Torque = dL - dt

1.5.11 · Physics › Rotational Mechanics

YEH HAI KYA?

Key word hai same point — $\vec{\tau }$ aur $\vec{L}$ dono ek hi origin ke baare mein measure hone chahiye (aur woh origin inertial ya non-accelerating hona chahiye, ya phir centre of mass).

YEH SACH KYUN HAI? (Scratch se derive karo)

Hum Newton's second law se shuru karte hain aur rotational law build karte hain — koi memorising nahi.

Step 1 — Ek particle ke angular momentum se shuru karo. $\vec{L}=\vec{r}\times \vec{p}$ Yeh step kyun? Yeh definition hai; sab kuch ise differentiate karne se nikalega.

Step 2 — Time ke saath differentiate karo (cross products ke liye product rule): $\frac{d\vec{L}}{dt}=\frac{d\vec{r}}{dt}\times \vec{p}+\vec{r}\times \frac{d\vec{p}}{dt}$ Yeh step kyun? $\vec{L}$ time ke through $\vec{r}$ aur $\vec{p}$ dono se depend karta hai, isliye dono differentiate hote hain.

Step 3 — Pehla term khatam karo. $\frac{d\vec{r}}{dt}\times \vec{p}=\vec{v}\times (m\vec{v})=m(\vec{v}\times \vec{v})=0$ Yeh step kyun? Kisi bhi vector ka cross product khud apne aap se zero hota hai ($\sin 0°=0$). Toh velocity-with-momentum kuch contribute nahi karta.

Step 4 — Doosre term par Newton's law use karo. $\frac{d\vec{p}}{dt}=\vec{F}\Rightarrow \vec{r}\times \frac{d\vec{p}}{dt}=\vec{r}\times \vec{F}=\vec{\tau }$ Yeh step kyun? Newton's 2nd law deta hai $\vec{F}=d\vec{p}/dt$; aur $\vec{r}\times \vec{F}$ torque ki definition hai.

Result: $\frac{d\vec{L}}{dt}=\vec{\tau }$

EK RIGID BODY TAK KAISE SCALE KARTA HAI? (Particles ka system)

Bahut saare particles ke liye, sab par sum karo: ${\vec{L}}_{\text{tot}}=\sum {\vec{L}}_i$, toh $\frac{d{\vec{L}}_{\text{tot}}}{dt}={\sum }_i{\vec{\tau }}_i={\vec{\tau }}_{\text{ext}}+\underset{=0}{\underbrace{{\vec{\tau }}_{\text{int}}}}$

Ek rigid body ke liye jo fixed axis ke around spin kar rahi hai, $\vec{L}=I\vec{\omega }$ ke saath (jahan $I$ us axis ke baare mein moment of inertia hai): ${\vec{\tau }}_{\text{ext}}=\frac{d(I\vec{\omega })}{dt}=I\frac{d\vec{\omega }}{dt}=I\vec{\alpha }(\text{if }I\text{ constant})$ Yahi famous $\tau =I\alpha$ hai — rotational $F=ma$. Lekin $\tau =dL/dt$ zyada general hai: yeh tab bhi hold karta hai jab $I$ change ho (jaise ek spinning skater apne arms andar kheench le).

Sabse important consequence

Worked Examples

Common Mistakes (Steel-manned)

Active Recall — Flashcards

Connections

• Angular Momentum — woh quantity $\vec{L}=\vec{r}\times \vec{p}$ jis par yeh law govern karta hai.
• Newton's Second Law — linear parent $\vec{F}=d\vec{p}/dt$ jisse humne derive kiya.
• Moment of Inertia — $L=I\omega$ mein $I$.
• Conservation of Angular Momentum — $\tau =0$ wala special case.
• Cross Product — woh maths jo explain karta hai kyun sirf perpendicular force twist karta hai.
• Torque and Angular Acceleration (tau = I alpha) — constant-$I$ wala special case.

Concept Map