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

1.6.2 · Physics › Oscillations & Waves

WHAT is the equation?

Newton se yeh equation kyun banti hai: Spring par mass ke liye, Hooke's law restoring force $F = - k x$ deta hai. Newton kehta hai $F = m \overset{x}{¨}$ . Toh $m \overset{x}{¨} = - k x \Rightarrow \overset{x}{¨} = - \frac{k}{m} x .$ $ω^{2} \equiv k / m$ define karo (positive, isliye $ω$ real hai) aur tum $\overset{x}{¨} = - ω^{2} x$ par aa jaate ho. Har SHM system (pendulum, LC circuit, floating block) isi ek shape par collapse ho jaata hai — isliye hum ise ek baar solve karte hain.

HOW to solve it from scratch (derivation, no guessing)

Formula Solution ki energy-trick derivation

Step 1 — $\overset{x}{˙}$ se multiply karo taaki total derivatives saamne aayein. $\overset{x}{˙} \overset{x}{¨} = - ω^{2} x \overset{x}{˙}$ Yeh step kyun? $\overset{x}{˙} \overset{x}{¨} = \frac{d}{d t} (\frac{1}{2} \overset{x}{˙}^{2})$ aur $x \overset{x}{˙} = \frac{d}{d t} (\frac{1}{2} x^{2})$ . Multiply karne se dono sides ko cleanly integrate kar sakte hain (yeh energy conservation hi hai disguise mein).

Step 2 — $t$ ke upar integrate karo: $\frac{1}{2} \overset{x}{˙}^{2} = - \frac{1}{2} ω^{2} x^{2} + C \Rightarrow \overset{x}{˙}^{2} = ω^{2} (A^{2} - x^{2}),$ constant ko $C = \frac{1}{2} ω^{2} A^{2}$ likhte hain taaki $\overset{x}{˙} = 0$ jab $x = \pm A$ . Kyun? Tab $A$ automatically maximum displacement (amplitude) ban jaata hai — woh turning point jahan speed zero ho jaati hai.

Step 3 — Variables separate karo: $\frac{d x}{A ^{2} - x ^{2}} = \pm ω d t .$ Kyun? Yeh integral $arccos$ / $arcsin$ ki definition hai — sinusoid naturally saamne aata hai, humne assume nahi kiya tha.

Step 4 — Integrate karo: $arccos (\frac{x}{A}) = ω t + ϕ \Rightarrow x (t) = A cos (ω t + ϕ)$ Do constants $A$ aur $ϕ$ woh do integration constants hain jo ek 2nd-order ODE mein hone chahiye.

Initial conditions se constants fix karna: $x (0) = A cos ϕ, v (0) = \overset{x}{˙} (0) = - A ω sin ϕ .$ Solve karo: $A = x_{0}^{2} + \frac{v _{0}^{2}}{ω ^{2}}, tan ϕ = - \frac{v _{0}}{ω x _{0}} .$ $v_{0} / ω$ kyun? Velocity, amplitude $\times ω$ hoti hai, isliye velocity ko position se compare karne ke liye tumhe $ω$ se divide karna padega taaki units same (metres) ho jayein.

The companion quantities (sab derived hain, memorised nahi)

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

Worked examples

Forecast-then-Verify

Recall Forecast first, then open

Q: Amplitude $A$ double karo. Kya hoga (a) period $T$ ka, (b) max speed ka? Forecast karo, phir check karo: (a) $T = 2 π / ω$ mein koi $A$ nahi → period unchanged (SHM isochronous hai!). (b) $v_{m a x} = A ω$ → double ho jaayega. Intuition trap "bada swing = slower" galat hai: woh zyada door jaata hai lekin utne hi time mein, toh usse tez move karna hi padega.

Common mistakes (Steel-manned)

Flashcards

What differential equation defines SHM?

\overset{x}{¨} + ω^{2} x = 0

, i.e.

\overset{x}{¨} = - ω^{2} x

(acceleration

\propto -

displacement).

SHM ka solution sinusoidal kyun hona chahiye?

Yeh woh akela function hai jiska 2nd derivative apne aap ka negative constant times version hota hai.

\overset{x}{¨} = - ω^{2} x

ka general solution kya hai?

x = A cos (ω t + ϕ)

, with constants

A

(amplitude) aur

ϕ

(phase).

Exactly do arbitrary constants kyun?

Yeh ek 2nd-order ODE hai; motion fix karne ke liye initial position aur initial velocity chahiye.

x_{0}, v_{0}, ω

A

express karo.

A = x_{0}^{2} + v_{0}^{2} / ω^{2}

x_{0}, v_{0}, ω

tan ϕ

express karo.

tan ϕ = - v_{0} / (ω x_{0})

x = A cos (ω t + ϕ)

se velocity aur acceleration?

v = - A ω sin (ω t + ϕ)

a = - A ω^{2} cos (ω t + ϕ) = - ω^{2} x

Max speed aur max acceleration?

v_{m a x} = A ω

(centre par),

a_{m a x} = A ω^{2}

(extremes par).

ω

ke terms mein period aur frequency?

T = 2 π / ω

f = ω /2 π

Ideal SHM mein period amplitude par depend karta hai kya?

Nahi —

T = 2 π / ω

mein koi

A

nahi (isochronous).

Position ka function bana ke speed (bina time ke)?

v = \pm ω A^{2} - x^{2}

Spring ke liye

ω

kya hai?

ω = k / m

Recall Feynman: ek 12-saal ke bachche ko samjhao

Socho ek ball hai jo table ke centre par ek rubber band se bandhi hai. Jitna zyada kheeencho, utna zyada band use wapas kheenchta hai. Toh woh middle overshoots karta hai, band use doosri taraf se wapas kheenchta hai, aur woh hamesha side to side hilta rehta hai. Agar tum uski position film karo aur draw karo, toh ek smooth wave milegi — exactly cosine ki shape. Wave ki height hai kitna pehle kheeencha tha ( $A$ ), aur wave kahan se shuru hoti hai depend karta hai ki tum side se chode ya middle se push kiya (woh hai $ϕ$ ). Cool magic yeh hai: kitna wide kheeencho woh nahi badalta ki ek poora wiggle kitna time leta hai.