Foundations — Velocity and acceleration in SHM — v = ω√(A² − x²)

1.6.4 · D1 · Physics › Oscillations & Waves › Velocity and acceleration in SHM — v = ω√(A² − x²)

Yeh page topic ka har symbol bilkul zero se build karta hai — pehle plain meaning, phir ek picture, phir woh reason ki topic iske bina kaam nahi kar sakta. Hum koi bhi letter apne section se pehle introduce nahi karte. Upar se neeche padho; har block sirf wahi use karta hai jo pehle aa chuka hai, aur end tak tum khud target relation assemble kar loge.

1. Oscillation khud — hum dekh kya rahe hain?

Ek chhoti bead imagine karo jo ek seedhe groove mein aage-peeche slide kar rahi hai. Ek special jagah hai — beech — jahan bead baar baar lauti aati hai. Is topic mein sab kuch us beech se measure hota hai.

Figure dekho: groove horizontal hai, beech ko ek chhote circle se mark kiya gaya hai, aur bead ko ek baar left mein aur ek baar right mein dikhaya gaya hai. Yeh topic us bead ke safar ki kahani hai.

2. $x$ — displacement, "main kahan hoon" wala number

Topic ko yeh kyun chahiye: poora point speed aur acceleration kisi given jagah par dhundhna hai. Bina labelled position axis ke koi "jagah" hi nahi hai. Note karo sign pehle se hi direction bata deta hai, jise hum acceleration ke liye reuse karenge.

3. $A$ — amplitude, "duniya ki seema"

Bead hamesha ke liye nahi bhaagti; woh bahar jaati hai, slow hoti hai, rukti hai, aur wapas aati hai. Jitni door woh kabhi bhi jaati hai, us distance ka ek naam hai.

Figure dekho: dono turning points $x=+A$ aur $x=-A$ par baithe hain. Unke beech ka shaded band woh poora territory hai jahan bead reh sakti hai.

4. Space ko ek spinning circle mein badalna — reference circle

Yeh woh trick hai jo SHM ko aasaan banati hai. Ek point lo jo radius $A$ ke circle ke ird-gird steadily ghoom raha hai. Side se light daalo aur neeche groove par uski shadow dekho. Woh shadow exactly SHM perform karta hai.

Figure dekho: violet dot circle ke ird-gird constant speed se jaata hai. Uski horizontal shadow (magenta) $-A$ aur $+A$ ke beech aage-peeche slide karti hai — yahi bead hai. Circle ka radius amplitude $A$ ke barabar hai. Yeh Reference circle (projection of uniform circular motion) hai.

Topic ko yeh kyun chahiye: yeh ek aage-peeche ki motion (describe karna mushkil) ko ek steady rotation mein convert kar deta hai (angles se aasaani se describe hoti hai). Neeche har symbol actually is circle par koi angle ya turning rate hai.

5. Angle $\theta$, aur dono functions $\sin \theta$ aur $\cos \theta$

Yeh batane ke liye ki dot circle par kahan hai, humein ek angle chahiye — aur uske coordinates padhne ka ek rule.

Section 4 ki figure dobara dekho: orange arc $\theta$ hai, right-pointing zero-line se counter-clockwise khul raha hai. Yeh convention fix karna hi hai jo agli dono definitions ko unambiguous banata hai.

6. $\omega$ — angular frequency, "circle kitni tez spin karta hai"

Dot ek steady rate se ghoomta hai. Us rate ka ek naam hai.

Picture: ek fast spinner (bada $\omega$) shadow ko quickly aage-peeche hilata hai; ek slow spinner ek lazy sway banata hai. $\omega$ poori oscillation ka pace set karta hai — dekho Angular frequency ω and time period T.

7. $\varphi$ — phase, "dot kahan se shuru hua"

Do beads ek hi $A$ aur $\omega$ se swing kar sakti hain phir bhi out of step ho sakti hain — ek beech mein jab doosri edge par ho. Jo cheez unhe alag karti hai woh unka starting angle hai.

Picture: clock shuru karne se pehle poore circle ko $\varphi$ se rotate kar lo. Yeh Phase and phase difference ka foundation hai. Agar abhi confusing lag raha hai, toh har jagah $\varphi =0$ set karo — bead simply beech se bahar jaate hue shuru hoti hai.

8. $v$ — velocity, aur woh chain rule jo use deta hai

Derivative kyun? "Rate of change" exactly wahi hai jo ek derivative measure karta hai: thodi si time $dt$ mein $x$ kitna move karta hai. Koi aur tool instantaneous speed nahi deta; ek poore swing par average speed yeh fact hide kar deta ki bead beech mein sabse fast hoti hai.

Yahan Section 5 ka $\cos \theta$ asli kaam karta hai: sine ka derivative cosine hota hai, isliye velocity naturally cosine carry karta hai. Yeh akeli line hi baad mein $\cos \theta =\frac{v}{A\omega }$ likhne deti hai.

Circle par picture: shadow sabse fast chalti hai jab dot top ya bottom cross karta hai (uska horizontal motion sabse zyada hai) — exactly jab shadow beech mein hoti hai, $x=0$. Toh speed centre par maximum, edges par zero.

9. $a$ — acceleration, ek baar aur differentiate karke

Second derivative kyun? SHM mein velocity khud constant nahi hai — bead beech ki taraf fast hoti hai aur edges ki taraf slow. Us changing-ness ko capture karne ke liye hum Section 8 ki velocity differentiate karte hain.

Minus sign kyun matter karta hai: yeh kehta hai acceleration hamesha displacement ke opposite point karti hai — beech ki taraf wapas. Woh restoring behaviour SHM ki definition hai. Dekho Simple Harmonic Motion — definition a = −ω²x.

10. Pythagorean identity — woh bridge jo time hata deta hai

Ab hamare paas do equations hain jo dono ek hi angle $\theta =\omega t+\varphi$ chhupaate hain: $\sin \theta =\frac{x}{A}\text{(Section 5 se)},\cos \theta =\frac{v}{A\omega }\text{(Section 8 se)}.$ Hum $v$ sirf $x$ ke terms mein chahte hain — hidden angle (aur isliye time) ke bina. Us ke liye tool woh identity hai jo $\sin$ aur $\cos$ ko link karti hai.

Picture: radius $1$ ke reference circle par, ek point ke coordinates $(\cos \theta ,\sin \theta )$ hain. Centre se uski distance hamesha $1$ hai, aur us right triangle par Pythagoras exactly ${\cos }^2\theta +{\sin }^2\theta =1$ deta hai. Yeh Pythagorean identity sin² + cos² = 1 hai.

11. Energy words — kinetic aur potential (woh see-saw)

Target relation secretly energy book-keeping hai, isliye do aur words:

Figure dekho: jaise $x$ badhta hai, potential bar utha aur kinetic bar utni hi amount se gira — ek see-saw jiska total height kabhi nahi badalta. Woh constant total exactly wahi hai isliye $v$ exactly utna hi shrink karta hai jitna $x$ badhta hai, reproducing $v=\pm \omega \sqrt{A^2-x^2}$.

Yeh foundations topic ko kaise feed karte hain

Neeche har arrow bas yeh kehta hai "pehle zaroori hai." Ise order ki checklist ki tarah padho: equilibrium → displacement → amplitude → circle → angle aur uske do functions → spin rate → phase → dono derivatives → the identity → final relation, with energy as the parallel route.

Equipment checklist

Right side cover karo aur khud test karo. Agar koi bhi answer shaky lage, main note se pehle woh section dobara padho.

Connections

• Simple Harmonic Motion — definition a = −ω²x
• Energy in SHM — kinetic and potential
• Phase and phase difference
• Reference circle (projection of uniform circular motion)
• Angular frequency ω and time period T
• Pythagorean identity sin² + cos² = 1