Foundations — Angular displacement θ, angular velocity ω, angular acceleration α

1.5.2 · D1 · Physics › Rotational Mechanics › Angular displacement θ, angular velocity ω, angular accelera

Is page pe assume kiya gaya hai ki tumhe kuch nahi pata. Hum har letter, ratio aur picture build karte hain jis par parent note Angular displacement, velocity, acceleration rely karta hai, is order mein jahan har idea apne pehle wale idea par rest karta hai.

0. "Rigid body" ek axis ke around ghoomna kya hota hai?

Kisi bhi symbol se pehle, mental picture fix karo.

Ek rigid body woh object hota hai jiska shape kabhi nahi badalta — uske upar bane kisi bhi do dots ke beech ki distance hamesha fixed rehti hai. Ek ghoomta bicycle wheel, merry-go-round, clock ki sooiya: sab rigid hain.

Ek axis woh fixed line hai jiske around object ghoomta hai — centre ke through ek pin. Axis par ke points bilkul nahi hilte; usse door ke points sabse zyada hilte hain.

Topic ko yeh kyun chahiye: kyunki rotational mechanics ka poora trick yeh hai ki metres mein measure karna unfair hai — rim ke paas wala red dot tezi se dauda karta hai jabki hub ke paas wala dot dheere chhenta hai. Hume ek aisi quantity chahiye jo woh share karein. Woh quantity hai angle.

1. Circle, radius $r$, aur arc length $s$

Ghoomte body par ek point lo. Jab woh turn karta hai, woh ek circle trace karta hai.

• $r$ = radius = axis se us point tak seedhi-line distance (metres mein, m).
• $s$ = arc length = woh curved distance jo point actually circle ke along travel karta hai (metres mein, m).

Dono kyun chahiye: $r$ kisi given point ke liye fixed hota hai; $s$ badhta rehta hai jaise point move karta rehta hai. Inke beech ka ratio hume abhi angle dene wala hai.

2. Angle $\theta$ — aur radians kyun

Yeh hai key move. Ghoomte point se kata hua wedge dekho. Woh wedge kitna "open" hai? Wahi openness hai angle $\theta$ (Greek letter theta, sirf "angle" ka ek naam).

Hum ise degrees mein measure kar sakte hain, lekin circles ke liye ek smarter unit hai jo exactly circles ke liye bani hai.

Woh definition literally ek ratio hai:

$\theta =\frac{s}{r}(\text{arc length}÷\text{radius})$

Yeh ratio angle ko encode kyon karta hai: ek bada wedge same $r$ ke liye zyada lamba arc $s$ sweep karta hai, toh ratio $s/r$ badhta hai. $r$ se divide karne par "circle kitna bada hai" waali information hat jaati hai, sirf openness bachti hai — aur exactly yahi ek angle hota hai.

Poora circle: poora ghoom ke arc circumference hai $s=2\pi r$. Toh ek full turn hai: $\theta =\frac{2\pi r}{r}=2\pi \text{ rad}=360^{\circ }.$

3. "Rate of change" ka matlab kya hota hai (kisi bhi $\omega$ se pehle)

Parent note "rate of change" jaisi phrases aur symbols $\Delta$, $\frac{d}{dt}$ use karta hai. Inhe samajhte hain.

• $\Delta$ (Greek delta) ka matlab hai "change in" — final value minus starting value. $\Delta \theta ={\theta }_{\text{end}}-{\theta }_{\text{start}}$.
• $t$ = time, seconds mein (s).
• Ek rate jawaab deta hai "yeh quantity per second kitna change hoti hai?" — tum change lete ho aur use isme kitna time laga us se divide karte ho: $\frac{\Delta \theta }{\Delta t}$.

Picture: $\theta$ ko page ke upar aur time $t$ ko page ke across plot karo. Ek interval mein, rate do points ko join karne wali line ki steepness hai — rise over run.

Topic ko yeh kyun chahiye: angular velocity sirf "angle ka rate of change" hai, aur angular acceleration hai "us velocity ka rate of change." Dono graph par steepnesses hain.

Hamesha $\Delta$ kyun use nahi karte? Kyunki object ki spin rate moment to moment vary kar sakti hai. $\frac{d\theta }{dt}$ abhi sach bolta hai; $\frac{\Delta \theta }{\Delta t}$ sirf ek interval-over-interval average deta hai. Jab spin rate constant hoti hai, dono equal hote hain.

4. Topic mein use hone wala Greek alphabet

Teen Greek letters poore topic ko carry karti hain. Yeh sirf labels hain — inhe kaise bolein aur padhein yeh hai.

• $\omega =\frac{d\theta }{dt}$ — angle kitni tezi se grow karta hai. Units rad/s.
• $\alpha =\frac{d\omega }{dt}$ — spin rate khud kitni tezi se grow karta hai. Units rad/s².

Yeh exist kyun karte hain: yeh position $x$, speed $v$, aur acceleration $a$ ke angular twins hain — same teen ideas, lekin metres ki jagah angle mein measure kiye gaye, taaki rigid body ke sab points ek set of numbers share karein.

5. Radius — metres mein wapas jaane ka bridge

Har point $\theta ,\omega ,\alpha$ share karta hai. Lekin rim ke paas wala point phir bhi zyada tez feel hota hai. Woh kahan se aata hai? Radius se.

$\theta =s/r$ ko rearrange karne par master bridge milta hai:

$s=r\theta .$

Ek angular quantity ko $r$ se multiply karo aur tum linear (metres) world mein wapas aa jaate ho:

• distance: $s=r\theta$
• speed: $v=r\omega$
• along-path acceleration: $a_t=r\alpha$

Parent note $v=r\omega$ aur $a_t=r\alpha$ ko $s=r\theta$ ko differentiate karke derive karta hai — woh sirf "dono sides ka rate of change lo" hai, $\frac{d}{dt}$ tool use karke jo humne §3 mein define ki.

6. Topic jo ek aur distinction assume karta hai: tangential vs centripetal

Circle par ek point do bilkul alag directions mein accelerate kar sakta hai:

• Tangential — path ke along, point ko tez ya dheema karta hai. Yeh hai $a_t=r\alpha$.
• Centripetal — centre ki taraf inward point karta hai, sirf path bend karta hai (direction change karta hai, speed nahi). Yeh hai $a_c={\omega }^{2}r$.

Yeh dono perpendicular hain. Hum yeh abhi sirf isliye flag kar rahe hain taaki symbols $a_t$ aur $a_c$ baad mein surprise na karein; poori kahani Centripetal Acceleration and Force mein hai.

7. Yeh foundations topic ko kaise feed karte hain

Equipment checklist

Right side cover karo aur khud test karo — tum parent note ke liye ready ho jab har jawaab instantly aaye.

Connections

• Parent topic — jahan yeh symbols kaam aate hain.
• Linear Kinematics Equations — $x,v,a$ ki duniya jise yeh angular quantities mirror karti hain.
• Uniform Circular Motion — woh case jab $\alpha =0$, constant $\omega$.
• Centripetal Acceleration and Force — inward $a_c={\omega }^{2}r$ jo humne §6 mein flag kiya.