Jab rocket thode se angle of attackα par fly karta hai (uski axis aur airflow ke beech ka angle), toh guzarti hawa ek net sideways aerodynamic force produce karti hai, jo normal forceN hai, jo CP par act karti hai.
Yeh force CG ke baare mein ek torque create karta hai (kyunki rocket CG ke around pivot karta hai). Woh torque tilt ko theek karta hai ya aur bura karta hai yeh is baat par depend karta hai ki kaun sa point peeche hai:
CP, CG ke peeche → force tail ko sideways push karta hai, nose ko wapas hawa ki taraf swing karta hai → restoring → stable. ✅
CP, CG ke aage → force nose ko aur bahar push karta hai → tilt aur buri ho jaati hai → unstable. ❌
Notation note. Poori derivation mein, ℓ ko CG→CP separation maano (ek single signed lever arm: positive jab CP, CG ke peeche ho). Nose se measure ki gayi absolute positions Xcp aur Xcg likhi jaati hain, toh ℓ=Xcp−Xcg. Lever arm (ℓ) aur nose-referenced positions (X) ko alag symbols rakhne se confusion nahi hoti.
Step 1 — Geometry set up karo.
Maano CG pivot hai. CP, axis ke saath CG ke ℓ distance peeche hai. Angle of attack α par, airspeed V hai, air density ρ hai, reference area A hai.
Yeh step kyun? Torque banane ke liye humein ek lever arm aur ek force chahiye; lever arm CG→CP distance ℓ hai.
Step 2 — Aerodynamic normal force likho.
Sideways (normal) force sab aerodynamic forces ki tarah scale karti hai — dynamic pressure 21ρV2 aur area A ke saath, aur ek coefficient ke saath jo α ke saath badhta hai:
N=21ρV2ACNαα
jahan CNα=∂CN/∂αnormal-force curve slope hai.
Yeh step kyun? Chhote α ke liye, CN≈CNαα (linear) hota hai, kyunki tilt ko double karne se sideways air push roughly double ho jaati hai.
Step 3 — CG ke baare mein moment lo.M=−Nℓ=−21ρV2ACNαℓα
Yeh step kyun? Torque = force × perpendicular lever arm. Minus sign key hai: agar ℓ>0 (CP, CG ke peeche), toh positive tilt α ek negative (opposing) moment deta hai → yeh α ko wapas zero par push karta hai.
Step 4 — Non-dimensionalise karo: Static Margin.
Lever arm ℓ=Xcp−Xcg ko body diameterd se divide karo. Kyunki ek "caliber" ka matlab hi ek body diameter hai, diameter yahan sahi reference length hai (rocket length L se divide karna ek valid dimensionless number deta, lekin calibers mein measure kiya margin nahi).
Non-dimensionalise kyun karo? Taaki ek bade aur chhote rocket ko ek hi scale par compare kiya ja sake; "calibers of margin" ek universal design number hai.
Nahi — yeh angle of attack aur Mach number ke saath shift karta hai; CG bhi fuel burn hone par shift karta hai.
Fins tail par kyun lagti hain?
CP ko peeche, CG ke peeche kheenchne ke liye.
Ek weather-vane hawa ki taraf kyun point karta hai?
Uska surface area (air pressure/drag) zyaadatar pivot ke peeche hota hai, weight ki wajah se nahi.
Recall Feynman: explain to a 12-year-old
Ek dart seedha uda jaata hai kyunki uske peeche feathers hote hain. Agar koi gust dart ko sideways kar de, toh hawa un peeche wale feathers par sabse zyaada push karti hai aur pointy nose ko wapas hawa ki taraf swing kar deti hai. Ek rocket ek giant dart hai: uski fins feathers hain. Jab tak "hawa pakadne wali jagah" (CP) "balance ki jagah" (CG) ke peeche hai, rocket hamesha apni nose sahi taraf wapas ghuma leta hai. Fins aage lagao aur dart ulta ghoom jaayega!
