Foundations — Rolling KE = ½mv² + ½Iω²

1.5.14 · D1 · Physics › Rotational Mechanics › Rolling KE = ½mv² + ½Iω²

Ye note parent topic ka "prerequisite unlock" hai. Parent jis bhi letter, squiggle, ya idea pe rely karta hai, wo sab yahan define kiya gaya hai — ek picture se shuru karke, kabhi kisi aisi symbol se nahi jo tune abhi tak dekhi nahi.

---

1. Mass — letter $m$ (aur $M$)

Picture:

Upar chalk ke wheel ko dekho. Uspe dots bichhe hue hain — har dot ek tukda $m_i$ hai. Parent note energy ko tukde-tukde jodta hai, isliye hume ek single tukde ko naam dena aana chahiye, tabhi hum unhe jod sakte hain.

Topic ko iske liye zaroorat kyun hai: kinetic energy ek particle se shuru hokar build hoti hai aur phir sum ki jaati hai. "Ek tukda $m_i$" ke idea ke bina kuch bhi sum karne ke liye nahi hai.

Recall

$m_i$ ke neeche chhota $i$ kya matlab rakhta hai? Ek label — "tukda number $i$". Ye hume bahut saare chunks mein se ek specific chunk ki baat karne deta hai.

---

2. Summation sign $\sum$

Picture: ye badi "add machine" hai. Tum ise ek rule dete ho (jaise $\frac{1}{2}{m}_i{v}_i^2$) aur ye wheel ke har dot ko visit karke total result deta hai.

Topic ko iske liye zaroorat kyun hai: poori derivation "$KE={\sum }_i(\text{tukde }i\text{ ki energy})$" hai. $\sum$ wo tool hai jo hazaron tiny energies ko ek number mein badalta hai.

---

3. Speed $v$, velocity $\vec{v}$, aur chhota arrow

Picture:

Ek moving dot se khincha hua blue arrow: uski length speed hai, uski direction wo taraf hai jidhar wo point karta hai. Speed bas arrow ki length hai, direction ignore karke.

Topic ko iske liye zaroorat kyun hai: key move ${\vec{v}}_i={\vec{v}}_{cm}+{\vec{u}}_i$ (true motion = slide + spin) ek arrow addition hai. Ye tabhi meaningful hota hai jab directions attached hon.

---

4. Centre of mass aur uski velocity ${\vec{v}}_{cm}$

Picture:

Wheel ke hub ko ek yellow dot se mark kiya gaya hai; yellow arrow ${\vec{v}}_{cm}$ hai, ground ke saath aage point kar raha hai. Har tukde ki motion is special point ke relative measure ki jaati hai.

Topic ko iske liye zaroorat kyun hai: König's theorem — poori derivation ka engine — literally "har velocity ko cm-motion + cm-se-dekhi-motion mein split karo" hai, aur ye clean split ${\sum }_i{m}_i{\vec{u}}_i=0$ pe rely karti hai.

---

5. Relative velocity ${\vec{u}}_i$ (cm se dekhi motion)

Picture: figure s03 mein, hub ke around curling pink arrows ${\vec{u}}_i$ hain — pure spin, koi forward drift nahi, kyunki hum pehle hi forward drift ${\vec{v}}_{cm}$ ghata chuke hain.

Topic ko iske liye zaroorat kyun hai: parent derivation exactly is line se shuru hoti hai. Isme har symbol ab build ho chuka hai.

---

6. Dot product $\vec{a}\cdot \vec{b}$

Topic ko iske liye zaroorat kyun hai: distributive rule use karke $({\vec{v}}_{cm}+{\vec{u}}_i)\cdot ({\vec{v}}_{cm}+{\vec{u}}_i)$ expand karne se exactly teen terms milte hain — ${\vec{v}}_{cm}\cdot {\vec{v}}_{cm}$ (slide), $2{\vec{v}}_{cm}\cdot {\vec{u}}_i$ (cross), ${\vec{u}}_i\cdot {\vec{u}}_i$ (spin). Distributive rule nahi toh expansion nahi.

---

7. Angular velocity $\omega$

Picture: figure s03 mein $\omega$ se labeled pink circular arrow spin rate dikhata hai. Hub se $r_i$ door ek tukde ki spin-speed $u_i=|{\vec{u}}_i|=\omega r_i$ hoti hai — door matlab tez, jaise clock ki tip apne middle se aage bhaagti hai. (Yahan $u_i$ Section 5 ke spin arrow ${\vec{u}}_i$ ki length hai.)

Topic ko iske liye zaroorat kyun hai: spin energy $\omega$ ke terms mein likhi jaati hai, aur — neeche rolling assumption add karne ke baad — $\omega$ slide speed se jod diya jaata hai. Poori build ke liye Angular Velocity and Angular Acceleration dekho.

Recall Hub se door ek tukda spin ke dauran tez kyun move karta hai?

Kyunki $u_i=\omega r_i$: same spin rate $\omega$ ek bada arc sweep karta hai jab $r_i$ bada hota hai.

---

8. Axis se doori $r_i$ aur radius $R$

Picture: figure s03 ek pink tukda dikhata hai jisme hub tak ek dashed line $r_i$ length ki hai, aur contact point tak seedha $R$ draw kiya gaya hai.

Topic ko iske liye zaroorat kyun hai: $r_i$ spin-energy sum ke andar aata hai (har tukda $\frac{1}{2}{m}_i{r}_i^2{\omega }^2$ contribute karta hai); $R$ uss $r_i$ ki special value hai ground touch karne wale rim tukde ke liye, aur ye spin ko slide se next section mein link karta hai.

---

9. Rolling without slipping — link $v_{cm}=\omega R$

Topic ko iske liye zaroorat kyun hai: ye single equation parent ko two-part KE ko ek speed se rewrite karne deta hai, jo incline problems ko solve karne laayak banata hai.

---

10. Moment of inertia $I_{cm}$ aur shape number $\beta$

---

Prerequisite map

Har foundation aage feed karta hai: tukde + sum $M$ aur $I$ build karte hain; cm $\sum m_i{\vec{u}}_i=0$ deta hai; ye plus dot-product distributive rule König split build karte hain; no-slip $v_{cm}=\omega R$ add karta hai aur shape $\beta$ add karta hai; saath milke topic formula banate hain, jise height + gravity incline pe power deti hai.

---

Equipment checklist