Asymmetric Thrust

The moments about the CG

Fail one engine and two moments appear about the normal axis, and they add. The live engine's thrust, acting off-centreline, yaws the nose toward the dead engine; the dead side's drag, worst of all a windmilling propeller, pulls the same way from the other wing. The moment arithmetic is thrust times arm, so the engine's distance from the CG decides the severity: wing-mounted engines far outboard hand the rudder a large moment, while tail-mounted engines close to the centreline make the asymmetric case comparatively mild. Between jets and propeller aircraft the difference is everything that the propeller module added: the jet's failure costs thrust and adds intake drag, but the propeller aircraft:

loses the slipstream lift over the dead wing (a rolling moment on top of the yaw).

suffers the windmilling drag until the propeller is feathered.

has its live engine's thrust line shifted by P-factor.

That shift is why the propeller twin, not the jet, has a critical engine: with both propellers clockwise (seen from behind), the left engine's failure leaves the long-arm right engine, the worst case. Counter-rotating propellers abolish the distinction, and jets never had it.

Achieving directional balance

The methods for balancing the yaw, roughly in the order a crew reaches for them:

rudder — the primary answer, a fin moment opposing the asymmetry.
feathering the failed propeller — removing the windmilling half of the problem at its source.
a small bank toward the live engine — the next section's subject, easing the rudder's task.
when performance allows it, reducing the live engine's thrust — shrinking the moment itself, the trade being climb performance for controllability.

On the ground the list shortens to rudder and nosewheel steering, and certification, as the VMCG section shows, refuses to count the nosewheel.

The sideways balance: wings level or wing down

The rudder solves the yaw and creates a new problem: its side force, pointing toward the dead side, must itself be balanced along the lateral axis, and there are two ways to do it:

Wings level — the aircraft accelerates sideways until a steady sideslip toward the dead engine builds, and the fuselage and fin side force from that slip balances the rudder's. It works, but the slip costs drag, the sideslip works against the fin (below), and (the exam's favourite trap) the slip ball sits centred while the aircraft is genuinely sideslipping: with the wings level there is no lateral acceleration for the ball to show, and it reports coordination that does not exist.
Wing down — bank a few degrees, up to about 5° toward the live engine, and the tilted lift's sideways component balances the rudder force instead. Zero sideslip becomes possible: least drag, least rudder demand, the certification assumption. Now the ball hangs toward the live engine, displaced perhaps half its width, telling the truth about the bank while the aircraft flies its cleanest asymmetric state.

The fin, the sideslip, and the hazards at VMC

The fin flies this whole case at an angle of attack assembled from two inputs: the rudder deflection and the sideslip. A sideslip toward the dead engine strikes the fin on the side the rudder is already loading: it adds to the fin's α, pushing the surface toward its own stall while also demanding still more rudder. At VMC everything is at its worst at once. The syllabus lists the hazards by name:

the aircraft's α is high (low speed).
the sideslip is large.
the loads on the fin are near their greatest.
the fin's α sits close to the fin stall, beyond which directional control is lost abruptly, with no aerodynamic warning to precede it.

This is why the wing-down, zero-sideslip technique is not a refinement but the method: it reserves the fin's angle of attack for the rudder's job alone.

Crosswind, mass and the shape of the problem

Two circumstances resize the case. In free flight a steady crosswind by itself does nothing: the aircraft moves with the air mass and feels no moment. It matters where the flight is tied to the runway, on the take-off roll and while tracking the centreline after a failure. There a crosswind helps or hurts by which side it blows from: from the live-engine side, the weathercock tendency yaws the aircraft toward the wind, the same direction the rudder is already fighting for, and eases the task; from the dead-engine side, the weathercock adds to the asymmetric yaw and the rudder must hold both. And mass reshapes the problem in an unexpected direction: a light aircraft flies its departures and approaches at low speeds, close to the minimum control speeds, where the rudder's authority is smallest against the same engine-out moment. Heavy, the aircraft's problem is performance (climbing on one engine); light, the problem is controllability. The certification speeds below exist for exactly the light, slow corner of the envelope.

VMC, VMCL, VMCG

VMC (airborne) — the minimum speed at which, after a sudden failure of the critical engine, control can be maintained and straight flight held. Determined under the worst certification conditions: maximum take-off thrust on the remaining engine(s), the most unfavourable (aft) CG, take-off configuration, the failed propeller windmilling (unless auto-feathering), and no more than 5° of bank toward the live engine.
VMCL (approach and landing) — the same idea in the landing configuration with go-around thrust: the minimum speed at which control holds when the critical engine fails during approach, with the additional demonstrated requirement of roll capability: the aircraft must still be able to roll away from the dead engine (20° of bank within 5 seconds) for the go-around manoeuvre.
VMCG (on the ground) — the minimum speed on the take-off roll at which, when the critical engine suddenly fails, directional control can be kept using the rudder alone, with no credit for nosewheel steering (a wet or contaminated runway may give the nosewheel nothing to work with), while deviating no more than 30 ft from the runway centreline.

The CG's influence runs the same way through all three: the rudder's moment is its force times its arm from the CG, and an aft CG shortens that arm: less rudder moment from the same deflection, so control is lost at a higher speed: aft CG raises VMC, VMCL and VMCG, which is why all three are determined at the most unfavourable, aft position. (Forward CG, for once, is the friendly direction.)

Density: the moment shrinks with the thrust

The final influence closes the course where it began, with the atmosphere. The asymmetric moment is only as large as the thrust that drives it, and thrust falls with air density: hot days, high airfields, high altitude. Less thrust on the live engine means a smaller moment to balance, so VMC, VMCL and VMCG all reduce as thrust reduces. The minimum control speeds are lowest exactly where the climb performance is worst, and highest (most limiting) on the cold, dense day when the engines deliver everything. The same logic applies to deliberately reduced-thrust take-offs: less asymmetry to fear at the cost of runway. Controllability and performance trade against each other through the same thrust, the closing symmetry of the whole subject.