Simple harmonic motion — definition, restoring force F = −kx

1.6.1 · Physics › Oscillations & Waves

Motion ko scratch se DERIVE kaise karte hain?

Hum sirf $F = - k x$ se start karte hain aur bina koi answer memorize kiye $x (t)$ tak pahunchte hain.

Step 1 — Newton's law likho. $m \frac{d ^{2} x}{d t ^{2}} = - k x .$ Yeh step kyon? Force = mass × acceleration, aur acceleration position ka second time-derivative hota hai.

Step 2 — $ω^{2} = k / m$ define karo taaki clean ho jaye. $\frac{d ^{2} x}{d t ^{2}} = - ω^{2} x .$ Yeh step kyon? Ab yeh ek pure math question hai: "kaunsa function do derivatives ke baad minus ek constant times itself ke equal hota hai?"

Step 3 — Ek aisi function guess karo jiska 2nd derivative sign flip kar de. Try karo $x = A cos (ω t + ϕ)$ . $\frac{d x}{d t} = - A ω sin (ω t + ϕ), \frac{d ^{2} x}{d t ^{2}} = - A ω^{2} cos (ω t + ϕ) = - ω^{2} x . ✓$ Yeh step kyon? Sine aur cosine sirf wohi smooth functions hain jo do baar differentiate karne ke baad khud mein (sign ke saath) return karte hain — isliye yeh zaroor SHM describe karte hain.

Step 4 — Physics padhlo.

$A$ = amplitude (max displacement), initial conditions se set hota hai.
$ϕ$ = phase constant, yeh set hota hai ki woh kahan se shuru hua.
Velocity: $v = - A ω sin (ω t + ϕ)$ , max speed $v_{m a x} = A ω$ .
Acceleration: $a = - A ω^{2} cos (ω t + ϕ)$ , max $a_{m a x} = A ω^{2}$ .

Figure — Simple harmonic motion — definition, restoring force F = −kx

Worked Examples

Common Mistakes (Steel-manned)

Quick Recall

Recall Self-test (hide karo aur answer doo)

SHM define karne wali do conditions kya hain? → equilibrium ki taraf restoring force aur $\propto$ displacement.
Speed max kahan hai / acceleration max kahan hai? → speed max $x = 0$ par; acceleration max $x = \pm A$ par.
Kya $T$ , $A$ par depend karta hai? → Nahi.
$F = - k x$ aur $F = ma$ se $ω$ derive karo. → $ω = k / m$ .

Recall Feynman: 12-saal ke bacche ko explain karo

Socho ek ball ek smooth bowl ke bottom mein rakh rakhi hai. Agar tum use nudge karo, woh roll karke wapas aati hai. Jitna zyada tum use side mein push karo, utna zyada bowl use ghar push karta hai — do guna door, do guna push. Kyunki push itne neatly badhta hai, ball bilkul steady rhythm mein aage-peeche rock karti hai, jaise ek clock. Choti nudge aur badi nudge dono aage-peeche jaane mein same time lete hain — yahi steady rhythm "simple harmonic motion" hai. Ek spring ke saath weight bhi exactly yahi dance karta hai.

Connections

Energy in SHM — $\frac{1}{2} k A^{2}$ kahan se aata hai
Simple Pendulum — small-angle approximation deta hai $F = - (m g / L) x$
Hooke's Law — $F = - k x$ ka origin
Damped Oscillations — ek velocity term $- b v$ add karo
Uniform Circular Motion — SHM uski shadow hai (projection)
Taylor Series — kyun har valley bottom ek parabola hai

SHM ke liye defining condition kya hai?

Restoring force (ya acceleration) equilibrium ki taraf directed ho aur displacement ke proportional ho:

F = - k x

a = - ω^{2} x

F = - k x

mein minus sign ka kya matlab hai?

Force displacement ki direction se opposite hai (restoring), koi fixed negative value nahi.

Mass–spring SHM ke liye angular frequency ka formula?

ω = k / m

a = - \frac{k}{m} x

a = - ω^{2} x

se match karne par milta hai.

Kya SHM ka period amplitude par depend karta hai?

Nahi.

T = 2 π m / k

mein koi amplitude nahi hoti (isochronism).

Cycle mein speed maximum kahan hoti hai aur zero kahan?

Max speed

x = 0

par (

v_{ma x} = A ω

); zero speed

x = \pm A

par.

Acceleration maximum kahan hoti hai?

Extremes

x = \pm A

par (

a_{ma x} = A ω^{2}

), jahan force sabse badi hoti hai.

Displacement

x

par speed ka expression?

v = ω A^{2} - x^{2}

(energy conservation se).

SHM oscillator ki total mechanical energy?

E = \frac{1}{2} k A^{2} = \frac{1}{2} m v^{2} + \frac{1}{2} k x^{2}

(constant).

Nature mein SHM itni common kyon hai?

Kisi bhi stable equilibrium ke paas

U \approx \frac{1}{2} k x^{2}

(parabola) hota hai, isliye

F = - d U / d x = - k x

Kya

x = A cos (ω t + ϕ)

ek valid SHM solution hai? Batao kyon.

Haan; iska second derivative

- ω^{2} x

hai, jo

a = - ω^{2} x

satisfy karta hai.

Simple harmonic motion — definition, restoring force F = −kx

SHM exist kyon karta hai?

SHM kya hai? (definition)