1.5.13
Jab ek wheel bina slip ke roll karta hai , tab wheel ka woh point jo ground ko touch kar raha hai — ek instant ke liye — bilkul still khada hota hai . Woh slide nahi karta. Yahi ek fact force karta hai ek relationship between kitni fast center move karta hai (v ω v = R ω
Definition Rolling without slippingRadius R contact point ka velocity surface ke relative zero ho . Koi skidding nahi hoti (jaise ice par car) aur koi spinning-in-place nahi hoti (jaise mud mein phansa wheel). Ground contact point ko "grab" kar leta hai.
Do motions ek saath hoti hain:
Translation — center of mass aage speed v c m Rotation — body apne center ke baare mein angular speed ω
No-slip condition hi hai jo in dono otherwise independent motions ko link karti hai.
Agar aap ek wheel ko angle θ distance travel karta hai woh uski arc length ke barabar honi chahiye jo usne unroll ki.
Step 1 — Unrolling ki Geometry.
Wheel ko ek chhote angle θ s = R θ
Yeh step kyun? Arc length = × s
Step 2 — Center utni hi distance move karta hai.
x c m = s = R θ
Yeh step kyun? "No slip" ka matlab literally hai rim length unrolled = ground distance covered — kuch bhi skidding mein nahi jaata.
Step 3 — Time ke respect mein differentiate karo.
d t d x c m = R d t d θ ⇒ v c m = R ω
Yeh step kyun? d t d x c m = v c m d t d θ = ω R
Step 4 — Acceleration ke liye ek baar aur differentiate karo.
d t d v c m = R d t d ω ⇒ a c m = R α
Intuition Wheel par kisi bhi point ki velocityHar point ki velocity = (center ki velocity) + (center ke baare mein rotational velocity).
v p o in t = v c m + ω × r
Diagram padho (jab v c m = R ω
Point
Translation
Rotation
Total speed
Bottom (contact)+ v − R ω v − R ω = 0
Center + v 0 v
Top + v + R ω v + R ω = 2 v
Intuition Sabse bada faydaContact point instantaneously rest par hota hai , aur top point 2 v — center ki speed se double! Isliye ek rolling wheel ko contact point ke baare mein pure rotation ki tarah treat kiya ja sakta hai (instantaneous axis of rotation ).
Worked example Example 1 — Bicycle wheelRadius R = 0.35 m v = 5 m/s ω
Step 1: Constraint use karo v = R ω ⇒ ω = v / R = 5/0.35 = 14.3 rad/s Kyun? No slipping ⇒ v ω
Step 2: Top point ki speed = 2 v = 10 m/s Kyun? Top = translation v R ω = v
Worked example Example 2 — Accelerating car wheelEk car wheel (R = 0.30 m a = 2 m/s 2
Step 1: Differentiated constraint: a = R α Step 2: α = a / R = 2/0.30 = 6.67 rad/s 2 Kyun? Agar slip hoti, toh a = R α
Worked example Example 3 — Forecast-then-VerifyForecast: Ek ball v Naive guess: "v Verify: Bottom point: v c m − R ω = v − v = 0 Zero. Contact point momentarily frozen hai. Tumhara forecast galat tha — yahi toh no-slip ka poora point hai.
Common mistake "Contact point wheel ke saath move karta hai, isliye uski speedv Kyun sahi lagta hai: Contact point wheel ka part hai, jo aage move kar raha hai — surely woh bhi move karta hoga. Flaw yeh hai: us instant par, rotation bottom point ko exactly R ω = v backward drag karta hai, forward v Fix: Total bottom-point velocity = v − R ω = 0 axis hai, koi moving point nahi.
v = R ω Kyun sahi lagta hai: Itna clean formula hai. Flaw yeh hai: agar wheel slip kare (car ice par apne wheels spin kare, ya ek skidding ball), toh v = R ω ω v ≈ 0 Fix: Equation ek constraint hai jo sirf rolling without slipping ke liye hold karta hai .
Common mistake "Friction negative work karta hai aur ek perfectly rolling ball ko slow kar deta hai."Kyun sahi lagta hai: Friction usually energy dissipate karta hai. Flaw yeh hai: pure rolling mein contact point move nahi karta, isliye static friction zero work karta hai (W = F ⋅ d d = 0 Fix: Rolling without slipping mechanical energy conserve karta hai.
Recall Feynman: ek 12-saal ke bachche ko explain karoEk toy car socho. Tyre par chalk se ek dot lagao. Jab car roll karti hai, chalk dot sirf ek tiny instant ke liye road ko touch karta hai — aur us instant mein woh still khada hota hai , jaise ek paon zameen par plant hota hai jab tum chalte ho. Tumhara paon slide nahi karta; woh jagah par ruka rehta hai jabki baaki tum uske upar aage swing karte ho. Tyre bhi same karta hai: bottom still rehta hai, middle normal speed se move karta hai, aur bilkul top do-guni speed se aage zoom karta hai! Woh "bottom still rehta hai" rule hi reason hai ki car ki speed (v ω v = R ω
"Bottom Zero, Center Vee, Top is Twice" — 0 , v , 2 v R oll = R adius inhe link karta hai: v = R ω R
"Rolling without slipping" physically contact point par kya require karta hai? Contact point ka velocity ground ke relative zero hota hai (woh slide nahi karta).
Rolling constraint ke velocity, acceleration, aur displacement forms batao. v c m = R ω a c m = R α x c m = R θ
Arc length se v = R ω No-slip ka matlab distance moved = arc unrolled, x = R θ v = R ω
v 2 v v R ω = v
Bottom (contact) point ki speed? 0
Pure rolling mein static friction koi work kyun nahi karta? Contact point move nahi karta (
d = 0 ), isliye
W = F ⋅ d = 0 .
v = R ω Jab body slip/skid kare (jaise ice par spinning tyre): tab v = R ω
Ek wheel R = 0.5 ω = 4 v v = R ω = 2
v_point = v_cm + omega x r
