Visual walkthrough — Static margin = (XCP − XCG) - d — must be positive (at least 1 caliber)

---

Step 1 — Draw the rocket and name two points

WHAT. A rocket is a long tube with a nose and fins. We mark two special points along it, both measured with a ruler starting at the nose tip and running toward the tail (this "aft-positive" direction is our number line).

• The centre of gravity (CG) is the balance point — if you laid the rocket on one finger, this is where it sits level. Its distance from the nose is $X_{CG}$.
• The centre of pressure (CP) is where all the pushing of the air can be imagined to act as one single force. Its distance from the nose is $X_{CP}$.

WHY. Both distances start from the same ruler-zero (the nose tip). If we measured them from different places, their difference would be meaningless. Using one datum makes $X_{CP}-X_{CG}$ a real, physical gap you could measure with a tape.

PICTURE. Look at the orange rocket. The violet dot is CG, the magenta dot is CP. The two brackets under the body show the distances $X_{CG}$ and $X_{CP}$ growing from the nose.

---

Step 2 — Tilt the rocket: the angle of attack $\alpha$

WHAT. In steady flight the air streams straight down the body. Now a gust nudges the nose. The body axis and the oncoming air (the "relative wind") no longer line up — the small angle between them is the angle of attack $\alpha$. (See Angle of Attack.)

WHY. Nothing interesting happens at $\alpha =0$: air slides past symmetrically and pushes nowhere in particular. Stability is a question about what happens after a disturbance, so we must tilt first, then ask "does the rocket come back?"

PICTURE. The navy dashed line is the body axis; the orange arrows are the incoming wind. The wedge between them, shaded violet, is $\alpha$. We choose nose-up as the positive direction — increasing $\alpha$ means the nose lifts.

---

Step 3 — The air answers with a normal force $N$

WHAT. When the body is tilted, the streaming air strikes the flank and produces a force perpendicular to the body axis. We call it the normal force $N$ ("normal" is the old word for "at a right angle"). By the meaning of CP, this whole force acts at the CP.

WHY this tool — a single force at one point? The air actually pushes on every square centimetre of the surface, a messy distributed load. That is hopeless to reason about directly. The trick of the centre of pressure is to replace the entire mess by one equivalent arrow placed at $X_{CP}$ — same total push, same turning effect. One arrow we can do torque with; a pressure cloud we cannot.

PICTURE. The magenta arrow $N$ sprouts from the CP, at a right angle to the navy body axis, pointing toward the side the wind came from.

---

Step 4 — How big is $N$? It grows with the tilt (small-angle law)

WHAT. For small tilts the force is proportional to the tilt:

$N=\underset{\text{dynamic pressure}}{\underbrace{\frac{1}{2}\rho v^2}}\underset{\text{reference area}}{\underbrace{A}}\underset{\text{force-factor per radian}}{\underbrace{C_{N\alpha }}}\underset{\text{tilt in radians}}{\underbrace{\alpha }}$

Term by term:

• $\rho$ = air density (thicker air pushes harder),
• $v$ = speed through the air; the $v^2$ says doubling speed quadruples the push,
• $A$ = a chosen reference area (fixed for the rocket),
• $C_{N\alpha }$ = the normal-force slope — force gained per radian of tilt; it is positive (see Normal Force Coefficient),
• $\alpha$ = our tilt, in radians.

WHY this step. We do not need the messy full curve — only two facts survive into the stability argument: $N>0$, and $N$ grows as $\alpha$ grows. Everything in the bracket except $\alpha$ is a fixed positive number, so $N\propto \alpha$. The straight line is the honest small-angle picture.

PICTURE. A straight rising tangent line of $N$ against $\alpha$ through the origin, with the true curve bending away past the small-angle window; the slope block is labelled $\frac{1}{2}\rho v^{2}A{C}_{N\alpha }$.

---

Step 5 — Force × lever arm = torque about the CG

WHAT. A free-flying rocket has nothing holding it, so it spins about its own CG. A force that acts at a distance from the pivot creates a turning effect — a torque. The lever arm here is the gap between where the force acts (CP) and the pivot (CG):

$\text{lever arm}=X_{CP}-X_{CG}$

$\boxed{\tau =-N(X_{CP}-X_{CG})}$

Term by term:

• $N$ = the magenta force from Step 4,
• $(X_{CP}-X_{CG})$ = distance from pivot to force = the lever arm,
• the leading minus sign encodes "this force turns the tail around," which in our nose-up-positive convention is a nose-down (negative) rotation when the arm is positive.

WHY the minus. Picture the CP behind the CG (positive arm) and $N$ pushing sideways there. Grab the tail and shove it — the nose swings the opposite way, back toward the wind. That is a rotation opposing a positive $\alpha$, so it must carry a negative sign. Without the "$-$" the algebra would falsely predict the tilt growing.

PICTURE. The rocket pinned at the violet CG. The magenta force at the CP and the curved navy arrow show the tail swinging one way, nose the other — restoring, because CP is aft.

---

Step 6 — Substitute and read off the sign

WHAT. Put Step 4's $N$ into Step 5's torque:

\;=\; -\,k\,\alpha\,(X_{CP}-X_{CG})$$ **WHY.** Now the *entire* behaviour lives in the signs. $k>0$ is a fixed positive, so the sign of $\tau$ is decided by the lever arm $(X_{CP}-X_{CG})$, the sign of $\alpha$, and the overall minus. - $X_{CP}-X_{CG} > 0$ (CP **behind** CG): $\tau = -(\text{positive})\,\alpha$ → **opposite sign to $\alpha$** → torque always drives $\alpha$ back toward zero → **restoring → stable** ✅ - $X_{CP}-X_{CG} < 0$ (CP **ahead** of CG): $\tau = -(\text{negative})\,\alpha$ → **same sign as $\alpha$** → torque grows $\alpha$ → **tumbling** ❌ - $X_{CP}-X_{CG} = 0$: $\tau = 0$ → **neutral**, drifts ⚠️ > [!intuition] The mirror case: nose-**down** disturbance ($\alpha<0$) > We tested a nose-up gust ($\alpha>0$). What about a nose-*down* one? Set $\alpha<0$. With CP aft (positive arm) we get $\tau=-k(\text{positive})\alpha = -k\times(\text{negative}) > 0$ — a **positive**, nose-**up** torque that again pushes $\alpha$ back toward zero. The restoring behaviour is perfectly **symmetric**: the sign of $\tau$ is always *opposite* to the sign of $\alpha$ whenever CP is behind CG, so the rocket rights itself from a tilt in **either** direction. No new figure needed — it is figure s06's stable rocket seen in a mirror. **PICTURE.** Three little rockets side by side — stable (arrow curls back to axis), unstable (arrow flings away), neutral (no arrow) — one per sign of the lever arm. This connects directly to the [[Rocket Stability Criterion]] and to [[Weathercocking]] when the margin gets large. --- ## Step 7 — Make it a pure number: divide by the diameter **WHAT.** The gap $X_{CP}-X_{CG}$ is a length in cm. Divide by the body diameter $d$ (one "caliber") to get the ==static margin==: $$\text{SM} \;=\; \frac{X_{CP}-X_{CG}}{d}\qquad(\text{a plain number, no units})$$ **WHY divide.** A big rocket and a small rocket with the same *number of body-widths* of margin fly alike, even though their centimetre gaps differ. Making it dimensionless lets one rule — "$\text{SM} \ge 1$ caliber" — govern every rocket. See [[Fin Design]] for how fins push $X_{CP}$ aft to raise this number. **PICTURE.** The lever gap shown, then the same gap measured in units of $d$: the ruler is re-ticked in calibers, reading "1 cal, 2 cal, 3 cal." --- ## Step 8 — The edge case: why $\ge 1$ caliber, not just $>0$ **WHAT.** Mathematically any $\text{SM}>0$ is stable. In reality both points *move*: $X_{CP}$ drifts **aft in the transonic regime** then **forward supersonically** (see [[Transonic Aerodynamics]]), and the CG shifts as propellant burns. A rocket that is barely positive at launch can slip to **negative** mid-flight. **WHY the cushion.** We demand at least **1 caliber** so that when the points wander, the margin still stays above zero everywhere in the flight. But do not over-do it: past ~2 calibers the rocket becomes **over-stable** and weathercocks hard into crosswinds, arcing away and losing altitude. **PICTURE.** A margin-vs-Mach curve dipping through the transonic zone; a green band marks the safe $1$–$2$ caliber corridor, red zones above and below. --- ## The one-picture summary Everything above compressed: rocket tilted by $\alpha$, force $N$ at the CP, pivot at the CG, restoring curl when CP is aft, and the caliber ruler turning the gap into $\text{SM}=(X_{CP}-X_{CG})/d$. > [!recall]- Feynman retelling — the whole walkthrough in plain words > Take a rocket and mark two dots with a ruler that starts at the nose: the balance dot (CG) and the air-push dot (CP). Now a gust tips the nose a tiny bit — that little lean is $\alpha$, measured in radians. The moving air feels the tilt and shoves sideways on the body, and we pretend that whole shove is one arrow sitting at the air-push dot. The harder you tilt, the harder the shove — a straight line, trustworthy as long as the tilt stays small (under about ten degrees). Because the rocket floats free, it spins about the balance dot, and how fast it spins up depends on how heavy-and-long it is (its moment of inertia). The shove, sitting a little *behind* the balance dot, grabs the tail and swings it — which points the nose right back into the wind. Tip the nose the *other* way and the same thing happens mirror-image, so it recovers from a tilt in either direction. Flip the dots so the push is in *front* of the balance point, and the same shove flings the nose further off — that's a tumble. Measure the gap between the dots in rocket-widths and you get the static margin; keep it at least one width (but not too many) so it stays safely positive even as the dots drift around during flight. > [!mnemonic] > **Tilt → Push → Pivot → Come-back.** Four words, whole derivation. Push behind pivot = come-back = stable.