Foundations — Gyroscope in spacecraft attitude control — preview

1.5.17 · D1 · Physics › Rotational Mechanics › Gyroscope in spacecraft attitude control — preview

Yeh page har symbol aur picture build karta hai jis par parent note Gyroscope in spacecraft attitude control — preview depend karta hai. Hum assume karte hain ki tumne pehle kabhi arrow-with-a-hat ya $d/dt$ nahi dekha. Hum har ek ko earn karenge.

0 · Vector kya hota hai, aur yeh chota sa arrow $\stackrel{⃗}{}$ kyun?

Kuch quantities ko sirf ek number chahiye: 20° ka temperature, 3 kg ki mass. Doosron ko ek number aur ek direction chahiye: "5 metre jao — lekin kis taraf?"

Figure s01 neeche ek single vector ko do jagah par draw kiya hua dikhata hai.

Figure s01 dekho. Cyan arrow $\vec{A}$ hai. Ise bina ghoomaye idhar-udhar slide karo (amber copy) aur woh still same vector hai — sirf length aur direction matter karta hai, woh kahan draw hai nahi. Yeh choti si baat hi woh poora reason hai ki ek spinning wheel compass ki tarah kaam kar sakta hai: uska angular-momentum arrow space mein apni jagah par tika rehta hai.

0b · Vectors ke saath do kaam: stretch karna aur add karna

Kisi bhi vector se kuch multiply karne se pehle humein yeh samajhna hoga ki uska kya matlab hai, aur do arrows ko saath add karne ka kya matlab hai. (Hum chhota arrow $d\vec{L}$ baad mein, §5 mein, milenge, jab ke hum define kar lenge ki chhote $d$ ka kya matlab hai — isliye abhi ke liye hum do ordinary arrows add karte hain.)

Figure s04 head-to-tail rule ko uss special case ke liye dikhata hai jis ki humein sabse zyada zarurat padegi: ek bada arrow jiski tip par ek tiny arrow right angle par chipka ho.

Figure s04 dekho. Kyunki woh chota added arrow (amber) bade cyan arrow ki tip par right angle par chipka hai, nayi dashed arrow ki almost same length hai lekin direction thodi turn ho gayi hai. Woh picture apne dimaag mein rakh lo — yeh precession ka poora raaz hai, jise hum §5 mein unlock karenge jab $d$ notation aa jaayegi.

1 · Angular speed $\omega$ aur angular velocity $\vec{\omega }$

Ek car ka speedometer measure karta hai ki woh aage kitni tezi se ja rahi hai. Spinning ke liye humein ek alag meter chahiye: object kitni tezi se ghoomta hai.

Lekin "spinning" mein secretly ek direction bhi hoti hai: kaun sa axis, aur kaunsi taraf. Isliye hum scalar ko vector mein promote karte hain.

Radians kyun, degrees kyun nahi? Radians "natural" angle unit hain jahan arc-length $=$ radius $\times$ angle bina kisi extra conversion factor ke. Woh clean relationship exactly woh hai jo hum baad mein use karenge — yeh compute karne ke liye ki ek arrow ki tip sideways kitni door slide karti hai. Degrees ek badsorat $\pi /180$ le aate.

2 · Moment of inertia $I$ — rotational "heaviness"

Ek bhaari shopping trolley ko chalane ke liye tum zor se dhakka dete ho; uski mass straight-line motion mein changes ka virodh karti hai. Spinning ka apna "heavy to get going" version hota hai.

Figure s02 equal mass ki do wheels ko compare karta hai lekin unka mass placement alag-alag hai.

Figure s02 mein doono wheels ki same mass hai, lekin daayein wali ki mass rim tak baahir push ki gayi hai. Uss wali ka $I$ zyada hai — woh better gyroscope hai, kyunki door ki mass spin changes ka zyada virodh karti hai.

3 · Angular momentum $\vec{L}=I\vec{\omega }$ — "spin ki matra", ek arrow ke roop mein

Ab doono ko combine karo: ek spinning object ek certain amount of spinning motion carry karta hai. Woh amount angular momentum hai.

Axis ke along kis taraf? Wahi right-hand rule jo $\vec{\omega }$ ke liye hai: apni right hand ki ungliyan us taraf curl karo jis taraf wheel ghoomta hai; tumhara thumb $\vec{L}$ ke along point karega.

Figure s03 spin-arrow ko axis ke along khada hua dikhata hai.

Is quantity ke baare mein aur depth Angular Momentum mein hai.

4 · Torque $\vec{\tau }$ — ek push nahi, ek twist

Ek plain force kisi cheez ko straight line mein push karta hai. Wrench ghoomane ke liye tumhe ek force pivot ke side mein apply karni hoti hai — ek twist.

Torque ko spin se connect karne wala rulebook Torque and Newton's Second Law for Rotation mein hai.

5 · Rate of change $\frac{d\vec{L}}{dt}$ — "yeh kitni tezi se change ho raha hai?"

Parent note ki poori derivation ek expression par tiki hai: $\vec{\tau }=\frac{d\vec{L}}{dt}$. Us fraction ko samajhte hain.

Updated spin-arrow head-to-tail sum $\vec{L}+d\vec{L}$ hai — yeh exactly woh picture hai jo Figure s04 mein §0b se hai, ab arrows par real names ke saath. Tipping torque ke liye, $d\vec{L}$ us taraf point karta hai jis taraf torque point karta hai, jo $\vec{L}$ ke sideways hai — isliye sum ki almost same length hai lekin turned direction. $\vec{L}$ girne ki jagah ghoomta hai. Yeh precession ka beej hai.

6 · Cross product, aur $\vec{\tau }=\vec{\Omega }\times \vec{L}$ ko derive karna

Parent $\vec{\tau }=\vec{\Omega }\times \vec{L}$ quote karta hai. Hum ise sirf quote nahi karenge — hum ise build karenge. Pehle tool.

Figure s06 $\theta$ angle ko tail-to-tail maapa hua aur perpendicular output arrow dikhata hai.

Ab doosra turning-rate milte hain. Wheel apni axis ke baare mein tezi se $\omega$ rate par spin karta hai; lekin puri axis bhi ek circle mein dheere-dheere walk karti hai. Woh slow walk apna khud ka rotation hai, apne khud ke angular-velocity vector ke saath.

Ek rotating vector kaise change hota hai — key link. Maano $\vec{L}$ ki length fixed rehti hai lekin pura arrow precession $\vec{\Omega }$ se rotate ho raha hai. $d\vec{L}/dt$ kya hoga?

Isliye $\vec{L}$ ki tip, $\vec{\Omega }$ se ghoomayi jaati, is velocity se move karti hai $\frac{d\vec{L}}{dt}=\vec{\Omega }\times \vec{L}.$ Padhte hain: $\vec{L}$ mein change $\vec{\Omega }$ aur $\vec{L}$ dono ke perpendicular hai (ek sideways slide), size $\Omega L\sin \theta$ ke saath — exactly ek rotating arrow. Ab §5 se ek law $\vec{\tau }=d\vec{L}/dt$ ke saath combine karo:

Figure s05 $\vec{L}$ ki tip ko circle trace karte hua dikhata hai, $d\vec{L}/dt=\vec{\Omega }\times \vec{L}$ uske tangent ke saath.

7 · Conservation of angular momentum — pointing trick

Prerequisite map — ise kaise padhein

Neeche wala diagram ek dependency chart hai: arrow "$X\to Y$" matlab "Y samajhne se pehle X chahiye." Top boxes se shuru karo (pure ideas jinhein kuch nahi chahiye) aur arrows follow karo neeche; har path eventually bottom par gyroscope topic mein funnel ho jaata hai. Ise top-to-bottom padhna exactly woh order hai jismein is page ne symbols introduce kiye, aur dikhata hai kyun har ek ko agle se pehle aana pada.

Equipment checklist

Predict each answer before revealing.