Foundations — Torque = dL - dt

1.5.11 · D1 · Physics › Rotational Mechanics › Torque = dL - dt

Pehle aapko us one-line law par trust karne ke liye har symbol earn karna hoga. Neeche, har piece ko zero se build kiya gaya hai — pehle plain words mein, phir picture, phir yeh topic is cheez ke bina kyon nahi chal sakta. Upar se neeche padho: har block sirf wahi cheezein use karta hai jo uske upar define ho chuki hain.

Yeh page parent topic ka ground floor hai. Yahan kuch bhi assume nahi kiya gaya — hum ek page par ek arrow se shuru karte hain.

1. Ek arrow: vector kya hota hai

Ek plain number (jaise "5 kg") ek scalar hai — sirf size, koi direction nahi. Lekin "5 m/s east ki taraf move karna" ko bhi direction chahiye, isliye yeh ek vector hai.

Neeche ki figure dekho. Left mein, blue arrow ki length size hai aur yeh kahan point karta hai woh direction hai — woh single arrow ek saath do pieces of information carry karta hai. Right mein, "5 kg" labelled yellow dot ek scalar hai: ek bare number jiske paas koi direction nahi hai.

2. Position vector $\vec{r}$ — "main kahan hoon, kahan se measure karke?"

Key subtlety: $\vec{r}$ ka tab tak koi matlab nahi jab tak aap $O$ naam nahi lete. Origin badle, aur $\vec{r}$ badal jaayega. Isliye parent note baar baar "usi point ke baare mein" chilaata rehta hai — har rotational quantity ek chosen $O$ ke relative measure ki jaati hai.

Neeche ki figure mein, blue arrow $\vec{r}$ yellow origin $O$ se green object tak jaati hai. Ab red dashed arrow $\vec{r}{}^{\prime }$ dekho: yeh same object tak pahuncha hai lekin ek alag origin $O^{\prime }$ se shuru hota hai — isliye iski alag length aur direction hai. Same object, alag $\vec{r}$, sirf isliye kyunki humne reference point badal diya.

3. Velocity $\vec{v}$ aur mass $m$

4. Linear momentum $\vec{p}=m\vec{v}$

5. Force $\vec{F}$, rate-of-change symbol, aur Newton's second law

Hume vector version chahiye kyunki core law $\vec{L}$ aur $\vec{p}$ differentiate karta hai, jo arrows hain, plain numbers nahi.

Full build ke liye Newton's Second Law dekho। Parent topic is line se shuru karta hai aur use rotate karta hai, isliye yeh ek hard prerequisite hai।

6. Cross product $\vec{a}\times \vec{b}$ — "twisting" parts ke liye ek machine

Yeh ek genuinely naya tool hai, isliye hum ise carefully build karte hain। Zyada ke liye Cross Product dekho।

$\sin \theta$ kyun, na ki $\cos \theta$? Kyunki $\sin \theta$ measure karta hai ki do arrows kitne perpendicular hain:

• Agar arrows same taraf point karein ($\theta =0°$): $\sin 0°=0$ → cross product zero hai।
• Agar arrows right angles par hain ($\theta =90°$): $\sin 90°=1$ → cross product biggest hai।
• Agar arrows opposite taraf point karein ($\theta =180°$): $\sin 180°=0$ → zero phir से।

Neeche ki figure exactly yahi dikhati hai: same do arrows ke teen panels $0°$, $90°$, $180°$ par, har ek ke neeche $\sin \theta$ print hai — $0$, phir $1$, phir phir $0$। Dekho kaise "twisting power" tab peak karti hai jab arrows ek dusre se square hote hain।

7. Angular momentum $\vec{L}=\vec{r}\times \vec{p}$

8. Torque $\vec{\tau }=\vec{r}\times \vec{F}$

Notice karo perfect parallel: $\vec{L}=\vec{r}\times \vec{p}$ aur $\vec{\tau }=\vec{r}\times \vec{F}$। Kyunki $\vec{F}=d\vec{p}/dt$ hai, torque ka $\vec{L}$ se wahi relation hai jo force ka $\vec{p}$ se। Yahi parallel poora topic hai।

9. Radians, moment of inertia $I$, angular velocity $\vec{\omega }$, angular acceleration $\vec{\alpha }$

Ek fixed axis ke baare mein rigid body ke liye, yeh spin ko neatly $\vec{L}=I\vec{\omega }$ ke roop mein package karte hain, jo $\vec{\tau }=I\vec{\alpha }$ lead karta hai (dekho tau = I alpha)। Rotational dictionary:

Foundations topic ko kaise feed karti hain

Sab kuch single boxed law mein funnel hota hai, phir rigid-body special case aur Conservation of Angular Momentum (kya hota hai jab twist zero ho) mein branch out hota hai।

Equipment checklist

Right side cover karo; kya aap aage badhne se pehle har ek answer de sakte ho?