Foundations — Relation to linear quantities - v = rω, a_t = rα, a_c = rω²

1.5.3 · D1 · Physics › Rotational Mechanics › Relation to linear quantities - v = rω, a_t = rα, a_c = rω²

Un teen formulas par trust karne se pehle, tumhe unke andar ke har letter ka poora knowledge hona chahiye. Yeh page har symbol ko bilkul zero se build karta hai — pehle plain words mein, phir ek picture, phir kyun yeh topic us cheez ke bina nahi chal sakta. Upar se neeche padho; har block uske upar wale par lean karta hai.

1. Axis aur rigid body

Picture (Figure 1): ek flat disk imagine karo jo apne center se ek nail ke through pin ki gayi ho. Nail hi axis hai. Disk par koi bhi point lo — jab disk turn karta hai, woh point nail ke center par ek perfect circle trace karta hai. Do alag alag points do alag alag size ke circles trace karte hain, lekin woh kabhi collide ya drift nahi karte, kyunki body rigid hai.

Kyun topic ko yeh chahiye: "rigid" silently hero hai. Yeh guarantee karta hai ki har point ka axis se distance $r$ time mein constant rehta hai. Parent page par har derivation mein, hum $r$ ko derivative se bahar kheenchte hain — woh move tabhi legal hai jab rigidity $r$ ko fixed rakhe.

Yahan bhi dekho Angular velocity and angular acceleration jahan yahi shared-rotation idea wapas aata hai.

2. Radius $r$

Picture: Figure 1 mein, nail se kisi point tak jaane wali straight spoke dekho. Uski length $r$ hai. Rim par ek point ka bada $r$ hoga; nail ke paas ke point ka chhota $r$ hoga; axis PAR ke point ka $r=0$ hoga.

Kyun topic ko yeh chahiye: $r$ poori dictionary hai. Core message "door = tez" literally $v=r\omega$ hai — same turning rate ke liye bada $r$, bada $v$. Agar per-point $r$ na ho, to angular aur linear language kabhi link nahi ho sakti.

3. Angle $\theta$ aur radian measure

Hum $\theta$ ko radians mein measure karne par zor dete hain, degrees mein nahi. Yahan kyun, scratch se build kiya gaya hai.

Picture (Figure 2): kisi point ke circular path lo. Straight length $r$ ko arc ke saath curved measuring tape ki tarah rakh do. Agar exactly ek $r$ fit hota hai, toh woh $1$ radian hai ($\approx 57.3°$). Ek full circle ki arc $2\pi r$ lambi hoti hai, isliye ek full turn $2\pi$ radians $=360°$ hota hai.

Kyun topic ko radians chahiye: rearrange karne par definition deta hai ==$s=r\theta$== — arc length equals radius times angle — lekin sirf radians mein. Degrees mein tumhe har jagah $\pi /180$ ke ugly factors carry karne padte. Radians isliye engineer kiye gaye hain taaki yeh cleanly padhe, aur $s=r\theta$ literally woh seed hai jise parent page differentiate karta hai har doosra formula paane ke liye.

Prerequisite deep-dive: Radian measure and arc length.

4. Arc length $s$ — path par linear position

Picture: Figure 2 mein, $s$ woh curved trail ki length hai jo point peeche chhodta hai. Yeh ek linear (metre) quantity hai, unlike $\theta$ jo ek angular (radian) quantity hai.

Kyun topic ko yeh chahiye: $s$ linking variable hai. Yeh ek taraf angular hai ($=r\theta$) aur doosri taraf linear (iske rate of change speed hai). Woh double life exactly yahi hai jo $r$ ko do worlds ke beech translate karne deti hai.

5. Rate of change: time par slope padhna

Parent page ki har derivation "differentiate with respect to time" phrase use karti hai. Symbols aane se pehle tumhe picture karna chahiye ki iska matlab kya hai.

Picture (Figure 3): kisi bhi quantity ko time ke against plot karo. Ek instant par, curve ko touch karti tangent line draw karo. Uski steepness (rise over run) us moment par rate of change hai. Steep upward line = tezi se badh raha hai; flat line = nahi badal raha; downward = ghat raha hai.

Kyun topic ko yeh chahiye — aur kyun YEH tool: "speed" aur "acceleration" rates hain. Position ko speed mein, ya speed ko acceleration mein convert karne ke liye, humein woh ek machine chahiye jo instantaneous rate-of-change report kare — woh machine time-derivative hai. Koi aur tool "yeh abhi kitni tezi se change ho raha hai?" ka jawab nahi deta. Isliye parent note $s=r\theta$ ko ek baar differentiate karta hai (speed paane ke liye) aur do baar (acceleration paane ke liye).

6. Angular quantities: $\omega$ aur $\alpha$

Ab jab "rate of change" ka ek picture hai, to angular quantities asaan hain.

Picture: merry-go-round par, $\omega$ woh single "turning speed" hai jo sabhi share karte hain — poori disk ke liye ek number. $\alpha$ tabhi nonzero hota hai jab koi disk ko tez kar raha ho (ya brake laga raha ho). Agar spin steady hai, $\alpha =0$.

Kyun topic ko yeh chahiye: yeh translator ke inputs hain. $\omega$ ko $r$ se multiply karo aur linear speed $v$ milti hai; $\alpha$ ko $r$ se multiply karo aur tangential acceleration $a_t$ milta hai. Yeh poori body mein shared hote hain, jo angular language ko itna simple banata hai.

Full treatment: Angular velocity and angular acceleration.

7. Linear quantities: $v$, $a_t$, $a_c$

Yeh woh hain jo formulas produce karte hain — private, per-point outputs.

Picture (Figure 4): circle par ek point par teen arrows draw karo — $\vec{v}$ tangent (circle ko graze karta hua), ${\vec{a}}_t$ usi tangent line ke along, aur ${\vec{a}}_c$ center ki taraf aimed. Tangent arrows inward arrow ke perpendicular hain.

Kyun topic ko yeh chahiye: Newton's laws, force, aur energy sab linear bolte hain. Spinning body par unhe use karne ke liye hume shared angular quantities ko har point ki linear quantities mein convert karna hoga — yahi poora purpose hai $v=r\omega$, $a_t=r\alpha$, $a_c=r{\omega }^2$ ka. Dekho Centripetal force and circular motion aur Uniform vs non-uniform circular motion.

8. Sines aur cosines — direction ke bookkeepers

Centripetal derivation point ki position ko $\vec{r}=r(\cos \theta ,\sin \theta )$ likhti hai. Tumhe pata hona chahiye ki $\cos$ aur $\sin$ ka matlab yahan kya hai.

Picture: jab $\theta$ $0$ se badhta hai, point counterclockwise chalta hai; horizontal axis par uska shadow $\cos \theta$ hai (1 se shuru, ghatta hai), vertical axis par uska shadow $\sin \theta$ hai (0 se shuru, badhta hai). $r$ se multiply karne par unit circle real circle tak scale ho jaata hai.

Kyun topic ko yeh chahiye: yeh direction ko numbers ke roop mein encode karte hain, taaki hum position vector ko differentiate karke dekh sakein ki uski direction kaise rotate hoti hai. Woh direction-tracking exactly wahi hai jo centripetal acceleration reveal karta hai — ek fact jo invisible hai agar tum sirf speed track karo.

Prerequisite map

Sab kuch parent topic mein funnel ho jaata hai.

Quick self-tests

Pehle predict karo, phir reveal karo.

Equipment checklist

Khud test karo — sirf aloud jawab dene ke baad reveal karo.