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

1.6.1 · D2 · Physics › Oscillations & Waves › Simple harmonic motion — definition, restoring force F = −kx

Hum bilkul paanch cheezein use karenge, aur main promise karta hoon ki har symbol use karne se pehle uski picture dikhaunga:

ek mass (ek blob), ek position line, ek spring;
slope ka idea (graph ki steepness);
curvature ka idea (graph kis taraf bend karta hai);
$sin$ aur $cos$ ki do shapes (drawn, assumed nahi);
aur ek spinning point ki shadow (ek circle jise tum edge-on dekhte ho).

Step 1 — Ek rule draw karo: "ghar se door ⇒ zyada push wapas"

KYA. Ek blob ko horizontal line par rakho. Ek khaas point mark karo aur use ghar kaho — woh jagah jahan spring relaxed hai aur zero force se push karti hai. Blob ki ghar se doori ko $x$ kaho. Daayein taraf positive $x$ hai, baayein taraf negative $x$ hai.

KYUN. Oscillation ke baare mein har claim ek ek physical fact se aani chahiye, toh aao algebra chhhune se pehle us fact ko visually fix kar lein: spring ka pull hamesha ghar ki taraf point karta hai, aur jitna door jaate ho utna zyada strong hota hai.

PICTURE. Figure mein, blob teen jagahon par baitha hai. Arrow (force) hamesha ghar ki taraf point karta hai, aur jab blob bhatakta hai toh uski length badhti hai.

Step 2 — Push ko "ye kaise bend karta hai" wali instruction mein badlo

KYA. Newton ka second law kehta hai push $=$ mass $\times$ acceleration. Acceleration wo hai ki velocity kitni tezi se change hoti hai. Hum ise "second derivative" symbol $\frac{d ^{2} x}{d t ^{2}}$ se likhte hain — ise padhte hain "position-vs-time graph ki curvature."

KYUN. Hum aakhirkar $x$ ko time ki function ke roop mein chahte hain — ek graph. Force acceleration control karta hai, aur acceleration kuch nahi sirf ye hai ki wo graph kitna tezi se bend karta hai. Toh $F = - k x$ asli mein time graph ki shape ke baare mein ek rule hai. Woh translation karna hi poora game hai.

PICTURE. Do chhote graph snippets: ek *neeche bend karta hai (jaise ek hill ka top — curvature negative), ek upar bend karta hai (jaise ek valley ka bottom — curvature positive). Second derivative bilkul yahi bend measure karta hai.

Step 3 — Rhythm control karne wale ek constant ka naam rakho

KYA. Clump $\frac{k}{m}$ ek single positive number hai. Hum ise $ω^{2}$ (omega-squared) naam dete hain taaki equation clean padhe.

KYUN. Nature ko parwah nahi ki hum ise " $k$ over $m$ " kehte hain; sirf combination matter karta hai. Ise ek naam dena, $ω^{2}$ , hume pehle se bata deta hai ki answer stiffness aur mass par sirf is ek number ke through depend karta hai — aur (spoiler) $ω$ baad mein rhythm ki speed niklegaa.

PICTURE. $ω = k / m$ label wala ek knob: stiffness $k$ badhaao, knob upar jaata hai; mass $m$ pile karo, knob neeche jaata hai.

Step 4 — Sirf do shapes draw karo jo question ka answer deti hain

KYA. Hume ek aisi shape chahiye jo bend ko do baar measure karne ke baad "khud mein wapas aaye, flip hokar." Ek circle par ek spinning point se $cos$ aur $sin$ draw karo: jab ek arm ghoomta hai, uski horizontal shadow $cos$ trace karti hai, uski vertical shadow $sin$ trace karti hai.

KYUN. Ye in shapes ko kyun aur parabola ya straight line ko kyun nahi? Kyunki $cos$ ka differentiate karna (slope lena) $- sin$ deta hai, aur dobara differentiate karne par $- cos$ aata hai — shape ek minus sign ke saath khud mein wapas aati hai. Ye bilkul Step 3 ka curvature rule hai. Koi aur elementary shape aisa nahi karti. Isliye SHM ek wave hai, kuch jagged nahi. (Ye spinning-point view Uniform Circular Motion hai edge-on dekha gaya.)

PICTURE. Ek circle par ek rotating arm; ek dashed line uski tip se horizontal axis par girta hai (cosine shadow) aur vertical axis par jaata hai (sine shadow). Shadow ko aage peeche sloshing dekhte raho — wahi oscillation hai.

Step 5 — Guess check karo: do baar differentiate karo aur sign flip dekho

KYA. $x = A cos (ω t + ϕ)$ propose karo. Uska slope do baar lo aur dekho ki hume $- ω^{2} x$ wapas milta hai ya nahi.

KYUN. Physics mein ek guess tab hi kisi kaam ka hota hai jab wo substitution survive kare. Hum ise curvature rule mein plug karte hain; agar dono sides match karein, toh guess ek genuine solution hai — luck se nahi, balki isliye ki shape uske liye banai gayi thi.

PICTURE. Teen stacked graphs, time mein aligned: position (cosine), phir uska slope = velocity (ek sine, par negative), phir uska slope = acceleration (ek flipped cosine). Notice karo ki top aur bottom graphs axis ke across mirror images hain — wahi $- ω^{2}$ flip hai.

Step 6 — Do free knobs padhna: amplitude $A$ aur phase $ϕ$

KYA. Solution mein do adjustable numbers hain, $A$ aur $ϕ$ . Ye tumhare blob ko shuru karne ke tarike se fix hote hain (wo kahan hai aur $t = 0$ par kitni tezi se chal raha hai).

KYUN. Ek second-derivative rule tumhara particular launch nahi jaanta — tumhe use do facts deni honi chahiye (start position, start velocity). $A$ aur $ϕ$ us information ke liye bilkul do slots hain. Same spring, infinitely many motions, sab same $ω$ ki cosines.

PICTURE. Same time axis par do cosine curves: ek tall (bada $A$ ), ek sideways shifted (alag $ϕ$ ). Time mein same wavelength (same $ω$ ), alag height aur alag starting point.

Step 7 — Edge cases cover karo taaki kuch surprise na kare

KYA. Special situations walk karo: blob kinare se rest se release kiya, blob ghar se kick kiya, aur degenerate "koi motion nahi."

KYUN. Jo derivation tum trust kar sako use har start handle karna chahiye, boring aur extreme dono. Aao verify karein ki cosine picture kabhi nahi tooti.

PICTURE. Teen panels: (a) $x = + A$ se rest mein release → pure cosine apne peak par shuru hoti hai; (b) ghar $x = 0$ se kick kiya → curve zero par shuru hoti hai aur upar uthti hai → ek sine (jo sirf $ϕ = - \frac{π}{2}$ wali cosine hai); (c) bilkul ghar par zero speed ke saath rakha → flat line, koi motion nahi.

Ek-picture summary

KYA. Sab kuch ek canvas par: spinning arm (circle), position line par slide hoti uski horizontal shadow (asli blob), aur cosine graph jo woh shadow paint karti hai jab time daayein flow karta hai.

KYUN. Ye single image poori derivation hai: circle → shadow → wave. Force ne graph ko wapas bend karaya (Step 2–3); sirf $cos / sin$ sahi se wapas bend karte hain (Step 4–5); ek spinning point bilkul wohi shadows produce karta hai (ye figure).

Recall Feynman retelling — ek dost ko batao

Ek fact se shuru karo: ek spring hamesha block ko ghar wapas kheenchti hai, aur jitna bahar ho utna zyada kheenchti hai — bilkul proportionally. Newton kehta hai force set karta hai ki position-vs-time graph kaise bend karta hai, toh hamara ek fact ban jaata hai: "jab bhi graph high ho, use neeche bend karo; jab bhi low ho, upar bend karo." Poochho ki kaun sa curve ye maanta hai. Straight line nahi; parabola nahi. Lekin cosine maanegi — uska slope do baar lo aur wo bilkul khud mein wapas aati hai, ulti, jo bilkul "height ke opposite bend karo" hai. Toh block ko cosine trace karna hi padega. Cosine kahan se aati hai? Ek circle par ghoomte point se: uski shadow wall par dekho aur shadow cosine mein aage peeche slide karti hai. Swing ka size ( $A$ ) aur kahan se shuru karta hai ( $ϕ$ ) depend karta hai ki tum use kaise chodh te ho, lekin rhythm ( $ω = k / m$ ) sirf spring aur mass par depend karta hai — kabhi nahi iti pe ki kitna kheencha. Bada pull, chhota pull: same tick-tock. Woh steady tick-tock, ek proportional pull-back se paida hua, simple harmonic motion hai.

Recall Walkthrough par self-test

Motion parabola kyun nahi ho sakti? → Uska second derivative ek constant hai, $- ω^{2} x$ nahi; wo height ke proportional bend back nahi karta.
$-$ sign graphs mein kahan end up hota hai? → Ye acceleration graph ko position graph ki mirror image mein flip kar deta hai.
$A$ aur $ϕ$ kya fix karta hai? → Starting position aur starting velocity.
$cos$ "circular motion edge-on dekha gaya" kyun hai? → Ye ek circle par steadily ghoomte point ki horizontal shadow hai.

Connections

Parent topic — SHM definition & $F=-kx$
Uniform Circular Motion — woh spinning arm jiska shadow hamara cosine hai
Energy in SHM — wahi motion $\frac{1}{2} k A^{2}$ ke through batayi gayi
Taylor Series — kyun pull kisi bhi valley bottom ke paas proportional hota hai
Hooke's Law — $F = - k x$ ka physical origin
Simple Pendulum — jahan proportionality sirf chhote angles ke liye hold karta hai
Damped Oscillations — kya hota hai jab hum $- b v$ term add karte hain