Load Factor (pull-up)⚠ unverified

Understanding the Parameters

• Load factor $n = L/W$ — the ratio of lift to weight, measured in "g." $n = 1$ is straight-and-level (wings carry exactly the weight); $n = 2$ means the wings carry twice the weight and everything aboard feels twice as heavy. It is the structural and physiological currency of manoeuvring flight.

• Why a pull-up adds g's — curving the flight path upward requires a centripetal force directed toward the centre of the arc (upward, at the bottom of a loop). The wing must supply that force on top of holding the aircraft up against gravity. So $L = W + mV^2/R$, and dividing by $W = mg$ gives $n = 1 + V^2/(Rg)$ — the "1" is the weight, the second term is the curving.

• Sensitive to speed and radius — because the centripetal term is $V^2/(Rg)$, load factor rises with the square of speed and inversely with radius: a tight ($R$ small), fast ($V$ large) pull-up piles on g quickly. Doubling the speed at the same radius quadruples the centripetal contribution. This is why high-speed dive recoveries and tight pull-ups are the classic way to over-g an airframe.

• Structural and human limits — the airframe has a limit load factor (e.g. +2.5 g for transports, +9 g for fighters) and an ultimate beyond which it breaks; pilots have physiological limits (grey-out/black-out around +4–5 g without a g-suit, more with). The pull-up load factor is what must be kept inside the V–n (flight envelope) diagram, whose curved left boundary is the accelerated stall ($n$ capped by $C_{L\max}$) and whose top is the structural limit.

• Relation to the level turn — a level turn's load factor is $n = 1/\cos\phi = \sqrt{1+(V^2/Rg)^2}$-type geometry (see Turn Radius); the pull-up is the vertical-plane case where lift and weight are aligned, so they add directly, giving the simpler $1 + V^2/Rg$. Both are the wing making centripetal force to bend the path.

Derivation (Approaching a Proof)

Consider the aircraft at the bottom of a vertical pull-up — curving upward with radius $R$ at speed $V$. Two vertical forces act: the wing's lift $L$ (upward) and the weight $W = mg$ (downward). The net upward force provides the centripetal acceleration $V^2/R$ needed to curve the path (Centripetal Force, Newton's Second Law):

$$L - W = \frac{m V^2}{R}.$$

Solve for the lift:

$$L = W + \frac{m V^2}{R} = mg + \frac{m V^2}{R}.$$

The load factor is lift over weight:

$$n = \frac{L}{W} = \frac{mg + mV^2/R}{mg} = 1 + \frac{V^2}{R\,g}. \qquad\blacksquare$$

The mass cancels — load factor is independent of how heavy the aircraft is, a purely kinematic-plus-gravity result. The "1" is the level-flight lift needed against gravity; the $V^2/(Rg)$ is the extra lift needed to curve the trajectory. (At the top of a loop, gravity points toward the arc's centre and helps, giving $n = V^2/(Rg) - 1$; this calculator is the bottom-of-loop case.)

Dimensional check. $\dfrac{V^2}{R\,g} = \dfrac{(\text{m/s})^2}{\text{m}\cdot\text{m/s}^2} = \dfrac{\text{m}^2/\text{s}^2}{\text{m}^2/\text{s}^2} = 1$ (dimensionless), so $n = 1 + (\text{dimensionless})$ is a pure number ✓ — expressed in multiples of g.

History and Development

• The V–n diagram. Load-factor limits are codified in the V–n (velocity–load-factor) diagram, the cornerstone of structural design and flight-envelope definition since the 1920s–30s. Its boundaries — the accelerated-stall curve, the structural limit, the never-exceed speed — bound every manoeuvre, including the pull-up this formula quantifies.

• G and the human body. Understanding manoeuvre g-loads drove aeromedical research: grey-out, black- out, and G-LOC (g-induced loss of consciousness) limits, and the invention of the anti-g suit (WWII) and straining manoeuvres, all respond to the load factors that pull-ups and turns generate.

• Dive-recovery structural failures. High-speed dive pull-outs producing excessive $V^2/R$ load factors were a notorious cause of in-flight structural failures in early high-performance aircraft, sharpening the distinction between the limit load factor (must sustain) and the ultimate (must not break below) that certification enshrines.

Related Concepts: Turn Radius, Stall Speed, Centripetal Force, Newton's Second Law, Lift Force, Static Margin

Notes: Bottom-of-loop pull-up load factor: wing supplies weight (1 g) plus centripetal force ($V^2/Rg$) → $n = 1 + V^2/(Rg)$. Rises with $V^2$ and $1/R$ — tight/fast pull-ups over-g quickly. Mass cancels. Keep inside the V–n diagram (accelerated-stall left boundary, structural-limit top; e.g. +2.5 g transport, +9 g fighter). Top-of-loop case is $V^2/(Rg) - 1$ (gravity helps). Vertical-plane companion of the level turn (Turn Radius). $g$ exposed as an input but constant. Defaults ⇒ $n\approx3.04$ g.