PART 66 MODULE 2 PHYSICS NOTES

Nature of Matter and SI Units

• Physics relevance: Understanding physics is essential for aircraft operation and design; many principles extend beyond exams to practical aerospace contexts.

• Matter definition: Anything that occupies space and has mass.

• Energy: The ability to do work; energy can be transformed but not created/destroyed (Law of Conservation).

• States of matter: Solid, liquid, gas. Changes between states involve energy transfer.

• SI Units (base quantities):

  • Length: metre (m)

  • Mass: kilogram (kg)

  • Time: second (s)

  • Temperature: kelvin (K) [historically Celsius and Fahrenheit used; SI uses K]

  • Electric current: ampere (A)

• Derived units: formed from base units (e.g., speed m/s, force N = kg·m/s^2, energy J = N·m = kg·m^2/s^2).

• Multiples and sub-multiples: used to avoid very large/small numbers in engineering practice.

• Matter and energy link: Matter changes state; energy is required or released during phase changes.

Base and Derived SI Units (Overview)

• Base units: m, kg, s, K, A. Other base quantities exist (e.g., luminous intensity, amount of substance), but the core five are highlighted here.

• Derived units are built from base units (e.g.,:

  • Speed: $v = \frac{\text{m}}{\text{s}}$

  • Acceleration: $a = \frac{\text{m}}{\text{s}^2}$

  • Force: $F = \text{kg}\cdot\text{m}/\text{s}^2 = \text{N}$

  • Energy: $E = \text{N}\cdot \text{m} = \text{kg}\cdot\text{m}^2/\text{s}^2 = \text{J}$

  • Power: $P = \frac{\text{J}}{\text{s}} = \text{W}$

• Common constants/examples: gravitational acceleration on Earth ≈ $g = 9.81\ \text{m}/\text{s}^2$; standard density of water ≈ $\rho_{water} = 1000\ \text{kg/m}^3$ at 4°C.

Matter and Energy

• Matter: anything with mass occupying space; air, water, fuel, and aircraft structures are matter.

• Energy: capacity to do work; can be stored (potential energy) or in motion (kinetic energy).

• Law of Conservation of Matter: Matter cannot be created or destroyed; it can change in form/state.

• Energy changes with state changes: burning coal converts solid to gas/volatile products, releasing heat energy.

Chemical Nature of Matter

• Atoms: fundamental units of elements; composed of subatomic particles: protons (positive), neutrons (neutral) in nucleus; electrons (negative) orbiting nucleus.

• Atomic number (Z): number of protons in nucleus; defines element identity.

• Atomic mass (A): total number of nucleons (protons + neutrons).

• Nucleus: positively charged; electrons circulate in electron shells (valence shells).

• Valence shell and bonding:

  • Atoms bond to fill incomplete valence shells via sharing electrons (covalent bonds).

  • Elements with full valence shells behave as insulators; electrons are less free to move.

  • Conductors have incomplete valence shells; free electrons enable electric current.

• Molecules: two or more atoms bonded chemically; properties of a molecule can differ from constituent atoms (e.g., H2O vs H2 and O2).

• Water (H2O): formed by sharing electrons between O and H; valence shell considerations govern bonding behavior.

• Mixtures vs compounds:

  • Mixtures: physically mixed, not chemically bonded; can be separated by physical means (filtration, evaporation).

  • Compounds: chemically bonded in fixed ratios; can only be separated by chemical reaction.

Physical Nature of Matter

• Three main physical states: solid, liquid, gas.

• General properties: density, compressibility, volume and shape characteristics, and inter-molecular forces (surface tension in liquids).

• Solids: definite shape and volume; incompressible relative to gases; high rigidity.

• Liquids: definite volume, assume container shape; incompressible with some surface tension; flow is governed by inter-murface bonding.

• Gases: no definite shape/volume; compressible; behavior governed by inter-molecular forces and external conditions (P, V, T).

Forces: Scalar vs Vector; Triangles, Equilibria, and Resolution

• Scalar vs Vector quantities:

  • Scalar: magnitude only (e.g., mass, time, temperature).

  • Vector: magnitude and direction (e.g., force, velocity).

• Force representation: as a vector with magnitude and direction; line of action matters.

• Triangle of Forces: vector addition by constructing a triangle, where the resultant is the third side; the equilibrant is the third side that balances the resultant.

• Graphical method prerequisites:

  • All vectors must be of the same type (all forces, or all velocities).

  • Use a common scale; reference datum direction is essential.

• Resultant vs Equilibrant:

  • Resultant is the vector sum of two or more forces.

  • Equilibrant is a single force equal in magnitude to the resultant but opposite in direction.

• Resolution of forces: a single force can be resolved into two perpendicular components: $F = \sqrt{(F<em>x)^2 + (F</em>y)^2}$; components: $F<em>x = F \cos\theta,\; F</em>y = F \sin\theta$.

• Coplanar forces: lines of action lie in a single plane; equilibrium implies concurrent lines of action; resultant and a couple can reduce non-concurrent coplanar forces to a single resultant plus a moment.

• Effect of an applied force: a force causes translation or rotation and may alter speed/direction; Newton’s second law links force, mass, and acceleration: $F = m a.$

• Equilibrium: net force (and net moment) equals zero; see beam examples for balancing forces and moments.

Moments, Centre of Gravity, and Couples

• Moment of a force: turning effect about a point; depends on magnitude, force direction, and perpendicular distance from pivot (moment arm) $M = F \times y$ where y is perpendicular distance from pivot to line of action.

• Units: Newton-meters (Nm).

• Clockwise vs anti-clockwise moments: balance requires total clockwise moments equal total anticlockwise moments.

• Beam examples illustrate balancing moments and vertical reactions at supports.

• Couples: two equal and opposite forces whose lines of action do not coincide; produces pure rotation with a moment equal to the product of force magnitude and separation distance: $\text{Moment of a couple} = F \cdot d$.

• Centre of Gravity (CG): point through which the total weight acts; balance about CG implies no net moment.

  • For a 3D body, CG is the point where the mass can be considered to act; for 2D lamina, CG is the centroid; for a rectangular lamina, it is at the intersection of diagonals.

  • In aircraft, CG is critical for weight and balance calculations to ensure stable flight characteristics.

• Methods to locate CG: static methods (moments) and geometric properties (centroid of shapes).

• Practical CG considerations in aviation: improper loading can cause nose- or tail-heavy aircraft and dangerous handling; frequent recalculation of CG is required after loading changes.

Stress, Strain, and Elasticity

• Stress: internal force per unit area within a material; average quantity: $\sigma = \frac{F}{A}$.

• Strain: deformation relative to original size; \varepsilon = \frac{\Delta L}{L}\, (\text{dimensionless, often expressed as %}).

• Five primary stress types:

  • Tension (tensile): pulls ends apart; eg. control cables.

  • Compression: squeezes together; eg. rivets under compression.

  • Torsion: twisting about the axis; diagonal tensile and perpendicular compressive stresses; eg. engine crankshaft.

  • Bending (flexural) stress: caused by a load at a distance; top surface often in tension, bottom in compression on a cantilever.

  • Shear: forces acting parallel to cross-section; eg. rivets, clevis bolts under load.

• Elasticity and Hooke’s Law: within the proportional limit, strain is proportional to stress; $\varepsilon = \frac{\sigma}{E}$ where $E$ is Young’s modulus; Hooke’s law in its simple form: $F = k x$ for a spring, with elastic potential energy $U = \tfrac{1}{2} k x^2$.

• Stress–strain relationship: linear region in a stress–strain graph; slope gives Young’s modulus; beyond elastic limit, material yields and does not return to original length.

• Significance: aircraft structures rely on elastic behavior to absorb loads without permanent deformation; misapplication leads to failure.

Kinematics (Linear, Angular) and Dynamics Overview

• Linear motion basics ( datum-based): displacement, distance, speed, velocity, acceleration; variables x and t as standard symbols; velocity and acceleration are vector quantities.

• Speed vs velocity: speed is magnitude only; velocity includes direction.

• Equations of linear motion (uniform acceleration):

  • $v = u + a t$

  • $s = ut + \tfrac{1}{2} a t^2$

  • $v^2 = u^2 + 2 a s$
    where $u$ = initial velocity, $v$ = final velocity, $a$ = acceleration, $t$ = time, $s$ = displacement.

• Examples of linear motion in aviation context: acceleration from rest, deceleration under constant retardation, distance traveled under given acceleration.

• Newton’s Laws (Overview from Module 3):

  • 1st Law (Inertia): a body at rest stays at rest and a body in motion stays in uniform motion unless acted upon by an external force.

  • 2nd Law: acceleration is proportional to the net external force and inversely proportional to mass: $F = m a$.

  • 3rd Law: action–reaction pairs are equal and opposite.

• Gravitational force: weight $W = m g$ with $g \approx 9.81\ \text{m/s}^2$ on Earth.

• Rotational motion basics: angular velocity $\omega$ (rad/s); linear velocity at a radius $r$ is $v = r\omega$; centripetal acceleration $a_c = \frac{v^2}{r} = r\omega^2$; centripetal force acts toward the center.

• Angular velocity unit and conversion: 1 revolution per second = $2\pi$ rad/s; conversion between rpm and rad/s if needed.

• Periodic and harmonic motion: pendulums and springs as classical examples; energy exchanges between kinetic and potential forms drive oscillations; period of a simple pendulum (small amplitude) $T = 2\pi\sqrt{\frac{L}{g}}.$; for a spring–mass system, SHM is described by Hooke’s law and energy exchange; damping and resonance considerations.

• Machines and Levers: mechanical advantage (MA) and velocity ratio (VR) concepts; levers (First, Second, Third Class) with appropriate geometries; MA relates load to effort via lever arms; VR relates input and output displacements/velocities.

Dynamics and Energy Concepts

• Mass vs Weight: Weight is the gravitational force acting on a mass; mass is the intrinsic amount of matter; mass is constant across locations; weight depends on gravity: $W = m g$.

• Work and Energy:

  • Work done by a force: $W = F d$ (with displacement along force direction); unit: Joule (J).

  • Kinetic energy: $K = \tfrac{1}{2} m v^2$.

  • Potential energy (gravitational): $U = m g h$.

  • Conservation of energy: total mechanical energy (kinetic + potential) in a closed system remains constant in the absence of non-conservative forces (like friction).

• Power and horsepower:

  • Power: $P = \frac{dW}{dt} = \frac{F d}{dt} = \frac{\Delta E}{\Delta t}$

  • Unit: Watt (W) = Joule per second; 1 horsepower ≈ 746 W.

  • Brake horsepower (BHP) and shaft horsepower (SHP) definitions relate to engine torque and output at different stages; these are practical measures in aviation powerplants.

• Energy forms: Electrical, chemical, heat, chemical, potential, kinetic, etc.; energy can be stored (e.g., springs, compressed fluids) or present as moving energy.

• Heat and thermal processes: Heat is energy in transit due to temperature differences; three transfer modes: conduction, convection, radiation.

• Internal energy and energy transfer: energy can transform from one form to another (e.g., chemical energy to kinetic or to heat via friction).

Fluid Dynamics

• Fluids: liquids and gases; liquids are generally incompressible; gases compressible; viscosity is the internal friction resisting flow.

• Buoyancy and Archimedes’ principle: when submerged, a body experiences an upward buoyant force equal to the weight of displaced fluid: $F<em>b = \rho</em>{fluid} g V_{sub}$.

• Density and specific gravity: ρ = m/V; relative density (specific gravity) RD = ρ/ρ_{water} (RD of water = 1); RD < 1 => floats; RD > 1 => sinks.

• Hydrometry: A hydrometer measures relative density of liquids via floating level.

• Streamline/laminar flow vs turbulence: laminar flow follows smooth paths; turbulence involves eddies; Venturi effect demonstrates Bernoulli’s principle in action.

• Pressure concepts in fluids:

  • Static pressure $P_s$ is pressure in a stationary fluid.

  • Dynamic pressure $P_d = \tfrac{1}{2} \rho v^2$ depends on fluid velocity.

  • Total pressure $P<em>t = P</em>s + P_d$.

  • In flight, static and dynamic pressures relate to instrument readings (Pitot-static system).

• Bernoulli’s principle: along a streamline, total mechanical energy per unit volume remains constant (for incompressible, non-viscous flow): $P + \tfrac{1}{2}\rho v^2 + \rho g h = \text{constant}$.

• Gas laws and engine cycles (brief): gas behavior under temperature/pressure changes drive engine cycles (Otto cycle for constant volume combustion; Brayton cycle for constant pressure combustion).

• Expansion of fluids and gases with temperature: liquids expand volumetrically; gases follow gas laws with T and P changes.

Thermodynamics

• Temperature and heat: temperature is a measure of kinetic energy; heat is energy in transit due to temperature differences; heat capacity and specific heat describe energy required to raise temperature.

• Specific heat (c): energy required to raise 1 kg of substance by 1 K: $c = \frac{\Delta Q}{m \Delta T}$; units J/(kg·K).

• Heat capacity (C): $C = m c$; for a given mass, the capacity to absorb heat is higher for substances with higher c.

• Latent heat vs sensible heat:

  • Sensible heat: energy causing a temperature rise without phase change.

  • Latent heat: energy absorbed/released during phase changes (e.g., Fusion, Vaporization).

• Phase changes: solid ↔ liquid (fusion), liquid ↔ gas (vaporization); latent heat associated with the phase transition at a constant temperature.

• Expansion concepts: linear expansion (solids): $\Delta L = \alpha L<em>0 \Delta T$; volumetric expansion for solids: $\Delta V = \beta V</em>0 \Delta T$ where $\beta = 3\alpha$; for liquids, expansion is volumetric; coefficients vary by material.

• Gas laws recap:

  • Boyle’s Law: at constant temperature, P ∝ 1/V; $P<em>1 V</em>1 = P<em>2 V</em>2$

  • Charles’ Law: at constant pressure, V ∝ T; $\frac{V}{T} = \text{constant}$ (T in Kelvin)

  • Combined Gas Law: $\frac{P V}{T} = \text{constant}$ (ideal gas behavior).

• Gas turbine cycles (overview): Brayton cycle (constant-pressure processes) for jet propulsion; Otto cycle (constant-volume combustion) for piston engines.

• Heat of combustion: energy released as heat when fuel combusts; typically measured in MJ/kg or BTU/lb.

Optics and Fiber Optics

• Light basics: light behaves as both particle and wave; speed in vacuum c ≈ $3 \times 10^8\ m/s$; refraction, reflection, dispersion.

• Reflection: incidence angle equals reflection angle; normal line is perpendicular to reflecting surface.

• Plane mirrors produce virtual, erect images of the same size as the object; curved mirrors (concave/convex) can form real or virtual images depending on object distance and focal length; focal length is distance from mirror to focus F.

• Refraction: bending of light at boundary; Snell’s Law: $n<em>1 \sin \theta</em>1 = n<em>2 \sin \theta</em>2$ where refractive index n is related to light speed: $n = \frac{c}{v}$.

• Total internal reflection and critical angle: occurs when light attempts to move from higher to lower refractive index; critical angle $\theta<em>C = \arcsin\left(\frac{n</em>2}{n_1}\right)$; beyond it, refraction would require >90°, thus reflection occurs.

• Lenses: convex and concave lenses; focal length depends on curvature; images can be real or virtual, erect or inverted.

• Fibre optics: benefits over copper cables (bandwidth, EMI immunity, lighter, no sparks, safer in hazardous environments).

  • Core and cladding: total internal reflection confines light in core.

  • Attenuation (loss) is wavelength-dependent; typical optical wavelengths: 850 nm, 1300 nm, 1550 nm.

  • Transmitter types: LED, VCSEL, LASER; single-mode vs multimode fibers; bandwidth differences arise from modal dispersion.

  • Cable construction often includes buffer, aramid yarn, coating, buffering to protect fiber in deployment.

  • Applications include data communications and aircraft systems.

Wave Motion and Sound

• Waves: disturbances that carry energy; light is electromagnetic wave; mechanical waves exist in solids, liquids, gases.

• Plane vs spherical waves: sources create wavefronts that propagate energy; water waves and rope waves are common analogies.

• Transverse vs longitudinal waves: transverse waves oscillate perpendicular to direction of travel (e.g., on a string); longitudinal waves oscillate parallel to travel (e.g., sound in air).

• Wave properties: frequency (f, Hz), wavelength (\lambda), velocity (v); relation $v = f \lambda$.

• Sound: a mechanical wave with speed ~$340\;\text{m/s}$ in air at room temperature; speed depends on temperature.

• Sound intensity and loudness: intensity relates to amplitude; measured in decibels (dB).

• Pitch: related to frequency; higher frequency means higher pitch.

• Interference: constructive (amplitudes add) and destructive (cancel) interference; standing waves arise from interference of reflected waves with the incident wave; nodes and antinodes form in strings and air columns.

• Doppler effect: change in observed frequency due to relative motion between source and observer; observed frequency increases when the source approaches and decreases when it recedes; general formula combines source and observer motions with ambient wave speed.

• Resonance and damping: systems can be driven at their natural frequencies; resonance can lead to large oscillations and potential damage in aircraft components.

• Human hearing and psychoacoustics (loudness, pitch, quality): loudness relates to amplitude and perception; pitch to frequency; quality to harmonic content, vibrato, attack/decay.

Gyroscopes and Dynamics of Rotating Systems

• Gyroscope fundamentals: momentum-based devices with rigidity in space and precession; a spinning rotor tends to maintain orientation in space.

• Rigidity in space: high angular momentum makes the spin axis resistant to external torques; apparent wander due to Earth’s rotation is a frame-of-reference effect.

• Precession: external torque causes the spin axis to precess about a vertical axis; input axis and output axis define the geometry of precession.

• Real wander/topple: friction and bearing torques cause non-ideal behavior that can mimic drift and topple in practice.

• Gyroscope classifications:

  • Space (free) gyros: freedom to rotate about three axes with no external constraining torque.

  • Tied (displacement) gyros: externally controlled to maintain a specific orientation (e.g., Directional Gyro).

  • Earth (attitude) gyros: gravity constrains spin axis to earth-vertical.

  • Rate gyros: measure rate of turn about a particular axis; coupled with restraining springs.

  • Rate-integrating gyros: integrate turn rate; used in inertial navigation to stabilize platforms.

• Applications: aircraft datums for pitch, roll, yaw; stabilization and navigation.

Friction and Tribology

• Friction is the resistive force that opposes motion between contacting surfaces; wear occurs via surface interactions.

• Tribology: study of friction, wear, and lubrication.

• Surface geometry: real area of contact is much smaller than apparent area; friction depends on real contact area and material properties.

• Coefficient of friction (µ): ratio of friction force to normal force; varies with surface pair and conditions; typical values: polished oil-metal surfaces < 0.1; glass on glass ≈ 0.4; rubber on concrete ≈ 1.0.

• Static vs dynamic friction: static friction higher than dynamic friction; initiation of motion requires greater force than maintenance of motion.

• Practical implications: design and operation consider friction, lubrication, and wear to minimize energy loss and component degradation.

Key Equations and Concepts (Selected Highlights)

• Force and motion:

  • Newton’s Second Law: $F = ma$

  • Centripetal: $a_c = \frac{v^2}{r} = r\omega^2$; centripetal force toward rotation center.

• Energy and work:

  • Work: $W = F d$

  • Kinetic energy: $K = \tfrac{1}{2} m v^2$

  • Potential energy (gravitational): $U = m g h$

  • Power: $P = \frac{W}{t}$ or $P = \frac{\Delta E}{\Delta t}$; 1 W = 1 J/s; 1 hp ≈ 746 W.

• Momentum and impulse:

  • Momentum: $p = m v$; impulse: $J = \Delta p = m \Delta v$; impulse = force × time.

• Pressure and buoyancy:

  • Pressure in fluids: static pressure Ps; dynamic pressure Pd = \tfrac{1}{2} \rho v^2; total pressure Pt = Ps + P_d.

  • Archimedes: buoyant force $F<em>b = \rho</em>{fluid} g V_{sub}$.

• Gas laws (overview):

  • Boyle’s Law: $P<em>1 V</em>1 = P<em>2 V</em>2$ at constant T.

  • Charles’ Law: $\frac{V}{T} = \text{constant}$ at constant P.

  • General Gas Law: $P V = n R T$ (Ideal Gas Law).

• Bernoulli’s Principle (incompressible, inviscid flow): $P + \tfrac{1}{2}\rho v^2 + \rho g h = \text{constant}$.

• Refractive index and optics:

  • Refraction: $n = \frac{c}{v}$; Snell’s Law: $n<em>1 \sin\theta</em>1 = n<em>2 \sin\theta</em>2$.

  • Critical angle: $\theta<em>C = \arcsin\left(\frac{n</em>2}{n_1}\right)$ for n1 > n2.

• Wave motion:

  • Wave speed: $v = f\lambda$; frequency in Hz, wavelength in meters.

• Harmonic motion and pendulums:

  • Pendulum period (small angle): $T = 2\pi\sqrt{\frac{L}{g}}$.

• Elasticity and material properties:

  • Hooke’s Law for springs: $F = kx$; elasticity energy $U = \tfrac{1}{2} k x^2$.

  • Young’s modulus: $E = \frac{\sigma}{\varepsilon}$ with $\sigma = \frac{F}{A}$ and $\varepsilon = \frac{\Delta L}{L}$.

• Fluid mechanics and buoyancy are central to aerodynamics and engine cooling systems; Bernoulli’s principle and Venturi effects underpin many aircraft systems.

Quick Reference: Common Aviation-Relevant Concepts

• CG and weight balance: CG must remain within manufacturer limits; incorrect loading affects stability and controllability.

• Aircraft surfaces and materials: stress/strain behavior informs fatigue life and structural integrity under cyclic loading.

• HVAC and engine cooling: convection and radiation govern heat transfer; conduction in fins is critical for air-cooled engines.

• Flight instruments rely on pressure measurements (static and dynamic) to infer airspeed and altitude.

• Gyroscopes underpin attitude indicators and directional gyros; precession and rigidity in space are key phenomena in inertial navigation.

• Fibres optics in avionic data links offer high bandwidth with EMI immunity and safety in hazardous environments.

• Wave and sound concepts inform cockpit communications, noise control, and acoustic design.

Relationships to Core Principles

• The material emphasizes how foundational concepts interlink across domains: mechanics underpin flight dynamics; thermodynamics and energy transformations govern propulsion and heat management; fluids govern aerodynamics; optics and waves underpin sensors and communications.

• Ethical and practical implications: safety-critical calculations (CG, stress, fatigue), reliability of devices (gyros, hydraulics), and safety-related design choices (materials, lubrication, insulation) rely on rigorous application of these physics principles.

Practice Questions (Representative Examples)

• If a steel cable with cross-sectional area A is placed under a tensile load F, compute the tensile stress: $\sigma = \frac{F}{A}$. Then compute strain given a measured extension ΔL: $\varepsilon = \frac{\Delta L}{L}$, and determine if the material is within its elastic limit using Young’s modulus.

• A 1.5 kg mass is moving at 4 m/s. What is its momentum? How much impulse is needed to stop it in 0.2 s if the stopping force is constant?

• A pivoted beam has reactions at the supports such that clockwise moments balance anti-clockwise moments. If a 125 N force acts 3 m from the pivot producing a clockwise moment, what force must act 2 m from the pivot to balance? (Use moment balancing: sum of clockwise moments = sum of anticlockwise moments.)

• A pendulum of length 2.0 m swings with small amplitude. What is its period? (Use $T = 2\pi\sqrt{L/g}$ with g ≈ 9.81 m/s^2.)

• In a Venturi tube, if the velocity doubles, how does static pressure change according to Bernoulli’s principle? (Explain qualitatively using $P + \tfrac{1}{2} \rho v^2 = ext{constant}$.)

• For a gas with n moles at temperature T, what is the relation between P, V, and T? (Use $P V = n R T$ and discuss implications of increasing temperature at constant volume vs constant pressure.)