Applied PhysicsApplied Physics

ContentsContents Rotational DynamicsRotational Dynamics Thermodynamics & EnginesThermodynamics & Engines

Rotational DynamicsRotational Dynamics Angular velocityAngular velocity: the angle of a circle (arc) mapped out : the angle of a circle (arc) mapped out

by a rotating object per second:by a rotating object per second:

ωω = = θθss-1-1

Angular displacementAngular displacement: : θθ

Angular velocityAngular velocity: : ωω = = θθtt-1-1

Angular accelerationAngular acceleration: : αα = = ΔωΔω//ΔΔtt

Rotational DynamicsRotational Dynamics Moment of InertiaMoment of Inertia: Inertia = objects have a degree of : Inertia = objects have a degree of

reluctance to move. Moment of inertia is this but in reluctance to move. Moment of inertia is this but in rotational movement. Objects oppose the movement of rotational movement. Objects oppose the movement of angular acceleration. The more they oppose, the angular acceleration. The more they oppose, the greater the moment of inertia (kgmgreater the moment of inertia (kgm22))

Circular disc:Circular disc: I = MrI = Mr22/2/2

Solid cylinder:Solid cylinder: I = MrI = Mr22

Solid sphere:Solid sphere: I = 2MrI = 2Mr22/5/5

Kinetic Energy:Kinetic Energy: EEKK = ½I = ½Iωω22

Rotational DynamicsRotational Dynamics TorqueTorque: Turning force: Turning force

Pulling force causes torque, T:Pulling force causes torque, T: T = FrT = Fr

In terms of inertia: In terms of inertia: T = IT = Iαα

Rotational Momentum & PowerRotational Momentum & Power Angular Momentum, (L)Angular Momentum, (L): momentum = mass x velocity. : momentum = mass x velocity.

Angular momentum occurs in rotational movementAngular momentum occurs in rotational movement

L (kgmL (kgm22ss-1-1) = I) = Iωω

angular momentum before = angular momentum afterangular momentum before = angular momentum after

ImpulseImpulse: change in momentum: change in momentum

Angular Impulse, Angular Impulse, ΔΔL: change in angular momentumL: change in angular momentum

ΔΔL = TL = TΔΔtt

(small torque for long duration = large torque for small (small torque for long duration = large torque for small duration)duration)

Rotational Momentum & PowerRotational Momentum & Power Work & PowerWork & Power::

Work done = force x perpendicular distance… so…Work done = force x perpendicular distance… so…

Work done = torque x angle rotatedWork done = torque x angle rotated W = TW = Tθθ

Power = force x speed… so…Power = force x speed… so…

Power = torque x angular velocityPower = torque x angular velocity P = TP = Tωω

11stst Law of Thermodynamics Law of Thermodynamics 11stst Law of Thermodynamics Law of Thermodynamics: Energy can be neither : Energy can be neither

created nor destroyed (conservation of energy)created nor destroyed (conservation of energy)

- Thus power generation processes and energy sources - Thus power generation processes and energy sources actually involve conversion of energy from one form to actually involve conversion of energy from one form to another, rather than creation of energy from nothinganother, rather than creation of energy from nothing

ΔΔQ = Q = ΔΔU + U + ΔΔWW

ΔΔU:U: Change in internal energy of the system Change in internal energy of the system

ΔΔQ: HQ: Heat transferred into/out of the systemeat transferred into/out of the system

ΔΔW:W: Work done by/on the system Work done by/on the system

11stst Law of Thermodynamics Law of Thermodynamics

Cylinder has area, A. A fluid is admitted at constant Cylinder has area, A. A fluid is admitted at constant pressure, ppressure, p

p = F/Ap = F/A && Wd = fd …Wd = fd …

rearrange:rearrange: F = pAF = pA Wd = pAdWd = pAd (Ad = volume, (Ad = volume, V)V)

Wd = pVWd = pV or or ΔΔWd = pWd = pΔΔVV

11stst Law of Thermodynamics Law of Thermodynamics pV = nRTpV = nRT (Ideal Gas Law)(Ideal Gas Law)

Boyle’s Law: pV = constantBoyle’s Law: pV = constant

- Temperature remains constant (isothermal)- Temperature remains constant (isothermal)

- pV = constant and p- pV = constant and p11VV11 = p = p22VV22

- - ΔΔU = 0 because the internal energy is dependent onU = 0 because the internal energy is dependent on

temperature, which does not changetemperature, which does not change

- - ΔΔQ = Q = ΔΔW. If the gas expands to do work W. If the gas expands to do work ΔΔW, & W, & amount of heat amount of heat ΔΔQ must be suppliedQ must be supplied

- compression or expansion produces the same graph- compression or expansion produces the same graph

11stst Law of Thermodynamics Law of Thermodynamics AdiabaticAdiabatic: no heat flow (: no heat flow (ΔΔQ=0) into or out of a systemQ=0) into or out of a system

For a change in pressure or volume in a system, the For a change in pressure or volume in a system, the temperature loss can be calculated:temperature loss can be calculated:

pp11VV11/T/T11 = p = p22VV22/T/T22

At high p, low V: adiabatic = valueAt high p, low V: adiabatic = value

expected for isothermal at high Texpected for isothermal at high T At low p, high V: adiabatic cutsAt low p, high V: adiabatic cuts

isothermal at low Tisothermal at low T

Equation for adiabatic line:Equation for adiabatic line:

pVpVγγ = k = k

γγ = C = Cpp/C/Cvv k = constantk = constant

Adiabatic compression

11stst Law of Thermodynamics Law of Thermodynamics IsovolumetricIsovolumetric:: pp11TT11 = p = p22TT22

IsobaricIsobaric:: VV11TT11 = V = V22TT22

Adiabatic compression

P-V diagrams & EnginesP-V diagrams & Engines Gases undergo changes that will eventually cause them Gases undergo changes that will eventually cause them

to return to the original state. An ideal gas undergoing to return to the original state. An ideal gas undergoing these changes has the properties shown below:these changes has the properties shown below:

- Isovolumetric changes between a & b and c & d- Isovolumetric changes between a & b and c & d

- Isobaric changes between b & c and d & a- Isobaric changes between b & c and d & a

P-V diagrams & EnginesP-V diagrams & Engines Thermal EfficiencyThermal Efficiency:: net work output ÷ heat inputnet work output ÷ heat input

Actual efficiency of the engine will be lower than the Actual efficiency of the engine will be lower than the value of thermal efficiency alone, due to frictional value of thermal efficiency alone, due to frictional losses within the engine. The efficiency of a car = losses within the engine. The efficiency of a car = approx. 30%approx. 30%

Petrol EnginePetrol Engine: Otto Cycle: Otto Cycle

P-V diagrams & EnginesP-V diagrams & Engines Diesel EngineDiesel Engine::

- Higher thermal efficiency that petrol engines- Higher thermal efficiency that petrol engines

- Heavier than petrol engines- Heavier than petrol engines

- More noise and incomplete combustion (pollution)- More noise and incomplete combustion (pollution)

Both EnginesBoth Engines::

power output: area of p-V loop x npower output: area of p-V loop x noo cylinders x n cylinders x noo cycles per cycles per secsec

maximum energy input: fuel calorific value x fuel flow ratemaximum energy input: fuel calorific value x fuel flow rate

22ndnd Law & Engines Law & Engines 22ndnd Law of Thermodynamics Law of Thermodynamics: Entropy of an isolated : Entropy of an isolated

system not in equilibrium will tend to increase over time, system not in equilibrium will tend to increase over time, approaching a maximum value at equilibriumapproaching a maximum value at equilibrium

- i.e. entropy increases & all processes tend towards - i.e. entropy increases & all processes tend towards chaoschaos

Temperature gradient: Heat flows from a region of hot Temperature gradient: Heat flows from a region of hot temperature to a region of cold temperaturetemperature to a region of cold temperature

All heat engines give up their energy to a cold reservoirAll heat engines give up their energy to a cold reservoir

QQinin:heat flow from the hot reservoir to the engine :heat flow from the hot reservoir to the engine

QQoutout: heat flow from the engine to the cold reservoir. : heat flow from the engine to the cold reservoir.

Work done by heat engine = QWork done by heat engine = Qinin – Q – Qoutout

Efficiency = W/QEfficiency = W/Qinin = (Q = (Qinin – Q – Qoutout)/Q)/Qinin

22ndnd Law & Engines Law & Engines Limitations to Thermal EfficiencyLimitations to Thermal Efficiency::

- in an engine:- in an engine:

1)1) TTHH cannot be too high cannot be too high components could melt components could melt

2)2) TTCC will be in the range of atmospheric temperatures will be in the range of atmospheric temperatures

3)3) Analysis of the engine cycle can help to improve Analysis of the engine cycle can help to improve efficiencyefficiency

4)4) Design of ports so that gas can get enter & exit with Design of ports so that gas can get enter & exit with min. resistancemin. resistance

5)5) Lubrication reduces friction in bearingsLubrication reduces friction in bearings

Therefore an engine will never work at its theoretical Therefore an engine will never work at its theoretical efficiencyefficiency

SummarySummary Rotational DynamicsRotational Dynamics Rotational Momentum & PowerRotational Momentum & Power 11stst Law of Thermodynamics Law of Thermodynamics P-V diagrams & EnginesP-V diagrams & Engines 22ndnd Law & Engines Law & Engines