Foundations — Conservation of angular momentum — conditions

1.5.12 · D1 · Physics › Rotational Mechanics › Conservation of angular momentum — conditions

Yeh toolbox page hai. Isse pehle ki tum kaho "angular momentum conserved hota hai jab net external torque zero ho", tumhe pata hona chahiye ki time, position, velocity, momentum, force, cross product, torque, moment of inertia aur angular velocity mein se har ek ka kya matlab hai aur har ek kaisa dikhta hai. Hum inhe ek ek karke build karenge, ek aisi order mein jahan koi cheez use hone se pehle define ho — koi symbol formula mein tab tak appear nahi karta jab tak uska apna section us par drawn na ho.

0. Time $t$ — "woh clock jiske against sab kuch measure hota hai"

Hume $t$ pehle chahiye kyunki conservation time ke baare mein ek statement hai: "spin ki amount jab $t$ aage badhta hai nahi badlti." Is page par change ki har rate — velocity, momentum ka badalna, aur eventually spin quantity ka khud badalna — ek "per-second" quantity hai, isliye clock ko baaki sab se pehle exist karna chahiye.

1. Position vector $\vec{r}$ — "object kahan hai, ek chosen dot se measure kiya hua"

Ek chosen dot ki zaroorat kyun hai? Kyunki "kiske baare mein spin?" ka koi jawab nahi hai jab tak tum yeh na kaho kis point ke baare mein. Poora topic is choice par depend karta hai — origin badle aur $\vec{r}$ badle, aur (jaise hi define hoga) $\vec{r}$ se bani spin quantity bhi badle.

Figure s01 exactly yahi dikhata hai: origin par ek black dot labelled "chosen dot", object ke liye ek aur black dot, aur red arrow $\vec{r}$ origin se object tak jaata hai. Picture clearly dikhati hai ki $\vec{r}$ origin se drawn hai — koi origin nahi, koi arrow nahi.

Tail (origin par) se head (object par) tak woh chota arrow poori picture hai: ek arrow ek vector hai — ek aisi cheez jisme size aur direction dono hote hain. Plain numbers (jaise "3 kg") ki koi direction nahi hoti; unhe scalars kehte hain.

2. Velocity $\vec{v}$ aur momentum $\vec{p}=m\vec{v}$ — "kitna tez, kis direction mein, rokna kitna mushkil"

Symbol $\frac{d\vec{r}}{dt}$ ka bas matlab hai "tiny sliver of time $dt$ mein $\vec{r}$ kitna badlta hai, us sliver se divide karke". Ise padho "position ki rate of change" — isse zyada kuch scary nahi.

3. Force $\vec{F}$ — "woh push ya pull jo motion ko change karta hai"

Force ko ab kyun laate hain? Kyunki Newton's second law kehta hai force precisely wahi hai jo momentum ko change karta hai:

Hume yeh line isliye chahiye kyunki torque (§6) force se build hota hai, aur poore topic ki master equation §5 mein bani spin quantity ke derivative mein $\vec{F}=\frac{d\vec{p}}{dt}$ daalne se aati hai.

4. Cross product $\vec{a}\times \vec{b}$ — "do arrows kitne right angles mein hain, ek naye arrow ke roop mein package kiya hua"

Yeh page par sabse important tool hai, isliye hum ise carefully build karte hain.

Exactly yahi tool kyun, aur ordinary multiplication kyun nahi? Kyunki topic poochh raha hai "ek force kitna twist karta hai push ke bajaye?" Twisting ek sideways effect hai. Factor $\sin \theta$ woh machine hai jo sideways-ness measure karta hai:

• jab do arrows parallel hain ($\theta =0$), $\sin 0=0$ → cross product zero vector hai. Kuch bhi sideways nahi, koi twist nahi.
• jab woh perpendicular hain ($\theta =90^{\circ }$), $\sin 90^{\circ }=1$ → cross product jitna ho sake utna bada hai. Sab sideways, maximum twist.

Figure s02 dono extremes side by side dikhata hai: left par, do black arrows ek doosre ke upar pade (parallel) labelled "sin0 = 0, cross = 0"; right par, ek black arrow $\vec{a}$ aur ek red arrow $\vec{b}$ right angle par milte hain, labelled "sin90 = 1, cross = biggest". Left se right padhne par, sideways-ness badhti hai.

Right-hand rule (direction). Apne right hand ki ungliyan pehle arrow $\vec{a}$ ke saath point karo, unhe doosre arrow $\vec{b}$ ki taraf curl karo; tumhara thumb $\vec{a}\times \vec{b}$ ke along point karta hai. Isliye spin quantity aur torque (aage banenge) motion ke plane mein nahi, spin axis ke along point karte hain.

5. Angular momentum $\vec{L}=\vec{r}\times \vec{p}$ — "chosen dot ke baare mein spin ki amount"

Ab har piece exist karta hai, isliye hum show ka star assemble kar sakte hain.

Figure s03 ek particle dikhata hai jo position $\vec{r}$ par hai (origin se black arrow) aur uska momentum $\vec{p}$ red arrow hai. Ek dashed line mark karta hai $\vec{p}$ ka woh part jo $\vec{r}$ ke sideways hai — yahi part spin drive karta hai. Agar $\vec{p}$ seedha $\vec{r}$ ke along point kare, toh sideways part (aur hence $\vec{L}$) vanish ho jaata.

Sirf $\vec{p}$ ka woh component jo $\vec{r}$ ke perpendicular ho woh spin mein contribute karta hai — ek particle jo seedha origin ki taraf ja raha ho ($\vec{p}\parallel \vec{r}$) ka $\vec{L}=0$ hai, exactly jaise ek pathar jo seedha tumhari aankh ki taraf gir raha ho woh tumhare around circle nahi kar raha. (Aur origin par ek particle ka $\vec{r}=0$ hai, toh $\vec{L}=0$ bhi — §1 ka edge case, ab precise hua.)

6. Torque $\vec{\tau }=\vec{r}\times \vec{F}$ aur master equation

Ab hum $\vec{L}$ ka changer derive kar sakte hain, upar build kiye gaye har tool ko use karke. $\vec{L}=\vec{r}\times \vec{p}$ se shuru karo aur time ke respect mein differentiate karo:

$\frac{d\vec{L}}{dt}=\frac{d\vec{r}}{dt}\times \vec{p}+\vec{r}\times \frac{d\vec{p}}{dt}$

Humne kya kiya: cross-product product rule apply kiya (§4, line three). Kyun: $\vec{L}$ do arrows ka product hai jo dono $t$ ke saath change hote hain.

$\frac{d\vec{r}}{dt}\times \vec{p}=\vec{v}\times (m\vec{v})=m(\vec{v}\times \vec{v})=0$

Humne kya kiya: $\frac{d\vec{r}}{dt}=\vec{v}$ (§2) aur $\vec{p}=m\vec{v}$ use kiya; phir $\vec{v}\times \vec{v}=0$ kyunki ek vector khud ke parallel hai (§4, line one). Toh pehla term vanish ho jaata hai.

$\vec{r}\times \frac{d\vec{p}}{dt}=\vec{r}\times \vec{F}=\vec{\tau }$

Humne kya kiya: Newton's second law $\vec{F}=\frac{d\vec{p}}{dt}$ (§3) use kiya. Jo bachta hai woh exactly torque hai.

Ek force jo seedha origin ki taraf point kare (ek central force, jaise Sun ki taraf gravity) mein $\vec{r}\parallel \vec{F}$ hota hai, toh $\sin \theta =0$, toh $\vec{\tau }=0$. Yeh ek akela fact Kepler's equal-area law hai.

7. $\vec{L}=\sum \vec{r}\times \vec{p}$ se $L=I\omega$ tak — rigid-body shortcut build karna

Ek rigid spinning body ke liye (jaise skater), hum har mass element par $\vec{r}\times \vec{p}$ add karte hain. Yahan dikhata hai ki tidy shortcut kaise appear hota hai.

Build, ek ek honest step mein. Mass elements ko $m_i$ label karo axis se positions ${\vec{r}}_i$ par. Total angular momentum har particle ki sum hai: $\vec{L}={\sum }_i{\vec{r}}_i\times (m_i{\vec{v}}_i).$

Ek fixed axis ke baare mein spin karte rigid body ke liye, har particle radius $r_i$ ke circle mein move karta hai (axis se uski doori) speed $v_i=\omega r_i$ par, sideways direction mein. Toh axis ke along uska chota angular momentum size mein $m_i{r}_i{v}_i=m_i{r}_i(\omega r_i)=m_i{r}_i^2\omega$ hota hai. Unhe add karo: $L={\sum }_i{m}_i{r}_i^2\omega =\left(\underset{\text{ise }I\text{ kaho}}{\underbrace{\sum _i{m}_i{r}_i^2}}\right)\omega .$

Woh bracketed sum — shape aur mass akele ki ek property, kitni tez spin karta hai us se independent — ise hi hum moment of inertia $I$ naam dete hain. Yeh naturally $r^2$ se fall out hota hai jo cross product plus $v=\omega r$ produce karte hain.

Figure s04 dikhata hai kyun $r^2$ dominate karta hai: ek black spin axis ek near mass (chota $r$, black) aur ek red far mass (bada $r$) ke saath. Wahi mass axis se do guna door move karo toh $I$ mein uska contribution char guna ho jaata hai.

Yeh skater ko speed up kyun karne deta hai? $\vec{\tau }=0$ ke saath, $L=I\omega$ fixed rehta hai. Baahein andar karo → mass chote $r$ par jaata hai → $I=\sum m_i{r}_i^2$ shrink hota hai → $\omega$ ko product constant rakhne ke liye barhna padta hai. Koi bahari twist ki zaroorat nahi; $I$ ke andar ka $r^2$ kaam karta hai.

Prerequisite map

Ise top to bottom padho: ek clock aur origin set karo; $\vec{r}$, phir $\vec{v}$, phir $\vec{p}$ build karo; $\vec{F}$ aur Newton's law laao; $\vec{r}$ aur $\vec{p}$ ko cross product se $\vec{L}$ paane ke liye feed karo, aur $\vec{r}$ aur $\vec{F}$ ko $\vec{\tau }$ paane ke liye; $\vec{L}$ differentiate karo master equation tak pahunchne ke liye; aur ek rigid body par $\vec{r}\times \vec{p}$ sum karne se $L=I\omega$ milta hai.

Equipment checklist

Connections

• Yeh note Hinglish mein →
• Moment of inertia — §7 mein build kiya gaya $I$.
• Torque — §6 mein build kiya gaya twist.
• Newton's third law — kyun internal torques cancel hote hain.
• Central forces — classic $\vec{\tau }=0$ case.
• Conservation of linear momentum — straight-line cousin ($p=mv$).
• Kepler's laws — jahan zero torque equal areas ban jaata hai.
• Rotational kinetic energy — yahan build kiya gaya $I$ aur $\omega$ use karta hai.