Nomenclature & Sign Conventions

S = c · b

[c] = m [b] = m

which multiplies out to m² — wing area.

The one that needs no arithmetic, and the one everything else is divided by.

AR = b² / S

[b] = m [S] = m²

which multiplies out to 1 — aspect ratio.

A length squared over an area: aspect ratio is a pure number, which is why it compares a glider with an airliner.

V = √( 2W / (ρ · S · C_L) )

[W] = kg·m·s⁻² [ρ] = kg·m⁻³ [S] = m² [C_L] = 1

the quantities combine to m²·s⁻², and the square root of that is m·s⁻¹ — airspeed.

Weight is a force, not a mass — that is the error this one catches. Put kilograms in where newtons belong and the root returns something that is not a speed.

q = ½ · ρ · V²

[ρ] = kg·m⁻³ [V] = m·s⁻¹

which multiplies out to kg·m⁻¹·s⁻² (Pa) — dynamic pressure.

Density times a speed squared is a pressure. Every aerodynamic force on the site is this multiplied by an area and a coefficient.

L = q · S · C_L

[q] = kg·m⁻¹·s⁻² [S] = m² [C_L] = 1

which multiplies out to kg·m·s⁻² (N) — lift.

Pressure times area is force, and the coefficient carries no dimensions at all — which is the whole reason coefficients exist.

Re = ρ · V · L / μ

[ρ] = kg·m⁻³ [V] = m·s⁻¹ [l] = m [μ] = kg·m⁻¹·s⁻¹

which multiplies out to 1 — a pure number.

Four dimensional quantities that cancel completely. A Reynolds number with units on it has been worked out wrongly.

Re = V · L / ν

[V] = m·s⁻¹ [l] = m [ν] = m²·s⁻¹

which multiplies out to 1 — a pure number.

The same number through kinematic viscosity, which is μ/ρ and carries the density already. Two roads, one dimensionless answer.

C_Di = C_L² / (π · e · AR)

[C_L] = 1 [e] = 1 [AR] = 1

which multiplies out to 1 — drag coefficient.

Dimensionless throughout. Nothing here can tell you whether π belongs in it — which is the limit of the method.

P = D · V

[D] = kg·m·s⁻² [V] = m·s⁻¹

which multiplies out to kg·m²·s⁻³ (W) — power.

Force times speed. The most useful identity on the site: it turns a drag count into watts and a watt-hour into a flight.

w = V · C_D / C_L

[V] = m·s⁻¹ [C_D] = 1 [C_L] = 1

which multiplies out to m·s⁻¹ — airspeed.

A speed multiplied by a ratio of two dimensionless numbers is still a speed. The glide ratio is that ratio upside down.

T = ( P · FM · √(2 · ρ · A) )^⅔

[P] = kg·m²·s⁻³ [ρ] = kg·m⁻³ [A] = m²

the quantities combine to kg³·m³·s⁻⁶, and the cube root of that is kg·m·s⁻² (N) — thrust.

The fractional powers are the interesting part: two thirds of a power times a third of a density times a third of an area lands exactly on a force.

T = 2 · ρ · A · v_i · (V + v_i)

[ρ] = kg·m⁻³ [A] = m² [V] = m·s⁻¹

which multiplies out to kg·m·s⁻² (N) — thrust.

Momentum flux: density times area times a speed squared is a force. That is the actuator disc in one line.

s = 1.44 · W² / (g · ρ · S · C_L · F)

[W] = kg·m·s⁻² [g] = m·s⁻² [ρ] = kg·m⁻³ [S] = m² [C_L] = 1 [T] = kg·m·s⁻²

which multiplies out to m — a length.

Two forces on top, and a length comes out. Worth checking by hand: it is the closed form the sheet does not use, and it is easy to drop a factor in.

M = (n · W / 2) · y

[n] = 1 [W] = kg·m·s⁻² [l] = m

which multiplies out to kg·m²·s⁻² (N·m) — a moment.

Force times arm. The load factor is a pure number, so it changes the answer without changing what the answer is.

A = M / (σ · h)

[M] = kg·m²·s⁻² [σ] = kg·m⁻¹·s⁻² [h] = m

which multiplies out to m² — cap area.

A moment over a stress and a depth. Cap area falls in direct proportion to depth, which the dimensions show before any arithmetic does.

δ = ∬ M / (E · I) · dx · dx

[M] = kg·m²·s⁻² [E] = kg·m⁻¹·s⁻² [I] = m⁴ [l] = m

which multiplies out to m — a length.

M/EI is one over a length — a curvature. Integrating it twice along the span is what turns it into a deflection, and the dimensions are why there are two integrations rather than one.

H = C_h · q · S_f · c_f

[C_h] = 1 [q] = kg·m⁻¹·s⁻² [S] = m² [c] = m

which multiplies out to kg·m²·s⁻² (N·m) — a moment.

Pressure, area, arm. The chord appears twice — once inside the area and once as the arm — which is why hinge moment goes as the square of the chord fraction.

V_H = S_h · l_h / (S_w · MAC)

[S] = m² [l] = m [c] = m

which multiplies out to 1 — a pure number.

An area times a length over an area times a length. Dimensionless, which is what lets one number describe a trainer and a transport.

lever = l_s · t / √(4B² + t²)

[l] = m [b] = m

which multiplies out to m — a length.

Length times length over a length. The root adds two squared lengths, so it is a length too — a sum whose terms did not match would be the error to look for.

E = N · V · Q

[V] = kg·m²·s⁻³·A⁻¹ [Q] = s·A

which multiplies out to kg·m²·s⁻² (N·m) — energy.

Volts times coulombs is joules. A milliamp-hour is a charge with the hours hidden in it, which is why watt-hours and amp-hours are not interchangeable.

ΔV = I · R

[I] = A [R] = kg·m²·s⁻³·A⁻²

which multiplies out to kg·m²·s⁻³·A⁻¹ (V) — voltage.

Current times resistance is a voltage. Everything the battery sheet says about sag rests on this one line.

R·I² − V·I + P = 0

[R] = kg·m²·s⁻³·A⁻² [I] = A

which multiplies out to kg·m²·s⁻³ (W) — power.

Every term is a power: I²R, VI and P alike. That is what makes it a quadratic in current rather than a division, and it is a dimensional argument before it is an electrical one.

ρ = p / (R · T)

[p] = kg·m⁻¹·s⁻² [R] = m²·s⁻²·K⁻¹ [T] = K

which multiplies out to kg·m⁻³ — air density.

The specific gas constant carries the temperature away and leaves a density. Its own units — J per kilogram per kelvin — are what make that work.