Visual walkthrough — Rolling without slipping — v = Rω condition

1.5.13 · D2 · Physics › Rotational Mechanics › Rolling without slipping — v = Rω condition

Kuch bhi shuru karne se pehle, teen seedhi baatein jo hum baar baar point karenge:

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Step 1 — Radian aakhir hota kya hai (taaki $s=R\theta$ ka matlab samjhe)

KYA. Isse pehle ki hum kahe "wheel ne angle $\theta$ se ghuma", humein agree karna hoga ki angle measure kaise karte hain. Hum radian use karte hain.

YEH KYUN, degrees nahi? Degrees arbitrary hain — kisine ek baar 360 choose kar liya tha. Radian isliye choose kiya gaya hai taaki angle aur arc-length ek hi measuring system mein hon: yahi woh choice hai jo rolling arithmetic ko clean banati hai. Yeh ek line mein matter karega.

PICTURE. Figure dekho. Wheel ka center lo. Ek radius ko kisi opening se sweep karo. Rim ke saath jo arc trace hoti hai (orange) uski ek length hoti hai. Radian define hota hai:

$\underset{\text{angle, radians mein}}{\underbrace{\theta }}=\frac{\stackrel{\text{rim ke saath arc length}}{\overbrace{s}}}{\underset{\text{radius}}{\underbrace{R}}}$

Rearrange karo, yeh poori derivation ka beej hai:

$\underset{\text{sweep mein rim ki length}}{\underbrace{s}}=\underset{\text{radius}}{\underbrace{R}}\underset{\text{sweep kiya gaya angle}}{\underbrace{\theta }}$

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Step 2 — Wheel ko road pe unroll karo

KYA. Ab wheel ko roll karne do. Jab woh angle $\theta$ se ghoomti hai, rim ko peel off hote aur ground pe flat press hote dekho, jaise tape unspool hoti hai.

KYUN. "Rolling without slipping" ka ek precise matlab hai: rim road pe slide nahi karti. Toh rim ka har chhota sa tukda road pe lay down ho jaata hai, ek baar use touch karta hai, aur kabhi drag nahi hota. Koi rim waste nahi hoti, koi skip nahi hoti.

PICTURE. Figure mein, Step 1 ki orange arc ko road pe ek orange straight segment mein "unroll" kiya gaya hai. Kyunki kuch slip nahi hua, woh road segment arc ke utni hi lambi hai:

$\underset{\text{green segment}}{\underbrace{\text{road cover ki gayi length}}}=\underset{\text{arc jo unroll hui}}{\underbrace{s}}=R\theta$

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Step 3 — Center bilkul utni hi door jaata hai

KYA. Wheel ke center (axle) ko track karo. Poochho: jab wheel $\theta$ se ghoomti hai, center kitna aage gaya?

KYUN. Center contact point ke bilkul upar baitha hai aur body ke saath saath chalta hai. Jab rim $R\theta$ length ki road lay down karti hai, poora wheel — center bhi — utni hi road length aage badh jaata hai. Kuch bhi sliding mein nahi gaya, isliye center ka travel aur road covered mein koi farq nahi.

PICTURE. Figure center ki start (gray dot) aur end (blue dot) mark karta hai. Unke beech ka blue horizontal arrow, ise $x_{cm}$ bulao ("center of mass ka $x$"), green road segment ke saath tip-to-tail line up karta hai:

$\underset{\begin{array}{c}\text{center ne aage badhi}\\ \text{doori}\end{array}}{\underbrace{x_{cm}}}=\underset{\text{road covered = arc unrolled}}{\underbrace{R\theta }}$

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Step 4 — Distances ko speeds mein badlo (derivative kyun lete hain)

KYA. Humhare paas distances ka relation hai ($x_{cm}=R\theta$). Humein speeds ka relation chahiye. Toh hum poochhte hain ki har side time ke saath kaise change hoti hai — woh operation derivative hai.

DERIVATIVE KI JAGAH KOI AUR TOOL KYUN NAHI? "Speed" by definition ek second mein ek distance kitni tezi se badhti hai yeh hai. Joh tool "ek distance" ko "woh distance kitni tezi se badhti hai" mein convert karta hai woh hai derivative $\frac{d}{dt}$ ("time $t$ ke saath change ki rate"). "Kitni tezi se?" ka jawab koi doosra tool nahi deta.

PICTURE. Figure do snapshots dikhata hai jo ek tiny time $dt$ ke antar pe hain. Us waqt ke us patthin mein center ek tiny $d{x}_{cm}$ se aage badha aur wheel ek tiny $d\theta$ se ghoomti. $x_{cm}=R\theta$ ke dono taraf $\frac{d}{dt}$ apply karo (aur $R$ constant hai, isliye woh sirf bahar baitha rehta hai):

$\underset{\text{center kitni tezi se chalta hai}}{\underbrace{\frac{d{x}_{cm}}{dt}}}=R\underset{\text{wheel kitni tezi se ghoomti hai}}{\underbrace{\frac{d\theta }{dt}}}$

Ab hum in do rates ko naam dete hain:

• $\frac{d{x}_{cm}}{dt}\equiv v_{cm}$ — center ki speed, metres per second.
• $\frac{d\theta }{dt}\equiv \omega$ (omega) — angular speed, radians per second (ek second mein kitne radians ka spin; dekho Angular velocity and angular acceleration).

$\boxed{\underset{\text{center ki speed}}{\underbrace{v_{cm}}}=R\underset{\text{spin rate}}{\underbrace{\omega }}}$

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Step 5 — Ek aur derivative: accelerations

KYA. Speeds bhi change ho sakti hain. Poochho ki speeds kitni tezi se badhti hain — ek baar aur differentiate karo.

KYUN. Same logic dobara: ek speed kitni tezi se badhti hai woh acceleration hai. Agar wheel speed up kar rahi hai (Rolling down an incline, ek accelerating car), toh humein constraint ka accelerating version chahiye.

PICTURE. Figure teen linked quantities ko ek ladder ki tarah stack karta hai: distance $\to$ speed $\to$ acceleration, har rung $\frac{d}{dt}$ button dabane se milti hai. $v_{cm}=R\omega$ differentiate karo:

$\underset{\equiv a_{cm}\text{(linear accel.)}}{\underbrace{\frac{d{v}_{cm}}{dt}}}=R\underset{\equiv \alpha \text{(angular accel.)}}{\underbrace{\frac{d\omega }{dt}}}$

$\boxed{a_{cm}=R\alpha }$

Yahan $\alpha$ (alpha) angular acceleration hai — $\omega$ khud kitni tezi se badhta hai, ${\text{rad/s}}^2$ mein.

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Step 6 — Iski use karo: bottom, center, aur top ki velocity

KYA. Wheel ke teen khaas points ki actual speed nikalne ke liye $v_{cm}=R\omega$ use karo. Har point ek saath do kaam karta hai: woh center ke saath aage bhi jaata hai aur center ke irgird ghoomta bhi hai.

KYUN. Kisi bhi point ki velocity in dono motions ka sum hai:

$\underset{\text{true velocity}}{\underbrace{{\vec{v}}_{\text{point}}}}=\underset{\text{sab v se aage jaate hain}}{\underbrace{{\vec{v}}_{cm}}}+\underset{\text{spin part, rim pe size }R\omega }{\underbrace{\vec{\omega }\times \vec{r}}}$

Spin part ki size rim pe $R\omega$ hai, lekin uski direction depend karta hai ki tum kahan ho: bottom pe woh backward point karti hai, top pe forward.

PICTURE. Figure mein, blue arrows shared forward ride hain ($v$); orange arrows spin contribution hain ($R\omega =v$). Inhe tip-to-tail add karo:

• Bottom (contact): forward $v$ + backward $R\omega$. Kyunki $R\omega =v$ hai, yeh cancel ho jaate hain → speed $=0$.
• Center: forward $v$ + spin $0$ (tum axis pe ho) → speed $=v$.
• Top: forward $v$ + forward $R\omega =v$ → speed $=2v$.

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Step 7 — Degenerate cases (agar sach mein slip ho toh?)

KYA. Hum un scenarios ko cover karna zaroori hai jahan $v=R\omega$ toot jaata hai, aur do boundary cases. Yeh equation ek constraint hai, nature ka koi law nahi — yeh sirf ek situation mein hold karta hai.

KYUN. Jo reader sirf saaf case dekhta hai woh formula galat jagah use kar dega. Yahan saare chaar regimes hain, har ek figure mein dikhaya gaya hai:

Case	Kya hota hai	Relationship
Pure rolling	rim flat lagti hai, koi slide nahi	$v=R\omega$
Skidding (car on ice, wheels barely turning)	wheel aage drag hoti hai, spin se zyada	$v>R\omega$
Spinning out (burnout, tyre jagah pe ghoomta hai)	tezi se ghoomta hai par body barely chalti hai	$v<R\omega$
$\omega =0$ (sliding block)	bilkul spin nahi, sirf slide ho raha hai	$v\ne R\omega$ jab tak $v=0$ bhi na ho

PICTURE. Figure contact point ka net velocity arrow (red) har case mein dikhata hai. Sirf pure rolling mein woh red arrow zero-length hai. Skidding mein woh aage point karta hai (rim road pe forward slide karti hai); burnout mein woh peeche point karta hai (rim car ke neeche backward slide karti hai).

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Ek-picture summary

Yahan poori derivation ek single frame mein compress ki gayi hai: wheel $\theta$ se ghoomta hai, uski orange arc $R\theta$ green road mein unroll hoti hai, center same blue distance $x_{cm}=R\theta$ travel karta hai, aur $\frac{d}{dt}$ ek baar press karne se $v=R\omega$ milta hai, do baar se $a=R\alpha$ — contact point frozen hai aur top $2v$ pe race kar raha hai.

Recall Feynman: poora walkthrough seedhe alfazon mein

Socho ki tum ek tape roll ko floor pe unroll kar rahe ho. Tum roll ko thoda sa ghumate ho, aur tape ki ek strip ground pe flat press ho jaati hai — aur kyunki tape skid nahi karti, floor pe jo strip hai woh bilkul utni hi lambi hai jitni roll se aayi thi. Woh length hai $R\theta$: radius times woh angle jo tumne ghuma diya. Ab yahan trick hai — roll ka center aage utni hi door gaya hai, kyunki poori cheez buss tape ki utni hi aage khiski jitni lay down hui. Toh "center kitna aage gaya" aur "kitna ghuma times radius" ek hi doori hain. Poochho ki un mein se har ek ek second mein kitni tezi se badhta hai — wahi derivative hai — aur nikalta hai "center ki speed = radius times spin rate", $v=R\omega$. Ek baar aur poochho accelerations ke liye aur milta hai $a=R\alpha$. Aur upar se cherry: bilkul neeche, tape us pal floor se chipki hoti hai, toh contact point bilkul still khada rehta hai, jabki roll ka top — aage bhi ja raha hai aur aage ghoom bhi raha hai — center ki speed se do guna tezi se zoom karta hai. Yahi poori kahani hai: koi sliding nahi, toh arc road ke barabar hoti hai, aur baaki sab sirf yeh dekhna hai ki woh doori kitni tezi se badhti hai.

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Connections

• Parent topic: the v = Rω condition
• Instantaneous axis of rotation — Step 6 se frozen contact point
• Angular velocity and angular acceleration — jahan se $\omega$ aur $\alpha$ aate hain (Steps 4–5)
• Static vs kinetic friction — kyun frozen contact point static friction use karta hai
• Rolling down an incline — jahan $a_{cm}=R\alpha$ (Step 5) kaam aata hai
• Kinetic energy of rolling bodies — translation + rotation combine karne ke liye $v=R\omega$ use karta hai
• Moment of inertia — rolling dynamics ka agla ingredient

Concept Map