Foundations — Angular momentum L = Iω (fixed axis), L = r × p (general)

1.5.10 · D1 · Physics › Rotational Mechanics › Angular momentum L = Iω (fixed axis), L = r × p (general)

Yeh page assume karta hai aap kuch nahi jaante. Hum har letter, arrow, aur symbol jo parent note use karta hai, ek aisi order mein banate hain jahan har piece sirf pehle se bane pieces par depend kare. Upar se neeche padho.

1. Ek point, ek origin, aur ek arrow: $O$ aur $\vec{r}$

Kuch bhi spin karne se pehle, hume yeh kehne ka tarika chahiye ki cheezein kahan hain.

Topic ko yeh kyun chahiye: angular momentum hamesha ek chosen point ke baare mein measure hota hai. Dot $O$ ko hatao, aur arrow $\vec{r}$ badal jaata hai, toh answer badal jaata hai. Isliye parent note baar baar "about a chosen origin" kehta rehta hai.

2. Kitni fast, kis taraf: velocity $\vec{v}$ aur speed $v$

Topic ko yeh kyun chahiye: angular momentum motion ki parwah karta hai. Lekin — baad ke liye crucial hint — yeh sirf motion ke us hisse ki parwah karta hai jo $O$ ke around curl karta hai, us hisse ki nahi jo seedha $O$ ki taraf ya door jaata hai.

3. Mass $m$ aur linear momentum $\vec{p}$

Seedhi line wali version ke liye Linear Momentum dekho jis par yeh poora topic build hua hai.

4. Angle $\theta$ aur lever arm $r_{\perp }$

Ab hamare paas do arrows hain jo us point par share hote hain jahan object hai: $\vec{r}$ ($O$ se object tak) aur $\vec{p}$ (object ka motion). Unke beech ka angle show ka star hai.

Hume measure karna hai "motion ka kitna hissa $O$ ke around jaata hai." Figure dekho: motion ki direction ke along ek seedhi line daalo. $O$ se us line tak ki sabse chhoti doori woh piece hai jo matter karti hai.

$\sin \theta$ kyun, kuch aur kyun nahi? Hum $\vec{r}$ ka woh piece chahte hain jo motion ke sideways ho — woh part jo lever ki tarah kaam karta hai. Right triangle mein, $\theta$ ke opposite wali side exactly woh sideways piece hai, aur "opposite over hypotenuse" $\sin \theta$ ki definition hai. Toh $r\sin \theta$ precisely lever arm nikalta hai.

• $\theta =90^{\circ }$: motion fully sideways hai, $\sin 90^{\circ }=1$, toh $r_{\perp }=r$ — maximum swirl.
• $\theta =0^{\circ }$ ya $180^{\circ }$: motion seedha $O$ ki taraf/se door hai, $\sin \theta =0$, toh $r_{\perp }=0$ — zero swirl. Kuch bhi around nahi jaata.

5. Cross product $\vec{r}\times \vec{p}$

Hamare paas do arrows hain aur ek angle. Hume ek aisi operation chahiye jo:

1. Automatically swirl-magnitude $rp\sin \theta$ nikale,
2. Zero de jab motion seedha toward/away ho ($\theta =0$ ya $180^{\circ }$),
3. Hume axis bataye jis par swirl hota hai (kis taraf turn karta hai).

Woh operation hai cross product. Isliye parent note ise use karta hai ordinary multiplication ki jagah.

Yeh exactly angular momentum ko kyun chahiye: length ke andar $\sin \theta$ automatically straight-in/straight-out motion ko delete kar deta hai (unka $\sin \theta =0$ hai), sirf going-around part bachta hai. Aur thumb-direction hume spin axis free mein de deta hai. Full machinery ke liye Cross Product dekho.

6. Angular speed $\omega$ aur circular motion

Poore spinning body ke liye hum har speck ki straight-line speed track karna band kar dete hain aur switch karte hain kitni fast yeh turn karta hai par.

7. Moment of inertia $I$ aur sum symbol $\sum$

Topic ko yeh kyun chahiye: jab aap $\vec{r}\times \vec{p}$ ko rigid body ke har speck par add karte ho (har ek ke saath $v_i=\omega r_i$), constant $\omega$ factor out ho jaata hai aur jo bachta hai woh exactly $\sum m_i{r}_i^2$ hai. Us lump ko naam $I$ dene se ek messy sum clean $L=I\omega$ mein badal jaata hai. Full detail Moment of Inertia mein.

8. Do headline symbols: $L$ aur $\vec{L}$

Ab har ingredient define ho gaya hai, hum finally woh do formulas likh sakte hain jis par parent note tika hai.

Upar sab kuch isliye assemble kiya taaki yeh do lines kuch concrete mean karein na ki sirf symbols yaad karne ke liye. $L$ aage kahan jaata hai iske liye Rotational Kinetic Energy aur Conservation of Angular Momentum dekho, aur use kya change karta hai iske liye Torque dekho.

9. Foundations topic ko kaise feed karte hain

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