Conservation of angular momentum — conditions

1.5.12 · Physics › Rotational Mechanics

KYA conserve ho raha hai?

YE KYUN conserve hota hai? (First principles se derivation)

Hum Newton's second law ke rotational analogue se shuru karte hain aur condition derive karte hain.

Step 1 — $L$ define karo aur differentiate karo. $L = r \times p$ Ye step kyun? Conservation ka matlab hai $\frac{d L}{d t} = 0$ , isliye hume time derivative compute karni hogi.

Step 2 — Cross products ke liye product rule apply karo. $\frac{d L}{d t} = \frac{d r}{d t} \times p + r \times \frac{d p}{d t}$ Ye step kyun? $L$ do time-dependent vectors ka product hai; dono change ho sakte hain.

Step 3 — Pehla term khatam karo. $\frac{d r}{d t} \times p = v \times (m v) = m (v \times v) = 0$ Ye step kyun? Kisi bhi vector ka khud ke saath cross product zero hota hai ( $v ∥ v$ ).

Step 4 — Torque identify karo. $\frac{d p}{d t} = F \Rightarrow r \times \frac{d p}{d t} = r \times F = τ_{n e t}$

Step 5 — Master equation. $\frac{d L}{d t} = τ_{e x t, n e t}$

Sirf EXTERNAL torque kyun? Particles ke ek system mein, internal forces Newton's-third-law pairs mein aate hain jo unhe join karne wali line ke along act karte hain. Har pair torque contribute karta hai $r_{1} \times f + r_{2} \times (- f) = (r_{1} - r_{2}) \times f$ . Kyunki $f$ along hai $(r_{1} - r_{2})$ ke (central forces), ye cross product $= 0$ hai. Toh internal torques cancel ho jaate hain aur sirf external torque bachta hai.

Conditions ki poori list

Figure — Conservation of angular momentum — conditions

Worked Examples

Common Mistakes

Recall Feynman: 12-saal ke bachche ko explain karo

Socho tum ek kursi pe spin kar rahe ho aur haath bahar farke bhaari kitaabein pakde ho. Kitaabon ko apne seene ke paas kheencho aur tum achanak tezi se ghoomne lago — kisine tumhe push nahi kiya! Woh "spin ki matra" (angular momentum) waisi hi rehna chahti hai jab tak baahir se kuch tumhe twist na kare. Friction ya koi kursi pakad le toh woh ek baahri twist hogi jo ise badal degi. Jab tak baahir se kuch tumhe twist na kare, tumhari same spin rehti hai: arms bahar = slow aur wide, arms andar = fast aur tight.

Active Recall

Angular momentum conserved hone ki exact condition kya hai?

System ke chosen axis ke baare mein net external torque zero ho.

Torque aur angular momentum ko kaunsi equation link karti hai?

\frac{d L}{d t} = τ_{e x t, n e t}

Internal forces total

L

kyun nahi badlti?

Ye collinear Newton's-third-law pairs mein hoti hain, toh unke torques cancel ho jaate hain:

(r_{1} - r_{2}) \times f = 0

Central force jaise gravity ke liye

L

kyun conserved hai?

Force origin se guzarti hai, toh

τ = r \times F = 0

kyunki

r ∥ F

hai.

Skater arms andar kheenchti hai:

ω

aur KE ka kya hota hai?

ω

badhta hai (kyunki

I ω

fixed hai); KE badhti hai kyunki muscles kaam karti hain — KE conserved NAHI hai.

Kya

L

ek axis ke baare mein conserved ho sakta hai lekin doosre ke baare mein nahi?

Haan — sirf un axes ke baare mein jahan external torque component zero ho.

Kya

L

ke conservation ke liye origin choose karna irrelevant hai?

Nahi —

L

aur

τ

dono origin pe depend karte hain; ek chuno jahan

τ_{e x t} = 0

ho.

Kaunsa Kepler law angular momentum conservation ka direct consequence hai?

Kepler's 2nd law (equal times mein equal areas swept hote hain).

Connections

Moment of inertia — $L = I ω$ mein woh $I$ .
Torque — iski absence hi conservation condition hai.
Newton's third law — isliye internal torques cancel hote hain.
Kepler's laws — constant $L$ se equal-area law.
Conservation of linear momentum — exact translational analogue.
Central forces — classic zero-torque scenario.
Rotational kinetic energy — isliye $L$ conserved ≠ KE conserved.