of structure only in the immediate neighbourhood of a few molecules. This is called short range order.  Gases – Molecules have much greater kinetic energy, are widely separated and experience very small attractive forces. The molecules move in straight lines until they collide with another molecule or the container wall. Elasticity  When an object is subjected to equal and opposite balanced forces, it undergoes a change in size, shape or both. If the object returns to its original shape and size when the forces are removed then it is said to behave elastically Stress & Strain  𝑠𝑡𝑟𝑒𝑠𝑠 =

𝐹

𝐴

 The units of stress: 1𝑁𝑚−2 = 1 𝑃𝑎𝑠𝑐𝑎𝑙 = 1 𝑃𝑎 ∆𝐿 𝐿−𝐿0  𝑡𝑒𝑛𝑠𝑖𝑙𝑒 𝑠𝑡𝑟𝑎𝑖𝑛 = = 𝐿0

𝐿0

 Tensile strain is positive

 Compressive strain is negative Hooke’s Law And Young’s Modulus  𝑠𝑡𝑟𝑒𝑠𝑠 ∝ 𝑠𝑡𝑟𝑎𝑖𝑛  Young’s modulus (𝛾): Units – 𝑁𝑚 −2  Young modulus – the force is perpendicular to the area 𝑡𝑒𝑛𝑠𝑖𝑙𝑒 𝑠𝑡𝑟𝑒𝑠𝑠 𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑣𝑒 𝑠𝑡𝑟𝑒𝑠𝑠  𝛾𝑡𝑒𝑛𝑠𝑖𝑜𝑛 = , 𝛾𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 = 𝑡𝑒𝑛𝑠𝑖𝑙𝑒 𝑠𝑡𝑟𝑎𝑖𝑛

𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑣𝑒 𝑠𝑡𝑟𝑎𝑖𝑛

Shear Stress And Strain  𝑠ℎ𝑒𝑎𝑟 𝑠𝑡𝑟𝑒𝑠𝑠 = 𝐹/𝐴  𝑠ℎ𝑒𝑎𝑟 𝑠𝑡𝑟𝑎𝑖𝑛 =

∆𝑥 𝐿0

 𝑠ℎ𝑒𝑎𝑟 𝑚𝑜 𝑑𝑢𝑙𝑢𝑠 = 𝐺 =

𝑠𝑡𝑟𝑒𝑠𝑠

𝑠𝑡𝑟𝑎𝑖𝑛

=

𝐹/𝐴

∆𝑥/𝐿0

Measurement Of Pressure Pressure is the same at equal depths in a stationary fluid of uniform density 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑎𝑡 𝑏𝑜𝑡𝑡𝑜 𝑚 𝑜𝑓 𝑅𝐻 𝑐𝑜𝑙𝑢𝑚 = 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑎𝑡 𝑏𝑜𝑡𝑡𝑜 𝑚 𝑜𝑓 𝐿𝐻 𝑐𝑜𝑙𝑢𝑚𝑛 𝑃 + 𝑝𝑔 𝑦𝐵 = 𝑃𝑎𝑡𝑚 + 𝑝𝑔𝑦_𝐴 𝑃 − 𝑃𝑎𝑡𝑚 = 𝑝𝑔(𝑦𝐴 − 𝑌𝐵 ) = 𝑝𝑔ℎ  𝑝 = 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑙𝑖𝑞𝑢𝑖𝑑 𝑖𝑛 𝑚𝑎𝑛𝑜𝑚𝑒𝑡𝑒𝑟

 ℎ = 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑖𝑛 ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑎𝑟𝑚𝑠 Gauge Pressure 𝑃𝑔𝑎𝑢𝑔𝑒 = 𝑝𝑔ℎ

Mercury Barometer (Measures A bsolute Pressure) 𝑃 = 𝑃𝑎𝑡𝑚 = 𝑝𝑔ℎ + 0Pa 𝑝 = 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑚𝑒𝑟𝑐𝑢𝑟𝑦 Buoyancy Archimedes’ principle: “when a body is immersed in a fluid, the fluid exerts an upward force in the body equal to the weight of fluid that is displaced by the body” 𝑏𝑢𝑜𝑦𝑎𝑛𝑡 𝑓𝑜𝑟𝑐𝑒 = 𝑤𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒 𝑓𝑙𝑢𝑖𝑑 (𝑝𝑓 𝑉𝑔) 𝑝𝑓 = 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑙𝑢𝑖𝑑 𝑉 = 𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑢𝑏𝑚𝑒𝑟𝑔𝑒𝑑 𝑝𝑜𝑟𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑜𝑑𝑦

The body comes to rest in equilibrium when:  𝑡𝑜𝑡𝑎𝑙 𝑤𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑜𝑑𝑦 = 𝑏𝑢𝑜𝑦𝑎𝑛𝑡 𝑓𝑜𝑟𝑐𝑒  𝑚𝑔 = 𝑝𝑓 𝑉𝑔

 𝑚𝑔 > 𝑝𝑓 𝑉𝑔 ⇒ 𝑏𝑜𝑑𝑦 𝑠𝑖𝑛𝑘𝑠  𝑚𝑔 < 𝑝𝑓 𝑉𝑔 ⇒ 𝑏𝑜𝑑𝑦 𝑟𝑖𝑠𝑒𝑠

 𝑚 = 𝑡𝑜𝑡𝑎𝑙 𝑚𝑎𝑠𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑜𝑑𝑦

Capillarity  Cohesive forces are attractive forces between like molecules  Adhesive forces are attractive forces between molecules that are different from each other  When the tube is dipped into the liquid, attractive forces between the water and the glass mean that the water is attracted up the sides of the glass.  In the centre of the tube, cohesive forces drag water upwards. Water-water forces pull up the centre, and water-glass forces pull up water along the side of the tube.  This phenomenon is only seen when the tube is very narrow. This is because the rising of the fluid in the tube only takes place until the upwards force is balanced by the mg of the fluid up the tube. If the tube was wider then the mg will be much bigger and you tend to not see this phenomenon

Fluid Dynamics Of Non-Viscous Fluids  Incompressible – the fluid has constant density throughout  Viscosity – internal friction in the fluid  Laminar flow – layers of fluid slide smoothly past each other and there is a steady state pattern. Laminar flow is characteristic of lower fluid velocities  Turbulent flow – irregular, complex flow with swirling, mixing and eddies. N o steady state pattern. Occurs at high velocities or where objects in the flow produce large changes in velocity. Equation Of Continuity For an incompressible fluid, since no fluid crosses the walls of the tube then, 𝑚𝑎𝑠𝑠 𝑓𝑙𝑜𝑤𝑖𝑛𝑔 𝑖𝑛 = 𝑚𝑎𝑠𝑠 𝑓𝑙𝑜𝑤𝑖𝑛𝑔 𝑜𝑢𝑡 If fluid density is constant (incompressible) then, 𝐴1 𝑣1 = 𝐴2 𝑣 2 We can express continuity equation in another way, 𝐴𝑣 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑎𝑙𝑜𝑛𝑔 𝑎 𝑔𝑖𝑣𝑒𝑛 𝑓𝑙𝑜𝑤 𝑡𝑢𝑏𝑒 ∆𝑉 = 𝐴𝑣 (𝑚3 𝑠 −1) 𝐹 = 𝑣𝑜𝑙𝑢𝑚𝑒 𝑓𝑙𝑜𝑤 𝑟𝑎𝑡𝑒 = ∆𝑡  When the pipes wider the fluid is moving more slowly  When the pipe gets narrower the fluid has to speed up Bernoulli’s Equation – Energy Conservation 1 2 1 2 𝑃1 + 𝑝𝑣1 + 𝑝𝑔ℎ1 = 𝑃2 + 𝑝𝑣2 + 𝑝𝑔ℎ2 2 2

Alternatively, 1 𝑃 + 𝑝𝑣 2 + 𝑝𝑔ℎ = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑎𝑡 𝑒𝑣𝑒𝑟𝑦 𝑝𝑜𝑖𝑛𝑡 𝑖𝑛 𝑓𝑙𝑢𝑖𝑑 2 𝑃 𝑖𝑠 𝑡ℎ𝑒 𝑎𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒

Torricelli’s Theorem 𝑣0 = √2𝑔(ℎ𝑠 − ℎ𝑜 ) Fluid Dynamics Of Viscous Fluids  Streamline - a curve whose tangent, at every point, is in the direction of the fluid velocity at that point. Where streamlines converge the velocity increases; where streamlines diverge the velocity decreases  Laminar and non-viscous - The streamlines are all parallel and they are not getting closer together or apart so the fluid speed is constant. It is constant at all radii and its constant as you move along the pipe  Laminar and viscous - When you have a viscous fluid, the fluid has to move at a speed of the wall that it encounters. So right at the wall of the pipe, the fluid has to be stopped as its velocity has to be zero. But there is fluid moving through the pipe, so as the fluid moves away from the wall the fluid has to be speeding up  Turbulent - The molecules go everywhere; they move in all directions. In general, they might go somewhere but not by in an orderly fashion Viscosity  Because of viscosity the force to right will be transmitted through the fluid and tend to drag the lower plate to the right. To keep the lower plate stationary, we must apply an equal and opposite force to the left  We are exerting a force to the right  That force to the right is going to make the top layer move  But the fluid layer at the top is going to be dragging the one underneath due to friction  And the one underneath and so on until the bottom will also be dragging  And if w e didn’t hold onto the bottom plate, it will eventually move and get dragged along the fluid  So what we have then is a force being applied on the top layer of the fluid, but to keep the bottom layer fixed we also have to apply an equal and opposite force  So our fluid layer is experiencing a sheer stress  𝑠𝑡𝑟𝑎𝑖𝑛 𝑟𝑎𝑡𝑒 =

1 𝑑𝑑′

∆𝑡 𝐿

=

1 ∆𝑥

∆𝑡 𝐿

=

𝑣

𝐿

Molecular Transport Phenomena Free diffusion is the slow process of movement of a substance due to a random thermal motion. A substance moves from a region where it has a high concentration to one where it has a lower concentration. The average distance moved by molecules of type A, 𝑥 𝑟𝑚𝑠, in substance B in time 𝑡 is, 𝑥 𝑟𝑚𝑠 = √2𝐷𝐴𝐵𝑡. Where 𝐷𝐴𝐵 is the diffusion constant for molecules A in substance B. Osmosis And Membrane

Dialysis

–

Diffusion

Through

A

 Semipermeable membrane – membrane which may have small pores and allow only one type of molecule to pass through, or which may dissolve or react with molecules passing through  Osmosis – transport of water through a semipermeable membrane from region of high water concentration to one of low water concentration (i.e. low solute concentration to high solute concentration). Can create a substantial pressure difference across the membrane  Osmotic pressure – the back pressure that stops osmosis if one solution is pure water. C an be large (e.g. sufficient to support 260m H2O for pure water sea water system).  Relative osmotic pressure – the back pressure that stops osmosis if neither solution is pure water  Dialysis – transport of any molecule other than water through a semipermeable membrane because of a concentration difference  Reverse osmosis (reverse dialysis) – transport of water (other molecule) through a membrane due to the back pressure. Occurs in the opposite direction to osmosis (dialysis)  Active transport – a process in which a living membrane expends energy to move a substance across it. Can occur in the opposite direction to osmosis THERMODYNAMICS Temperature  Temperature is the physical quantity (a number) we use to measure hot and cold. Measured with a thermometer. Units: Kelvin (K)  For a constant volume,

𝑇

𝑇𝑡𝑝

=

𝑃

𝑃𝑡𝑝

 The temperature, T, measured using a constant volume 𝑃 gas thermometer is therefore, 𝑇 = 273 .116 × 𝑃𝑡𝑝

 The triple point temperature, 273.16 K, is defined as the temperature of a mixture of water liquid, vapour and ice all in equilibrium  𝑇(𝑜𝐶 ) = 𝑇 (𝐾) − 273.15 Thermal Energy  Gases have energy due to the random translational motion of the atoms or molecules  Matter generally also has energy due to other random atomic or molecular motions, such as rotational or vibrational motion  This energy is called thermal energy

 𝑠𝑡𝑟𝑎𝑖𝑛 𝑟𝑎𝑡𝑒 𝛼 𝑠ℎ𝑒𝑎𝑟 𝑠𝑡𝑟𝑒𝑠𝑠  Constant of proportionality is viscosity, 𝜂:

 The “more motion” – the more thermal energy  The higher temperature (big T) will give away thermal temperature to the lower temperature (smaller T)

𝑣 𝐹 = 𝜂𝐴 𝐿  𝜂 is low for water (i.e. require a small force to keep the fluid moving)  𝜂 is high for treacle (i.e. require a big force to keep the fluid moving)

Heat Transfer And Heat

𝜂=

𝑠ℎ𝑒𝑎𝑟 𝑠𝑡𝑟𝑒𝑠𝑠 𝑠𝑡𝑟𝑎𝑖𝑛 𝑟𝑎𝑡𝑒

=

𝐹/𝐴

𝑣/𝐿

 Viscosity, 𝜂, is the measure of the friction in the fluid:

For laminar flow of an incompressible, viscous fluid in a pipe  Fluid flows from high pressure to low pressure  The greater the pressure difference the greater the volume flow rate, 𝐹 ∝ (𝑃2 − 𝑃1 )  Friction within the fluid and between the fluid and the 1 walls of the container resists the flow, 𝐹 ∝ 𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒  For laminar flow, resistance to the flow increases as: o The viscosity increases o The tube gets longer o The radius of the tube decreases

Laminar/Turbulent Flow  When the velocity of a fluid exceeds a critical value the flow becomes highly complex. Thus, R eynolds number, 𝑝𝑣𝑙 𝑅𝑒 = 𝜂

     

𝑝 = 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑣 = 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝐿 = 𝑐ℎ𝑎𝑟𝑎𝑐𝑡𝑒𝑟𝑖𝑠𝑡𝑖𝑐 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛 (𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 𝑖𝑛 𝑎 𝑝𝑖𝑝𝑒) 𝜂 = 𝑣𝑖𝑠𝑐𝑜𝑠𝑖𝑡𝑦 𝑅𝑒 < 𝑎𝑏𝑜𝑢𝑡 2000 𝑖𝑚𝑝𝑙𝑖𝑒𝑠 𝑓𝑙𝑜𝑤 𝑙𝑎𝑚𝑖𝑛𝑎𝑟 𝑅𝑒 > 𝑎𝑏𝑜𝑢𝑡 3000 𝑖𝑚𝑝𝑙𝑖𝑒𝑠 𝑓𝑙𝑜𝑤 𝑡𝑢𝑟𝑏𝑢𝑙𝑒𝑛𝑡

Laminar, Viscous Flow – Poiseuille’s Law  Poiseuille’s Law: 𝐹 =

(𝑃2−𝑃1)𝜋𝑟4 8𝜂

(𝑖𝑛 𝑚 3 𝑠 −1 )

 𝑟 = 𝑟𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑢𝑏𝑒  𝑙 = 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑢𝑏𝑒  𝜂 = 𝑡ℎ𝑒 𝑣𝑖𝑠𝑐𝑜𝑠𝑖𝑡𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑙𝑢𝑖𝑑  (𝑃2 − 𝑃1 ) = 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑑𝑟𝑜𝑝 𝑎𝑙𝑜𝑛𝑔 𝑡ℎ𝑒 𝑡𝑢𝑏𝑒

 Big object/small...