Stall Speed and Load Factor - ATPL 081 — Principles of Flight

The previous lesson ended at a fixed angle: the wing always stalls at its critical angle of attack. But pilots do not fly angles; they fly speeds, and the speed at which the critical angle arrives is anything but fixed. It moves with weight, with configuration, with power, with contamination, and, most dramatically, with the load factor, the quantity this lesson defines and then multiplies everything by. This is the module's calculation lesson: by the end, the stall speed for any mass, bank angle or load factor is two lines of arithmetic.

One angle, many speeds

Start from the lift equation at the moment of stall. Flying at the critical angle, the wing delivers its maximum lift coefficient, and in steady level flight that maximum must still carry exactly the weight. Solve for the speed and the familiar formula appears, now with its precise name:

Everything in this lesson is this formula read in different directions. Whatever raises the weight the wing must carry, or lowers the $C_{L\,max}$ it can produce, raises the stall speed, by the square root of the change.

The lettered stall speeds

Operations and certification name several stall speeds, and the exam expects the letters sorted:

$V_{S1g}$ — the 1g stall speed: the slowest speed at which lift still exactly equals weight, from the formula above. The physically cleanest definition: the wing at its ceiling with the load factor still 1.
$V_S$ (demonstrated) — slightly lower: in the classic certification demonstration the aircraft decelerates steadily until the nose drops. In the final moments the aircraft is already sinking: the wing momentarily carries a little less than the full weight (the load factor slips below 1), so the speed recorded at the break is a few knots BELOW the 1g stall speed.
$V_{S0}$ — stall speed in the landing configuration: gear down, landing flap, which is also the configuration with the lowest stall speed.
$V_{S1}$ — stall speed in a specified configuration: clean, or whatever configuration the case at hand defines.
$V_{SR}$ — the reference stall speed: modern (CS-25) certification's chosen basis: a calibrated stall speed that may not be less than the 1g stall speed. The scheduled take-off and approach safety speeds are then built on it: they are fixed percentages above $V_{SR}$ , which is exactly why the margins the next lessons discuss exist by construction.

What moves the stall speed

Before the load factor takes over, the exam's list of ordinary influences: each one acting through the weight side or the lift side of the formula:

Mass and wing loading — more mass is more weight to carry: the stall speed grows with the square root of the mass. Wing loading (weight per square metre of wing) says the same thing per unit of area: a heavily loaded wing stalls at a higher speed.
CG position — with a forward CG the tailplane must push down harder to balance, and the wing carries the weight PLUS that download: the effective weight rises, and the stall speed rises with its square root. An aft CG relieves the download and lowers the stall speed slightly.
Thrust component — at high angles of attack the thrust line points partly upward, so the engines carry a share of the weight directly: the wing needs less lift, and the power-on stall speed is lower than the power-off one.
Slipstream — on propeller aircraft the propwash blows accelerated air over the inner wing: extra dynamic pressure, extra lift at the same aircraft speed: the stall speed falls further. (Jet aircraft get the thrust-component effect but no wing-washing slipstream.)
Contamination — the degradation lesson's story arriving in a formula: ice or frost cuts $C_{L\,max}$ , and a smaller divisor under the square root means a higher stall speed, at an earlier, unannounced angle.

The load factor

Now the multiplier that dominates the rest of the lesson. The load factor compares what the wing is actually lifting with what the aircraft weighs:

In steady level flight lift equals weight and n = 1. It leaves 1 the moment the flight path curves, because curving a path takes force beyond what supporting the weight requires. In a level coordinated turn, the lift vector banks with the wings, and its vertical part alone must still hold the full weight. Tilted by the bank angle φ, the lift must therefore grow to $L = W/\cos\varphi$ , which is the whole result:

The same logic runs vertically:

Pull-up (levelling off from a dive, or hauling the nose up) — the flight path curves upward, and the wing must supply the weight plus the centripetal force curving the path: n rises above 1, sharply if the pull is aggressive.
Push-over (nosing over into a descent) — the curve is the other way: part of the weight is spent curving the path downward, the wing lifts less than the weight, and n falls below 1, reaching zero in the weightless float of a parabolic push.

The accelerated stall

Put the load factor into the stall formula and the lesson's central result appears. The wing at its critical angle produces lift proportional to the speed squared; the lift it must produce is now n times the weight. The stall speed therefore grows with the square root of the load factor:

A stall met at a load factor above 1 is an accelerated stall: the wing reaches its critical angle at a speed well above the 1g stall speed, because the extra lift demand drives the angle of attack up early. Nothing about the stall itself is different: the critical angle and the separation are the same; only the speed at which it arrives has changed. One naming warning the syllabus itself makes: the accelerated stall is sometimes loosely called a 'high-speed stall', but that name properly belongs to the shock stall of the high-speed module, a different phenomenon entirely. In a level turn, combining the two formulas gives the stall speed directly from the bank angle: $V_{S1g} \times \sqrt{1/\cos\varphi}$ , the curve below, nearly flat to 30° and vicious beyond 60°.

What the exam asks

Ratio calculations dominate: bank angle, load factor and mass, all through the same square root. The overview, following the learning objectives:

Explain $V_{S0}$ , $V_{S1}$ , $V_{SR}$ and $V_{S1g}$

Solve $V_{S1g}$ from the lift formula for varying $C_L$

Describe and explain the influences on stall speed: CG, thrust component, slipstream, wing loading, mass, contamination

Define the load factor n, and explain why it increases in a turn and a pull-up and decreases in a push-over

Describe the influence of n on stall speed: $V_{S,n} = V_{S1g}\sqrt{n}$ , and explain the accelerated stall (and why 'high-speed stall' properly means the shock stall)

Calculate the stall-speed change as a function of load factor, bank angle in a level turn, and gross mass