Visual walkthrough — Angular displacement θ, angular velocity ω, angular acceleration α

1.5.2 · D2 · Physics › Rotational Mechanics › Angular displacement θ, angular velocity ω, angular accelera

Hum ye, is order mein, earn karenge:

1. ek angle asal mein hota kya hai jab hum ek clean measuring stick (radian) par zor dete hain,
2. kaise ek turning angle ek point ko ek arc par kheenchta hai,
3. kaise "turning per second" ($\omega$) "metres per second" ($v$) ban jaata hai,
4. kaise "turning-speed ka badalna" ($\alpha$) "path ke saath speed up hona" ($a_t$) ban jaata hai,
5. edge cases: centre point, zero spin, aur reversing spin.

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Step 1 — Ek dot paint karo aur sahi sawaal poochho

YE SETUP KYO. Hum dot ko plain $x,y$ metres mein track karne ki koshish kar sakte the, lekin wo "kitna bahar" aur "kitna around" ko ek ugly changing pair mein mila deta. Rotation mein disc par har dot ke liye ek cheez common hoti hai: wo sab har second ghoomne ki utni hi maatra se swing karte hain. Toh hum ek aisa number invent karte hain jo sirf "around" part capture karta hai aur "out" part chhod deta hai. Woh number hai angle.

YE KAISA DIKHTA HAI. Figure mein fixed length cyan spoke hai jiska length $r$ hai. Amber wedge "ghoomne ki maatra" hai — woh cheez jo hum aage properly measure karne wale hain.

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Step 2 — Angle ke liye ek fair measuring stick chunna: radian

Term by term, bilkul wahan jo hain:

• $s$ — arc length: curved distance jahan dot actually rim par sawaar hui (metres).
• $r$ — spoke: centre se dot tak seedhi doori (metres).
• $\theta$ — unka ratio. Dono metres hain, toh metres cancel ho jaate hain: $\theta$ ek pure number hai. Hum us unit ka naam radian rakhte hain.

RATIO KYO, AUR DEGREES KYO NAHI? Degree ek arbitrary human choice hai — ek full circle ko 360 slices mein kaato "kyunki ancient astronomers ko 360 pasand tha". Geometry mein kuch bhi 360 ki parwah nahi karta. Lekin ratio $s/r$ picture se khud hi humpar thopa jaata hai: woh poochhtaa hai "hum ne kitne spoke-lengths ki curved edge sweep ki?" Jab arc exactly ek spoke ke barabar ho ($s=r$), angle exactly $1$ radian hota hai. Koi conversion factor kabhi paida hi nahi hota. Yehi poora kaaran hai kyun radians baad ke formulas ko clean banate hain.

YE KAISA DIKHTA HAI. Figure mein, amber arc aur cyan spoke same length mein drawn hain — yeh seedha "1 radian" angle ko picture karta hai.

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Step 3 — Circle ka ek poora chakkar definition ko check karta hai

YE KYO KARO. Yeh ek sanity check hai: ek definition tabhi bharosemand hoti hai jab ek jaana-pehchana case sahi nikle. Ek full turn chahiye ki ek clean number de.

${\theta }_{\text{full}}=\frac{s}{r}=\frac{2\pi r}{r}=2\pi \text{ rad}$

• $2\pi r$ — circumference (ek full loop ke liye arc),
• $r$ — upar aur neeche cancel ho jaata hai,
• $2\pi$ — answer: ek full turn $2\pi$ radians ($\approx 6.283$) hai, jise hum $360°$ bhi kehte hain.

YE KAISA DIKHTA HAI. Figure quarter-turns mark karta hai: $\frac{\pi }{2}$, $\pi$, $\frac{3\pi }{2}$, $2\pi$ loop ke around — taaki tum radians ko clock positions ki tarah dekh sako.

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Step 4 — Ek clock add karo: turning-per-second angular velocity $\omega$ ban jaata hai

Term by term:

• $\Delta \theta$ — extra angle swept (radians) — amber wedge jo do snapshots ke beech badh raha hai.
• $\Delta t$ — un do snapshots ke beech ka time (seconds).
• $\omega$ — radians per second: har second mein kitna turn pack hota hai.

RATIO $d\theta /dt$ KYO AUR KUCH AUR KYO NAHI? Hum rate chahte hain: "per unit time kitna angle". Ek rate hamesha change-in-thing over change-in-time hoti hai — exactly wahi hai jo $\Delta \theta$ ko $\Delta t$ se divide karna karta hai. $\Delta t$ ko zero ki taraf shrink karna (derivative $d\theta /dt$) ek moment par instantaneous spin deta hai, na ki ek chunk ka average.

YE KAISA DIKHTA HAI. Figure mein do dotted spokes — $t$ par position aur $t+\Delta t$ par — unke beech amber wedge $\Delta \theta$ label kiya hua. Har second same wedge matlab constant $\omega$.

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Step 5 — $\omega$ ko real speed mein badlo: $v=r\omega$ derive karo

$s=r\theta$ ko differentiate KYO KARO. Linear speed matlab "metres of arc per second", yaani $v=ds/dt$. Humara $s=r\theta$ pehle se Step 2 se hai. Kyunki $r$ hamare dot ke liye fixed hai, hum sirf dekhte hain ki $s$ kaise badhta hai jaise $\theta$ badhta hai:

$v=\frac{ds}{dt}=\frac{d}{dt}\left(r\theta \right)=r\frac{d\theta }{dt}=r\omega$

• $ds/dt$ — metres of arc per second = linear speed $v$,
• $r$ — bahar nikala kyunki yeh constant hai (dot andar ya bahar nahi jaata),
• $d\theta /dt$ — Step 4 se $\omega$ ke roop mein pehchana gaya.

$\boxed{v=r\omega }$

YE KAISA DIKHTA HAI. Figure mein ek inner cyan dot aur ek outer amber dot hai. Same wedge (same $\omega$), lekin outer arrow (uski velocity) lambi hai — lambi $r$ $v$ ko scale up karti hai. Velocity arrow circle ke tangent point karta hai (travel ki direction mein), spoke ke along nahi.

Recall

$v$ tangent kyun hai, spoke ke along kyun nahi? Kyunki dot ka agli instant ka motion us rim ke along hai jis par woh sawaar hai — spoke direction ($r$) hamare dot ke liye kabhi nahi badlti, isliye iske along koi motion nahi hota. ::: Velocity circle par dot pe tangent hoti hai.

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Step 6 — Spin ko khud change hone do: $a_t=r\alpha$ derive karo

$v=r\omega$ ko differentiate KYO KARO. "Path ke saath speed up hona" matlab "linear speed ke change ki rate", $a_t=dv/dt$. Humara $v=r\omega$ hai $r$ fixed ke saath:

$a_t=\frac{dv}{dt}=\frac{d}{dt}\left(r\omega \right)=r\frac{d\omega }{dt}=r\alpha$

• $dv/dt$ — metres-per-second per second: tangential acceleration $a_t$,
• $r$ — constant, bahar nikala,
• $d\omega /dt$ — $\alpha$ ke roop mein pehchana gaya.

$\boxed{a_t=r\alpha }$

YE KAISA DIKHTA HAI. Figure time $t$ par velocity arrow (chhoti) aur $t+\Delta t$ par (lambi) dikhata hai, dono tangent. Tips ke beech extra amber arrow $a_t$ hai, motion ke saath point karta hua — yeh speed ko lengthen karta hai.

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Step 7 — Edge aur degenerate cases (reader ko kabhi map se mat girne do)

Case A — centre dot ($r=0$). Dot ko pin par rakho. Toh $v=r\omega =0\cdot \omega =0$ aur $a_t=r\alpha =0$. YE KYO SENSE KARTA HAI: centre kabhi koi arc ride nahi karta — woh sirf jagah par pivot karta hai. Yeh abhi bhi har doosre dot ke saath same $\theta ,\omega ,\alpha$ share karta hai, lekin uski linear speed zero hai. Figure is dot ko fixed dikhata hai jabki ek outer dot uske around sweep karta hai.

Case B — steady spin ($\alpha =0$). Agar spin rate kabhi nahi badlti, $a_t=r\cdot 0=0$: path ke saath koi speeding up nahi. Lekin dot abhi bhi accelerate kar rahi hai — inward, $a_c$ ke zariye — kyunki uski direction ghoomti rehti hai. Yeh steady-spin case exactly Uniform Circular Motion hai.

Case C — reversing spin ($\alpha$ opposite to $\omega$). Agar $\alpha$ current spin ke against point kare (negative $\alpha$ jabki $\omega >0$), disc slow hoti hai, rukti hai, phir ulta ghoomne lagti hai. Signs, pictures nahi, yeh carry karte hain: $\omega$ $0$ tak shrink hota hai, pass karta hai, aur negative grow karta hai. Figure wedge ko shrink hote, phir doosri taraf reopen hote dikhata hai. (Yeh parent-note ke Example 3 ka fan hai, jahan $\alpha =-45{\text{rad/s}}^2$.)

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Ek-picture summary

Yeh single blueprint poora walkthrough compress karta hai: angle $\theta =s/r$ (Step 2), uski rate $\omega =d\theta /dt$ (Step 4), aur do bridges $v=r\omega$ (Step 5) aur $a_t=r\alpha$ (Step 6), tangential $a_t$ aur inward $a_c$ arrows dot par perpendicular dikhaye gaye hain.

Recall Poore walkthrough ki Feynman retelling

Main ek spinning disc par ek dot paint karta hoon, pin se $r$ doori par. Yeh measure karne ke liye ki woh around kitni gayi, main degrees count nahi karta — main wo curved trail measure karta hoon jo usne chhodi aur spoke length se divide karta hoon. Woh ratio radians mein angle hai, aur rearrange karne par kehta hai trail length sirf spoke times angle hai: $s=r\theta$. Ab main ek clock start karta hoon. Angle kitni tezi se badhta hai woh $\omega$ hai. Kyunki trail length $r$ times angle hai, aur $r$ kabhi nahi badlta, trail angle ke growth rate ke $r$ times ki speed se badhta hai — toh dot ki real speed $v=r\omega$ hai. Ek outer dot aur ek inner dot har second same maatra ghoomte hain, lekin outer wale ka $r$ zyada hai, toh woh zoom karta hai. Aakhir mein, agar spin khud speed up ho (woh $\alpha$ hai), dot ki speed path ke saath $a_t=r\alpha$ par climb karti hai. Special dots mujhe honest rakhte hain: centre dot ka $r=0$ hai isliye woh metres mein kabhi nahi move karta; ek steady spin mein $\alpha =0$ hai toh koi path-speeding nahi lekin phir bhi ek inward turning pull hai; aur ek spin jo khud se lad rahi ho ($\alpha$ $\omega$ ke against) slow hoti hai, rukti hai, aur reverse ho jaati hai. Teen bridges, ek radius, poore neeche same picture.

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Connections

• Parent topic note — woh summary jise yeh page visually derive karta hai.
• Uniform Circular Motion — Step 7B ka $\alpha =0$ case.
• Centripetal Acceleration and Force — inward $a_c$ arrow $a_t$ ke perpendicular.
• Linear Kinematics Equations — linear world jisse bridges connect karte hain.
• Torque and Angular Acceleration — woh cheez jo Step 6 mein hum ne use kiya $\alpha$ ko cause karti hai.
• Moment of Inertia — mass ka rotational analogue.
• Rolling Motion — jahan $v=r\omega$ straight-line travel se milta hai.