Foundations — SHM differential equation — solution - x = A cos(ωt + φ)

1.6.2 · D1 · Physics › Oscillations & Waves › SHM differential equation — solution - x = A cos(ωt + φ)

Hum strictly bottom-up build karte hain. Neeche koi bhi cheez pehle use nahi hogi jab tak draw na ho — isliye hum jaanbujhkar poora formula abhi nahi likhte; ise piece by piece assemble karte hain aur sirf bilkul end mein pura spell out karte hain.

0. Ek line par position: $x$

Topic ko yeh kyun chahiye. SHM ek story hai har instant par bead kahan hai iske baare mein. Woh "kahan" $x$ hai, aur baaki sab — speed, force, energy — yeh dekhne se banta hai ki $x$ kaise change hota hai.

1. Samay: $t$

Topic ko yeh kyun chahiye. Bead ki position ek number nahi hai — yeh ek movie hai. Hum $x(t)$ likhte hain ("$x$ of $t$") yeh kehne ke liye ki "time $t$ par position kya hai".

2. Kitna fast, kitna curved: $\dot{x}$ aur $\ddot{x}$

Ab hum poochhte hain: bead kitni fast aur kis direction mein move kar rahi hai? aur woh motion kaise change ho rahi hai? Upar ke do dots yahi matlab rakhte hain.

Topic ko yeh kyun chahiye. SHM ka dil $\ddot{x}$ ke baare mein ek rule hai: acceleration hamesha $x$ ke proportion mein ghar ki taraf point karta hai. Bina double-dot ke koi equation nahi. (Woh acceleration kya cause karta hai — ek force — yeh hum next section mein milenge.)

3. Acceleration kya cause karta hai: force $F$, mass $m$, aur Hooke's spring

$F=-kx$ kahan se aata hai ki poori kahani ke liye Hooke's Law and the spring force dekho.

Topic ko yeh kyun chahiye. $F$ ke dono expressions ko equal set karo: $m\ddot{x}=-kx\Rightarrow \ddot{x}=-\frac{k}{m}x.$ Woh $-k/m$ $x$ ke aage exactly woh shape hai "acceleration displacement ke proportion mein ghar ki taraf point karta hai." Toh Hooke's law mein minus sign SHM ka minus sign ban jaata hai — yahi reason hai ki motion wapas swing karta hai instead of bhagkar door jaane ke.

4. Tuning knob: $\omega$

Topic ko yeh kyun chahiye. $\omega$ woh akela dial hai jo poori oscillation ki speed control karta hai. Yeh period set karta hai, max speed, max acceleration — sab kuch.

5. Wave shape: $\cos$, $\sin$, aur angle $\theta$

$\cos (\dots )$ jaise symbols sense banane se pehle, tumhe pata hona chahiye ki $\cos$ kya hai.

Topic ko yeh kyun chahiye. Solution literally hai $x=(\text{ek size})\times \cos (\text{ek angle jo time ke saath badhta hai})$. Cosine isliye choose kiya jaata hai kyunki yeh woh shape hai jiska second derivative apna hi flipped copy hota hai — ek matra function jo "acceleration points home" satisfy karta hai.

6. Time ko angle mein convert karna: $\omega t+\varphi$

Yahan pieces click together hote hain.

Topic ko yeh kyun chahiye. Ek 2nd-order equation ko reality pin down karne ke liye do facts chahiye: kahan se shuru kiya aur kitna fast. Size $A$ (next section) "kitna bada" carry karta hai; $\varphi$ "cycle mein kahan" carry karta hai. Saath milke yeh motion fully specify karte hain.

7. Swing ka size: $A$

Topic ko yeh kyun chahiye. $A$ poori wiggle ko scale karta hai. $x=\pm A$ par bead momentarily ruk jaati hai (turning points); $x=0$ par woh sabse fast hoti hai. $A$ seedha stored energy se juda hai — Energy in SHM dekho.

8. Ek swing kitna time leta hai: period $T$ (aur frequency $f$)

Topic ko yeh kyun chahiye. $T$ aur $f$ ek oscillation ke everyday, measurable "kitni baar" hain — jo ek stopwatch read karta hai. Notice karo $T$ mein koi $A$ nahi hai: bada swing zyada time nahi leta (SHM isochronous hai). $T$, $f$ aur $\omega$ ek hi rhythm ke teen naam hain, loop-size $2\pi$ se linked.

Poora symbol-set saath laana

Sirf ab — har letter earn karne ke baad — hum parent note ka solution assemble karte hain:

$x(t)=\underset{\text{kitna door}}{\underbrace{A}}\cos (\underset{\text{time se angle}}{\underbrace{\omega t}}+\underset{\text{head start}}{\underbrace{\varphi }})$

Ab har letter defined hai: $x$ = signed position, $t$ = clock, $A$ = swing size, $\omega$ = turning rate, $\varphi$ = starting angle, $T$/$f$ = round-trip time aur rate, $\cos$ = circle-shadow shape, aur unke peeche $F,m,k$ = woh force, mass aur stiffness jisne motion create ki.

Foundations topic ko kaise feed karte hain

Neeche ka map har chain ke neeche se upar padha jaata hai: raw ideas left par hain aur combined ideas mein flow karti hain, aur har chain final solution node mein khaali hoti hai.

Map padhna, node by node. Left column raw stuff hai: position $x$, time $t$, spring force (Hooke), stiffness/mass pair, circle, aur radians. Arrows follow karo:

• $x\to \dot{x}\to \ddot{x}$ ($t$ rate feed karte hue): dots position aur time se bante hain.
• Hooke ka pull-back Newton ke $F=m\ddot{x}$ ban jaata hai, aur $\ddot{x}$ ke saath milke SHM equation node deta hai.
• $k$ aur $m$ combine hokar $\omega$ dete hain; circle plus radians cos shadow shape dete hain; $\omega$ aur $t$ badhta angle banate hain, jise $\varphi$ shift karke $\omega t+\varphi$ banta hai; $\omega$ aur $2\pi$ period $T$ set karte hain.
• Har chain bottom node mein khaali hoti hai — poora solution $x=A\cos (\omega t+\varphi )$. Agar left-column ka koi bhi box shaky hai, toh us box se nikalna wala arrow exactly woh gap hai jo parent note tackle karne se pehle close karna hai.

Equipment checklist

Khud test karo — right side cover karo aur zor se jawab do.