Brunei DCA CPL(A) Principles of Flight cheat sheet

ISA sea level: 1013.25 hPa, 15 °C, density 1.225 kg/m³; temperature lapse rate 1.98 °C per 1000 ft up to 36,090 ft (tropopause, −56.5 °C).

Density ratio σ = ρ/ρ0; for a given IAS, TAS rises as σ falls: TAS ≈ IAS/√σ, so TAS exceeds IAS at altitude.

Dynamic pressure q = ½ρV²; aerodynamic force scales directly with q, so doubling speed quadruples force.

Aerodynamic force resolves into lift perpendicular to relative airflow and drag parallel to it; the resultant acts at the centre of pressure.

Bernoulli: static pressure falls where local flow velocity rises; the pressure difference between upper and lower surface integrates to lift.

Continuity: mass flow constant in steady flow, so a streamline tube narrowing forces velocity up and static pressure down.

Relative airflow is opposite to the flight path; angle of attack is measured between chord line and relative airflow, not between chord and horizon.

Boundary layer: laminar flow has low skin friction but separates early; turbulent flow has higher friction but resists adverse pressure gradients and delays separation.

Chord line joins leading and trailing edge; camber line is the locus of points equidistant between upper and lower surfaces; max camber position affects pitching moment.

Angle of attack for zero lift is negative for a cambered aerofoil and roughly zero for a symmetric one; CL = a(α − α0).

CL–α curve is linear until about 15–16° for a plain aerofoil, then stalls with a rounded peak; camber raises CLmax and shifts α0 negative.

Thickness-to-chord ratio around 12–15 % gives favourable lift-to-drag; increasing thickness raises CLmax but also profile drag.

Centre of pressure of a cambered aerofoil moves forward as CL increases; at the design CL it may coincide with the aerodynamic centre at 25 % chord.

Aerodynamic centre is the point about which pitching moment is nearly constant with α, typically at 25 % chord for subsonic aerofoils.

Leading-edge radius controls stall behaviour: sharp leading edges stall abruptly and at low α; large radii give gentler, progressive stall.

Laminar-flow sections have maximum thickness near 40–50 % chord; surface roughness destroys the laminar bucket and increases drag.

L = ½ρV²S CL; lift depends on dynamic pressure, wing area and lift coefficient.

CL is dimensionless and determined by aerofoil shape, camber, angle of attack and Reynolds/Mach number.

Lift slope a ≈ 2π per radian (about 0.1 per degree) for thin aerofoils in incompressible flow.

Lift is generated by pressure distribution plus the downward momentum imparted to the air; both descriptions must agree on the same resultant force.

For steady level flight, L = W; increasing weight at fixed speed requires a higher CL, hence a higher angle of attack.

At fixed CL, lift is proportional to V²; at fixed IAS, a heavier aircraft must fly at a higher CL or higher TAS.

Induced angle of attack reduces the effective angle of attack, so CL for a given geometric α is lower on a finite wing than a 2-D section.

In a climb at constant speed, lift is slightly less than weight because the thrust component supports part of the weight.

Parasite drag Dp = ½ρV² f, rising with the square of speed; it dominates at high speed.

Induced drag Di = L²/(½ρV² π b² e); it falls as speed rises and dominates at low speed and high CL.

Total drag = parasite + induced; the minimum total drag occurs where parasite drag equals induced drag, at CL = √(CD0/(πAe)).

CD = CD0 + CL²/(πAe); aspect ratio A = b²/S; Oswald efficiency e is typically 0.7–0.85 for light aircraft.

Induced drag is proportional to 1/V² at constant weight, so it quadruples when speed is halved.

Interference drag arises where surfaces meet (wing–fuselage fillet); form drag from separation; skin friction from boundary layer shear.

Lift-to-drag ratio L/D = CL/CD; best L/D occurs at the same CL as minimum drag and gives maximum glide range.

Drag rises steeply in the transonic region due to shock-induced separation (wave drag), starting near the critical Mach number.

Power required Pr = D × V; thrust required Tr = D, minimum at best L/D speed.

For a propeller aircraft, Pr ∝ V³ in the parasite-dominated region and ∝ 1/V in the induced-dominated region; the curve is a minimum at best L/D speed.

Minimum thrust required occurs at maximum L/D; minimum power required occurs at a lower speed, where CL = √(3CD0/(πAe)).

For a jet, thrust is approximately constant with speed, so minimum thrust and minimum power speeds differ from the propeller case.

Rate of climb = (Pa − Pr)/W; excess power determines climb performance, so best climb angle and best climb rate occur at different speeds.

Glide: tan γ = 1/(L/D); minimum sink rate occurs at CL = √(3CD0/(πAe)), maximum range at maximum L/D.

Specific range for a propeller aircraft is maximum at best L/D; for a jet it is maximum at a higher speed near maximum L/D times a factor.

Shaft power = brake power; thrust horsepower THP = BHP × propeller efficiency η.

Propeller efficiency η = THP/BHP, typically 0.75–0.85 in cruise and near zero at zero airspeed.

Power available falls with altitude because density falls: normally aspirated engines lose about 3 % per 1000 ft.

Power available is largely constant with speed for a fixed-pitch propeller at constant rpm; a constant-speed propeller holds rpm and hence power more nearly constant.

Excess power = Pa − Pr; its maximum gives best rate of climb speed (Vy).

Best angle of climb (Vx) occurs at maximum excess thrust, which is at a lower speed than Vy for a propeller aircraft.

Absolute ceiling is where maximum rate of climb is zero (Pa = Pr at all speeds); service ceiling is where ROC falls to 100 ft/min.

Static stability: initial tendency to return to equilibrium; positive, neutral or negative. Dynamic stability describes the subsequent motion over time.

Longitudinal static stability requires the centre of gravity ahead of the aerodynamic centre (stick-fixed neutral point).

Static margin = (neutral point − CG)/mean aerodynamic chord; larger margin means stronger stability but heavier control forces.

Dihedral gives lateral static stability: sideslip produces a differential angle of attack and a restoring roll.

Directional stability is provided by the vertical fin; weathercock stability returns the aircraft to zero sideslip.

Dutch roll is a coupled lateral–directional oscillation, usually lightly damped; spiral divergence is a slow unstable lateral mode.

Phugoid is a long-period longitudinal oscillation of speed and altitude at nearly constant angle of attack; short-period mode is heavily damped.

Moving the CG aft reduces longitudinal static stability and can cause pitch-up; forward CG increases stability and elevator forces.

Pitch about the lateral axis by elevator; roll about the longitudinal axis by ailerons; yaw about the normal axis by the rudder.

Control effectiveness depends on dynamic pressure at the surface: deflection force ∝ ½ρV²S; effectiveness falls as IAS falls.

Adverse yaw: down-going aileron produces more induced drag, yawing the nose away from the turn; coordinated with rudder.

Differential aileron and Frise aileron reduce adverse yaw by reducing down-aileron drag.

Trim tabs hold a control position without pilot force; balance tabs reduce hinge moment; anti-balance tabs increase feel with speed.

Elevator authority is reduced at aft CG because a shorter moment arm from CG to tail is available; forward CG increases pitch-down authority.

Control reversal can occur at high speed when aileron twist overcomes the intended rolling moment; aileron effectiveness falls near the critical Mach number.

Hinge moment = ½ρV² × hinge moment coefficient × control area × chord; stick force per g is a certification-relevant gradient.

Stall occurs at the critical angle of attack (typically 15–16° for a plain wing), independent of speed, weight and attitude.

Stall speed Vs = √(2W/(ρS CLmax)); it rises with weight (Vs ∝ √W) and with altitude (TAS rises, IAS stays constant).

Load factor raises stall speed: Vs(n) = Vs × √n; at 60° bank, n = 2, so stall speed rises by 41 %.

A spin requires both stall and yaw: autorotation occurs when one wing is more deeply stalled than the other.

Incipient spin, fully developed spin and recovery phases; recovery is PARE - power off, ailerons neutral, rudder opposite, elevator forward.

Ailerons can aggravate a spin; the outboard wing may be less stalled and aileron deflection deepens the stall on one side.

Shock stall (high-speed stall) occurs when local flow exceeds Mach 1, causing separation and buffet, often at low IAS at high altitude.

Stall warning devices: reed/horn from a pressure switch on the leading edge, or stick shaker; required to warn before stall with adequate margin.

V-n diagram: boundaries are stall lines (CLmax), limit load lines, maximum speed and manoeuvring speed VA.

VA is the speed at which full control deflection produces the limit load factor; below VA the wing stalls before structural failure.

Limit load is the maximum expected load in service; ultimate load is 1.5 × limit load, and structure must not fail below ultimate.

Certification load factors for normal category: +3.8 g and −1.52 g; utility +4.4/−1.76; aerobatic +6.0/−3.0.

Gust load factor n = 1 + (½ρV a U)/(W/S); it increases with speed, gust velocity and wing loading, and decreases with wing loading.

Design cruising speed VC, design dive speed VD (VD ≥ 1.25 VC) and never-exceed VNE (below VD) define the speed limits.

Load factor n = L/W; in a level turn n = 1/cos φ, so 60° bank gives 2 g and 75° gives about 3.9 g.

Structural failure risk rises when VNE is exceeded in turbulence or when abrupt control inputs are made above VA.

Speed of sound a = √(γRT); at sea level ISA about 661 kt (340 m/s), falling with temperature to about 573 kt at the tropopause.

Mach number M = TAS/a; local Mach number on the wing is higher than flight Mach number because of acceleration over the surface.

Critical Mach number Mcrit is the flight Mach number at which local flow first reaches Mach 1; it falls as CL and thickness rise.

Above Mcrit, shock waves form, causing wave drag, buffet, trim change and loss of elevator effectiveness.

Sweep delays Mcrit: the effective Mach number normal to the leading edge is M cos Λ; sweep also reduces lift slope and increases tip stall tendency.

Supersonic flow: oblique/normal shocks, expansion fans; lift slope changes from 2π to 4/√(M²−1) per radian in linear theory.

Mach tuck: aft movement of the centre of pressure at high Mach causes pitch-down; Mach trim and high-speed elevator designs counter it.

Area rule reduces transonic wave drag by smoothing the cross-sectional area distribution along the fuselage.

Geometric pitch is the theoretical forward distance per revolution; effective pitch is the actual distance; slip is the difference.

Advance ratio J = V/(nD); thrust and power coefficients CT and CP are functions of J and blade angle.

Propeller efficiency η = (T V)/P = J × CT/CP; maximum efficiency occurs at a design advance ratio, typically 0.7–0.85.

Fixed-pitch propellers cannot maintain optimum blade angle across speed; constant-speed units vary pitch to hold rpm.

Blade element theory: each section produces lift and drag; thrust is the sum of lift components, torque the sum of drag components.

As forward speed rises at fixed rpm, blade angle of attack falls, thrust decreases and efficiency peaks then declines.

Windmilling drag is high when an engine fails; feathering aligns the blade with the relative airflow to minimise drag and stop rotation.

P-factor: at high angle of attack, the descending blade takes a greater bite, producing asymmetric thrust and yaw.

Critical engine is the one whose failure gives the greatest adverse yaw and control difficulty - for a clockwise-turning (from cockpit) propeller pair, the left engine.

VMC is the minimum control speed with the critical engine failed and the other at maximum take-off power, gear up, flaps take-off; rudder fully applied.

VMC must not exceed 1.2 Vs1 in the take-off configuration; VMCA is the air minimum control speed.

Asymmetric thrust produces a yawing moment about the vertical axis and a rolling moment; rudder and aileron are needed, and bank toward the live engine reduces sideslip.

Single-engine service ceiling is where rate of climb on one engine is 50 ft/min; absolute ceiling is zero rate of climb.

Drift-down: at speeds below VYSE, the aircraft cannot maintain altitude; best single-engine climb speed VYSE gives maximum rate of climb.

Zero sideslip (ball centred) minimises drag on one engine; bank of about 2–5° toward the live engine is typical.

V1, VR and V2 relate to take-off; V1 is the decision speed where continued take-off or stop is possible within the runway.

ISA sea level: 1013.25 hPa, 15 °C, density 1.225 kg/m³; temperature lapse rate 1.98 °C per 1000 ft up to 36,090 ft (tropopause, −56.5 °C).

Density ratio σ = ρ/ρ0; for a given IAS, TAS rises as σ falls: TAS ≈ IAS/√σ, so TAS exceeds IAS at altitude.

Dynamic pressure q = ½ρV²; aerodynamic force scales directly with q, so doubling speed quadruples force.

Aerodynamic force resolves into lift perpendicular to relative airflow and drag parallel to it; the resultant acts at the centre of pressure.

Bernoulli: static pressure falls where local flow velocity rises; the pressure difference between upper and lower surface integrates to lift.

Continuity: mass flow constant in steady flow, so a streamline tube narrowing forces velocity up and static pressure down.

Relative airflow is opposite to the flight path; angle of attack is measured between chord line and relative airflow, not between chord and horizon.

Boundary layer: laminar flow has low skin friction but separates early; turbulent flow has higher friction but resists adverse pressure gradients and delays separation.

Chord line joins leading and trailing edge; camber line is the locus of points equidistant between upper and lower surfaces; max camber position affects pitching moment.

Angle of attack for zero lift is negative for a cambered aerofoil and roughly zero for a symmetric one; CL = a(α − α0).

CL–α curve is linear until about 15–16° for a plain aerofoil, then stalls with a rounded peak; camber raises CLmax and shifts α0 negative.

Thickness-to-chord ratio around 12–15 % gives favourable lift-to-drag; increasing thickness raises CLmax but also profile drag.

Centre of pressure of a cambered aerofoil moves forward as CL increases; at the design CL it may coincide with the aerodynamic centre at 25 % chord.

Aerodynamic centre is the point about which pitching moment is nearly constant with α, typically at 25 % chord for subsonic aerofoils.

Leading-edge radius controls stall behaviour: sharp leading edges stall abruptly and at low α; large radii give gentler, progressive stall.

Laminar-flow sections have maximum thickness near 40–50 % chord; surface roughness destroys the laminar bucket and increases drag.

L = ½ρV²S CL; lift depends on dynamic pressure, wing area and lift coefficient.

CL is dimensionless and determined by aerofoil shape, camber, angle of attack and Reynolds/Mach number.

Lift slope a ≈ 2π per radian (about 0.1 per degree) for thin aerofoils in incompressible flow.

Lift is generated by pressure distribution plus the downward momentum imparted to the air; both descriptions must agree on the same resultant force.

For steady level flight, L = W; increasing weight at fixed speed requires a higher CL, hence a higher angle of attack.

At fixed CL, lift is proportional to V²; at fixed IAS, a heavier aircraft must fly at a higher CL or higher TAS.

Induced angle of attack reduces the effective angle of attack, so CL for a given geometric α is lower on a finite wing than a 2-D section.

In a climb at constant speed, lift is slightly less than weight because the thrust component supports part of the weight.

Parasite drag Dp = ½ρV² f, rising with the square of speed; it dominates at high speed.

Induced drag Di = L²/(½ρV² π b² e); it falls as speed rises and dominates at low speed and high CL.

Total drag = parasite + induced; the minimum total drag occurs where parasite drag equals induced drag, at CL = √(CD0/(πAe)).

CD = CD0 + CL²/(πAe); aspect ratio A = b²/S; Oswald efficiency e is typically 0.7–0.85 for light aircraft.

Induced drag is proportional to 1/V² at constant weight, so it quadruples when speed is halved.

Interference drag arises where surfaces meet (wing–fuselage fillet); form drag from separation; skin friction from boundary layer shear.

Lift-to-drag ratio L/D = CL/CD; best L/D occurs at the same CL as minimum drag and gives maximum glide range.

Drag rises steeply in the transonic region due to shock-induced separation (wave drag), starting near the critical Mach number.

Power required Pr = D × V; thrust required Tr = D, minimum at best L/D speed.

For a propeller aircraft, Pr ∝ V³ in the parasite-dominated region and ∝ 1/V in the induced-dominated region; the curve is a minimum at best L/D speed.

Minimum thrust required occurs at maximum L/D; minimum power required occurs at a lower speed, where CL = √(3CD0/(πAe)).

For a jet, thrust is approximately constant with speed, so minimum thrust and minimum power speeds differ from the propeller case.

Rate of climb = (Pa − Pr)/W; excess power determines climb performance, so best climb angle and best climb rate occur at different speeds.

Glide: tan γ = 1/(L/D); minimum sink rate occurs at CL = √(3CD0/(πAe)), maximum range at maximum L/D.

Specific range for a propeller aircraft is maximum at best L/D; for a jet it is maximum at a higher speed near maximum L/D times a factor.

Shaft power = brake power; thrust horsepower THP = BHP × propeller efficiency η.

Propeller efficiency η = THP/BHP, typically 0.75–0.85 in cruise and near zero at zero airspeed.

Power available falls with altitude because density falls: normally aspirated engines lose about 3 % per 1000 ft.

Power available is largely constant with speed for a fixed-pitch propeller at constant rpm; a constant-speed propeller holds rpm and hence power more nearly constant.

Excess power = Pa − Pr; its maximum gives best rate of climb speed (Vy).

Best angle of climb (Vx) occurs at maximum excess thrust, which is at a lower speed than Vy for a propeller aircraft.

Absolute ceiling is where maximum rate of climb is zero (Pa = Pr at all speeds); service ceiling is where ROC falls to 100 ft/min.

Static stability: initial tendency to return to equilibrium; positive, neutral or negative. Dynamic stability describes the subsequent motion over time.

Longitudinal static stability requires the centre of gravity ahead of the aerodynamic centre (stick-fixed neutral point).

Static margin = (neutral point − CG)/mean aerodynamic chord; larger margin means stronger stability but heavier control forces.

Dihedral gives lateral static stability: sideslip produces a differential angle of attack and a restoring roll.

Directional stability is provided by the vertical fin; weathercock stability returns the aircraft to zero sideslip.

Dutch roll is a coupled lateral–directional oscillation, usually lightly damped; spiral divergence is a slow unstable lateral mode.

Phugoid is a long-period longitudinal oscillation of speed and altitude at nearly constant angle of attack; short-period mode is heavily damped.

Moving the CG aft reduces longitudinal static stability and can cause pitch-up; forward CG increases stability and elevator forces.

Pitch about the lateral axis by elevator; roll about the longitudinal axis by ailerons; yaw about the normal axis by the rudder.

Control effectiveness depends on dynamic pressure at the surface: deflection force ∝ ½ρV²S; effectiveness falls as IAS falls.

Adverse yaw: down-going aileron produces more induced drag, yawing the nose away from the turn; coordinated with rudder.

Differential aileron and Frise aileron reduce adverse yaw by reducing down-aileron drag.

Trim tabs hold a control position without pilot force; balance tabs reduce hinge moment; anti-balance tabs increase feel with speed.

Elevator authority is reduced at aft CG because a shorter moment arm from CG to tail is available; forward CG increases pitch-down authority.

Control reversal can occur at high speed when aileron twist overcomes the intended rolling moment; aileron effectiveness falls near the critical Mach number.

Hinge moment = ½ρV² × hinge moment coefficient × control area × chord; stick force per g is a certification-relevant gradient.

Stall occurs at the critical angle of attack (typically 15–16° for a plain wing), independent of speed, weight and attitude.

Stall speed Vs = √(2W/(ρS CLmax)); it rises with weight (Vs ∝ √W) and with altitude (TAS rises, IAS stays constant).

Load factor raises stall speed: Vs(n) = Vs × √n; at 60° bank, n = 2, so stall speed rises by 41 %.

A spin requires both stall and yaw: autorotation occurs when one wing is more deeply stalled than the other.

Incipient spin, fully developed spin and recovery phases; recovery is PARE - power off, ailerons neutral, rudder opposite, elevator forward.

Ailerons can aggravate a spin; the outboard wing may be less stalled and aileron deflection deepens the stall on one side.

Shock stall (high-speed stall) occurs when local flow exceeds Mach 1, causing separation and buffet, often at low IAS at high altitude.

Stall warning devices: reed/horn from a pressure switch on the leading edge, or stick shaker; required to warn before stall with adequate margin.

V-n diagram: boundaries are stall lines (CLmax), limit load lines, maximum speed and manoeuvring speed VA.

VA is the speed at which full control deflection produces the limit load factor; below VA the wing stalls before structural failure.

Limit load is the maximum expected load in service; ultimate load is 1.5 × limit load, and structure must not fail below ultimate.

Certification load factors for normal category: +3.8 g and −1.52 g; utility +4.4/−1.76; aerobatic +6.0/−3.0.

Gust load factor n = 1 + (½ρV a U)/(W/S); it increases with speed, gust velocity and wing loading, and decreases with wing loading.

Design cruising speed VC, design dive speed VD (VD ≥ 1.25 VC) and never-exceed VNE (below VD) define the speed limits.

Load factor n = L/W; in a level turn n = 1/cos φ, so 60° bank gives 2 g and 75° gives about 3.9 g.

Structural failure risk rises when VNE is exceeded in turbulence or when abrupt control inputs are made above VA.

Speed of sound a = √(γRT); at sea level ISA about 661 kt (340 m/s), falling with temperature to about 573 kt at the tropopause.

Mach number M = TAS/a; local Mach number on the wing is higher than flight Mach number because of acceleration over the surface.

Critical Mach number Mcrit is the flight Mach number at which local flow first reaches Mach 1; it falls as CL and thickness rise.

Above Mcrit, shock waves form, causing wave drag, buffet, trim change and loss of elevator effectiveness.

Sweep delays Mcrit: the effective Mach number normal to the leading edge is M cos Λ; sweep also reduces lift slope and increases tip stall tendency.

Supersonic flow: oblique/normal shocks, expansion fans; lift slope changes from 2π to 4/√(M²−1) per radian in linear theory.

Mach tuck: aft movement of the centre of pressure at high Mach causes pitch-down; Mach trim and high-speed elevator designs counter it.

Area rule reduces transonic wave drag by smoothing the cross-sectional area distribution along the fuselage.

Geometric pitch is the theoretical forward distance per revolution; effective pitch is the actual distance; slip is the difference.

Advance ratio J = V/(nD); thrust and power coefficients CT and CP are functions of J and blade angle.

Propeller efficiency η = (T V)/P = J × CT/CP; maximum efficiency occurs at a design advance ratio, typically 0.7–0.85.

Fixed-pitch propellers cannot maintain optimum blade angle across speed; constant-speed units vary pitch to hold rpm.

Blade element theory: each section produces lift and drag; thrust is the sum of lift components, torque the sum of drag components.

As forward speed rises at fixed rpm, blade angle of attack falls, thrust decreases and efficiency peaks then declines.

Windmilling drag is high when an engine fails; feathering aligns the blade with the relative airflow to minimise drag and stop rotation.

P-factor: at high angle of attack, the descending blade takes a greater bite, producing asymmetric thrust and yaw.

Critical engine is the one whose failure gives the greatest adverse yaw and control difficulty - for a clockwise-turning (from cockpit) propeller pair, the left engine.

VMC is the minimum control speed with the critical engine failed and the other at maximum take-off power, gear up, flaps take-off; rudder fully applied.

VMC must not exceed 1.2 Vs1 in the take-off configuration; VMCA is the air minimum control speed.

Asymmetric thrust produces a yawing moment about the vertical axis and a rolling moment; rudder and aileron are needed, and bank toward the live engine reduces sideslip.

Single-engine service ceiling is where rate of climb on one engine is 50 ft/min; absolute ceiling is zero rate of climb.

Drift-down: at speeds below VYSE, the aircraft cannot maintain altitude; best single-engine climb speed VYSE gives maximum rate of climb.

Zero sideslip (ball centred) minimises drag on one engine; bank of about 2–5° toward the live engine is typical.

V1, VR and V2 relate to take-off; V1 is the decision speed where continued take-off or stop is possible within the runway.