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

1.5.10 · Physics › Rotational Mechanics

1. Angular momentum KYA hai?

Cross product KYU? Kyunki sirf wahi part of motion jo origin ke "around" jaata hai, count karta hai. Seedha $O$ ki taraf ya $O$ se door ki motion zero turning produce karti hai — aur cross product automatically parallel component ko khatam kar deta hai.

$|\vec{L}|=rp\sin \theta =p\cdot (r\sin \theta )=p\cdot r_{\perp }$

jahan $r_{\perp }=r\sin \theta$ perpendicular distance (yani "lever arm") hai $O$ se motion ki line tak.

2. $L=I\omega$ kaise $\vec{L}=\vec{r}\times \vec{p}$ se nikalta hai (poori derivation)

Setup. Ek rigid body lo jo ek fixed axis (maano $z$-axis) ke around rotate kar raha hai. Particle $i$ ko lo jo axis se $r_i$ perpendicular distance par hai. Kyunki yeh rigid hai, har particle ki same angular speed $\omega$ hai.

Step 1 — particle $i$ ki speed. Yeh $r_i$ radius ke circle mein travel karta hai, isliye $v_i=\omega r_i.$ Yeh step kyun? Circular motion ke liye, tangential speed $=$ radius $\times$ angular speed.

Step 2 — uska momentum tangential hai, $p_i=m_i{v}_i=m_i\omega r_i$. Velocity axis se radius vector ke perpendicular hai, isliye $\theta =90^{\circ }$ aur $\sin \theta =1$.

Step 3 — axis ke baare mein uska angular momentum. $L_{i,z}=r_i{p}_i\sin 90^{\circ }=r_i(m_i\omega r_i)=m_i{r}_i^2\omega .$ Yeh step kyun? Hum axis ke along component use karte hain, jo fixed-axis spin ke liye physically conserved/relevant hota hai.

Step 4 — sabhi particles par sum karo (axis component add up hota hai): $L_z={\sum }_i{m}_i{r}_i^2\omega =\left({\sum }_i{m}_i{r}_i^2\right)\omega .$

Step 5 — bracket ko moment of inertia $I={\sum }_i{m}_i{r}_i^2$ ke roop mein pehchano:

3. Torque se connection (rotational Newton's law)

$\vec{L}=\vec{r}\times \vec{p}$ ko differentiate karo: $\frac{d\vec{L}}{dt}=\underset{\vec{v}\times m\vec{v}=0}{\underbrace{\frac{d\vec{r}}{dt}\times \vec{p}}}+\vec{r}\times \frac{d\vec{p}}{dt}=\vec{r}\times \vec{F}=\vec{\tau }.$

4. Worked examples

extracts karta hai perpendicular part ko. $r_{\perp }=r\sin \theta$ (lever arm) ya $p_{\perp }=p\sin \theta$ use karo — kabhi bhi raw product $rp$ nahi.

5. Active recall

Flashcards

Connections

• Moment of Inertia — woh $\sum m_i{r}_i^2$ jo $\vec{r}\times \vec{p}$ ko $I\omega$ mein badalta hai
• Torque — $\vec{\tau }=\vec{r}\times \vec{F}=d\vec{L}/dt$
• Conservation of Angular Momentum — zero-torque ka consequence
• Cross Product — isliye direction aur $\sin \theta$ aate hain
• Linear Momentum — translational analogue $\vec{p}=m\vec{v}$
• Rotational Kinetic Energy — $KE=\frac{1}{2}I{\omega }^2=L^2/2I$
• Kepler's Second Law — equal areas = constant $L$ orbits ke liye

Concept Map