Principles of Flight: The Complete Study Hub

This is the one everyone's a little afraid of — not because the concepts are obscure, but because there's a lot of them, and they build on each other. Get the forces-in-a-climb picture wrong and the turning-flight maths never quite clicks either. So this hub takes it in the order it actually needs to be learned: high-speed aerodynamics, then the forces that govern climbing and turning, then what happens when a wing runs out of lift, then how the whole aircraft holds (or doesn't hold) an attitude on its own.

Transonic Flight & High-Speed Aerodynamics

Mach number is simply Mach = TAS ÷ LSS (local speed of sound). Three Mach numbers matter for how an aircraft behaves as it accelerates:

Mcrit — Critical Mach Number: the point at which airflow over any part of the airframe (usually the wing's upper surface, where local flow accelerates fastest) first reaches M1.0
Mdet — Detachment Mach Number: while decelerating, the shockwave detaches from the aircraft to form a bow wave ahead of the nose
Mfs — Free Stream Mach Number: the Mach number of the undisturbed airflow approaching the aircraft

As speed increases past Mcrit, shockwaves form on the wings and move aft. Push further and a detached bow shock forms ahead of the nose — below Mdet, this bow wave does the aircraft a favour, slowing the incoming air to subsonic speed before it ever reaches the structure.

That dip is the shock stall — it occurs when CL, as a function of Mach number, reaches its maximum for a given angle of attack. Once Mcrit is passed, shockwaves increase drag sharply — this is the Drag Divergence Mach Number (Mdr), sometimes called the Mach number for critical drag rise. The resulting wave drag has two components: energy drag (kinetic energy lost as heat, raising local shockwave temperature) and separation drag (from boundary layer separation behind the shockwave).

Swept wings carry higher drag at supersonic speed, but they buy something valuable at the stall: a swept wing has a reduced effective Aspect Ratio (AR = span ÷ mean chord), which gives it lift-curve characteristics similar to a genuinely low-AR wing — a gentler, more progressive loss of lift at the stall, rather than a sudden break.

Two design features manage wave drag directly: vortex generators, placed just ahead of the shockwave location, and area ruling — shaping the fuselage (the "Coke bottle" effect) to smooth the cross-sectional area distribution along the aircraft's length.

Forces in a Climb

In a climb, lift no longer directly opposes weight — weight develops an apparent-drag component acting rearward along the flight path.

W cos γ = Lift
W sin γ = Weight Apparent Drag (WAD)
Available Thrust = Drag + W sin γ
Excess Thrust (T − D) = W sin γ

γ is the flight path (climb) angle. Lift ends up less than weight in a climb — weight becomes the hypotenuse of the force triangle, with W cos γ and W sin γ as its two components.

The shortcut worth remembering: Load Factor in a climb = cos γ directly — the weight term cancels out algebraically, so you never actually need the aircraft's mass to find it.

Best flight path angle, from thrust and drag: sin γ = (T − D) ÷ Weight. Or, if you're given ratios instead of raw numbers: sin γ = (Thrust÷Weight) − (Drag÷Lift), approximating Lift ≈ Weight.

Climb Gradient

Climb Gradient (%) = sin γ × 100 — at small climb angles the hypotenuse and adjacent sides are near-equal, which is what makes this simplification valid.

For a one-engine-inoperative climb gradient on a multi-engine aircraft, the same relationship holds: sin γ = (Thrust − Drag) ÷ Weight, with Lift ≈ Weight at the low climb angles transport aircraft actually achieve.

Descent & Glide

In a descent, Weight Apparent Drag flips to become Weight Apparent Thrust (WAT) — weight now assists forward motion instead of opposing it, sized the same way: W sin γ. One fact worth holding onto for the exam: glide angle is independent of weight.

Turning Flight & Load Factors

In a level turn: Load Factor = 1 ÷ cos(angle of bank) — dependent only on bank angle, never on mass, speed, or aircraft type.

Radius (m) = TAS² ÷ (g × tan θ)

TAS goes in as m/s (knots × 0.514), g = 9.81 m/s² unless the question says otherwise. Radius increases with the square of speed, and — usefully — mass doesn't appear in the formula at all: any two aircraft flying the same speed and bank angle fly the same radius, regardless of size.

Rearranged for bank angle: tan θ = TAS² ÷ (Radius × g).

Rate 1 turn (the standard reference rate) is 3°/second, or 180°/minute. The bank angle required for it: AOB = (TAS in kt ÷ 10) + 7.

Higher TAS needs a higher bank angle to hold a given rate of turn — and a higher bank angle means a higher load factor. The effects are small at low bank angles (a standard 25° turn only needs about 10% extra lift, Load Factor ≈ 1.1) but grow rapidly: at 60° bank, Load Factor = 2, lift requirement doubles, and induced drag increases by a factor of 4. With a heavy aircraft, the sustainable bank angle can end up limited by available thrust before critical angle of attack is even reached.

Turn balance: a skidding turn has the nose pointing into the turn, ball displaced toward the higher wing — correct with rudder ("step on the ball") or reduce bank. A slipping turn has the nose pointing out of the turn — correct with opposite rudder or reduce bank. Unbalanced turns increase drag and passenger discomfort.

Stalling

The stall is caused by flow separation — once it begins, CL stops rising with angle of attack (alpha), and the stall progresses gradually as the separation point moves forward. On swept wings at high speed, a sharp, thin leading edge can trigger localized leading edge separation that reattaches before the main separation occurs further aft.

Roll control at the stall — low speed aileron reversal: ailerons work normally up to critical alpha, but downward aileron movement (to raise a wing) increases camber, which can push that wing's local alpha beyond critical — dropping the wing you were trying to raise. Near high alpha, rudder's secondary roll effect can be the safer control input.

Pitch control at the stall: most aircraft are designed so increasing rearward stick force is needed to hold high alpha, and at critical alpha a natural nose-down trim change occurs — pitching the aircraft down and reducing alpha back below critical automatically. Swept-wing transport aircraft must have the natural tip-stall tendency (which would otherwise pitch the nose up) designed out for certification.

Stalling in the final turn is the one worth taking most seriously: the wing needs extra lift to generate the turning force, so the stall comes at a higher speed than expected — and buffet can be masked by flap/gear turbulence, nose-up attitude masked by bank angle. Tightening the turn increases alpha further; since the outer wing moves faster than the inner, the inner stalls first, rolling the aircraft into an even deeper bank. Serious, serious trouble.

Accelerated stall (stalling in the manoeuvre): can happen at any speed — it depends on load factor, not airspeed alone. A tailplane stall is caused by negative alpha and gives no warning and no buffet. Canards (nose-mounted stabilising aerofoils) must stall before the main wing, forcing a pitch-down before the wing itself can stall.

Stall recovery: lower the nose, apply full power, and if a wing drops, apply opposite rudder.

Stalling speed scales with the square root of load factor:

Vs (new) = Vs (original) × √Load Factor

The same relationship works for weight changes: Vs ∝ √weight. A pushover (load factor reduced below 1.0g) has the opposite effect — it lowers Vs.

Other stalling-speed effects: at higher altitudes, TAS at the stall increases even though IAS stays roughly constant — worth remembering for stopping distances at high-altitude airports, especially "hot and high" fields where reduced air density compounds the effect. A more forward CG raises stalling speed (more nose-down tailplane lift needed, increasing the load on the main wing) — published Vs figures assume the forward CG limit. In jet aircraft, power-on stalling speed is lower than idle, because the vertical thrust component reduces the lift the wing needs to provide — so published figures assume zero thrust, the worst case.

Stability

Static stability is the aircraft's first response to being displaced from trim. Dynamic stability is what happens afterward, over time. Each can be positive (returns to original), neutral (stays where displacement ends), or negative (diverges further) — and increasing stability always costs manoeuvrability.

There can be no dynamic stability without positive static stability — the aircraft needs positive static stability just to begin returning toward its original attitude at all. If it does, it almost never goes straight back without a period of oscillation — the goal is positive dynamic stability, where that oscillation decreases to zero. Aerodynamic damping is what causes the amplitude to decrease; if damping is negative, amplitude increases instead — this is called divergence.

Two named oscillations worth knowing: the phugoid — a long-period pitch/speed exchange, e.g. a jet airliner slowly trading altitude for airspeed and back — and PIO (Pilot Induced Oscillation), where the pilot's own corrections worsen the oscillation. Recovery: let go of the controls.

Moment coefficients: Cm (pitch — nose-up positive), Cl (roll — right positive), Cn (yaw — right positive). CG position matters directly: aft CG is more unstable, forward CG is more stable. Trim changes affect "stick force stability," and manoeuvring stability is the stick force required to increase load factor by 1g.

At altitude, aerodynamic damping decreases, so stability decreases and stick force per g decreases too. A positive yaw angle gives negative sideslip (and vice versa) — "dihedral effect" is the general term for any design feature contributing to lateral stability.

Gust load factor — the load factor resulting from a sudden vertical gust changing angle of attack:

A couple of items from the source notes are flagged for a second look against your official course material before exam use: one worked mass-ratio example had partially obscured figures in the original handwriting, and the "anhedral" aside reads more like a personal mnemonic than a universal syllabus fact.

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