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

1.6.1 · D1 · Physics › Oscillations & Waves › Simple harmonic motion — definition, restoring force F = −kx

Is page par yeh assume kiya gaya hai ki tumne koi bhi notation pehle nahi dekhi. Hum har symbol ek picture se build karenge, ek aisi order mein jahan har naya idea sirf pehle se bane hue ideas par tike.

0. Stage: ek mass, ek spring, ek ghar ki jagah

Socho ek block ek smooth table par baitha hai, ek spring se wall se bandha hua. Akela chhodo toh woh ek jagah par chup baitha rehta hai. Thoda dhakelo, aur woh aage-peeche slide karta hai. Is topic mein sab kuch isi picture ke baare mein baat karta hai.

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

"Ghar" ki jagah kyun chahiye? Kyunki baaki har symbol kuch ghar ke relative measure karta hai. Agar zero par agree na karein toh "displacement" aur "restoring force" ka koi matlab hi nahi hoga.

1. $x$ — displacement (ghar se ek signed distance)

Picture: table ke saath ek ruler rakho, ghar par $0$ ke saath. Ruler par block ki position hi $x$ hai. Upar wali figure mein lal dot dekho — uska label $x$ seedha ruler se padha jaata hai.

Sign kyun matter karta hai: poora topic ek aisi force ke baare mein hai jo $x$ ko oppose karti hai. Agar $x$ ka koi sign na hota, toh hum kabhi "opposite" nahi keh sakte. Ek signed $x$ se ek formula dono taraf ko handle kar leta hai.

2. $A$ — amplitude (jitni door $x$ kabhi jaata hai)

Picture: neeche wali figure mein $- A$ aur $+ A$ par do dashed walls hain. Block unke beech bounce karta hai, har wall ko ek pal ke liye chhuta hai phir mur jaata hai.

Kyun chahiye: $A$ motion ka size set karta hai. Halki nudge chhota $A$ deti hai; zor se khinchne par bada $A$ milta hai. Gaur se dekho (hum dekhenge) ki rhythm $A$ par depend nahi karti — lekin phir bhi hum yeh symbol chahiye taaki swing ki width describe kar sakein.

3. Force $F$ aur restoring idea

Figure mein do coral arrows dekho: daayein taraf arrow baayein point karta hai, baayein taraf arrow daayein point karta hai. Dono ghar ki taraf nishana lagate hain. Woh "ghar ki taraf nishana" wala behaviour hi restoring word pakadta hai — aur isi liye block kabhi bhaagta nahi; use hamesha wapas kheencha ja raha hota hai.

4. $k$ — force constant (spring kitni stiff hai)

Picture: upar wali figure mein, block ko do guna door kheencho aur coral arrow do guna lamba ho jaata hai. Woh "arrow length distance ke saath badhti hai" exactly wahi hai jo $k$ measure karta hai — yeh proportionality hai kitna door aur kitna zor ke beech.

(Is proportional-spring law ka apna ghar hai: dekho Hooke's Law.)

5. Rate of change — velocity $v$ aur acceleration $a$

Hume language chahiye time ke saath position kaise badalti hai ke liye. Do words:

Picture: block ko dekho. Ghar par woh tezi se guzar raha hota hai (bada $v$ ). Walls $\pm A$ par woh ek pal ke liye ruk gaya hota hai ( $v = 0$ ) lekin sabse zyada kheeencha ja raha hota hai (bada $a$ ). Toh $v$ aur $a$ opposite moments par peak karte hain.

Yahan derivative kyun laate hain

"Kitni tezi se $x$ badalta hai" ko exact symbol mein convert karne ke liye hume ek mathematical tool chahiye: derivative. Hum ise choose karte hain — aur simple division $Δ x /Δ t$ ko nahi — kyunki velocity har instant par badalt rehti hai, isliye hume instantaneous rate chahiye, ek single point par slope, na ki time ke ek chunk par average.

Picture: $x$ ka time ke against graph par, har point par curve ka slope $v$ hai; woh slope khud kaise bend karta hai woh $a$ hai. Seedhi line ka acceleration zero hota hai; curving line accelerate karti hai.

6. Newton ka second law — bridge $F = ma$

Topic ko iska kyun chahiye: humhare paas force ka ek rule hai ( $F = - k x$ , §4) aur acceleration ka ek symbol ( $a = d^{2} x / d t^{2}$ , §5). Newton ka law woh bridge hai jo dono ko motion ke baare mein ek statement mein jodta hai: $ma = - k x .$ Is bridge ke bina, spring law aur motion do unconnected facts hoti.

7. $ω$ — angular frequency (rhythm number)

$ma = - k x$ ko rearrange karne par milta hai $a = - \frac{k}{m} x$ . Combination $k / m$ baar baar aata rehta hai, isliye hum ise ek naam dete hain.

Picture: socho ek spot ek circle ke around steady rate par ghoom raha hai; wall par uski chhaya bilkul hamari block jaisi aage-peeche slide karti hai. $ω$ hai kitne radians us circle ke har second mein sweep hote hain — isi liye ise angular frequency kaha jaata hai chahe hamari block straight line mein move kare. Yeh chhaya connection Uniform Circular Motion ki poori kahani hai.

8. Sine wave $A cos (ω t + ϕ)$ — motion ki shape

Picture: block ki position ko time ke against plot karo aur tum ek smooth wave paate ho jo hamesha upar aur neeche jaati hai — ek cosine curve, neeche draw ki gayi. Uski height $A$ hai, time mein repeat-length $T$ hai, aur $ϕ$ ise sideways shift karta hai.

Sine/cosine hi kyun aur kuch nahi

Hume ek aisi function chahiye jiska second derivative minus itself ke equal ho (kyunki §6 ne kaha $a = - ω^{2} x$ , yani acceleration, position ka $- ω^{2}$ times hai). Pucho: kaunsi shape, do baar differentiate hone par, khud wapas aati hai lekin sign flip hoke? Sirf sine/cosine family yeh karti hai — exactly isi liye SHM ek sine wave hai aur koi doosra curve nahi. Iska poora solving parent note ka kaam hai; yahan hum bas symbols se milte hain.

9. Energy symbols $K E$ , $U$ , $E$

Picture: ek valley shape $U = \frac{1}{2} k x^{2}$ . Block ek marble ki tarah is bowl mein roll karta hai: sides par upar = bahut stored energy, koi speed nahi; bottom par = koi stored energy nahi, top speed. Yeh "bowl" picture hi deep reason hai ki SHM har jagah hoti hai, aur yeh Energy in SHM ka subject hai aur (har valley bottom parabola kyun hoti hai iske liye) Taylor Series.

Prerequisite map

Equipment checklist

Daayein taraf chhupaao aur khud test karo — tum SHM ke liye ready ho jab har line instant ho.

x

kya measure karta hai, aur yeh signed kyun hai?

Ghar (equilibrium) position se distance, sign ke saath jo bataata hai kaunsi taraf; sign ek formula ko "force displasment ko oppose karta hai" kehne deta hai.

A

(amplitude) kya hai?

Ghar se sabse badi distance jo block reach karta hai; hamesha positive; motion

- A

aur

+ A

ke beech rehti hai.

Force ke liye "restoring" ka matlab kya hai?

Force hamesha ghar ki taraf wapas point karti hai — displacement ke opposite.

k

kya measure karta hai aur uski units kya hain?

Spring stiffness: displacement ke har metre par newtons of restoring force (N/m).

F = - k x

mein minus sign kyun hai?

Yeh ek direction rule hai —

F

x

ke opposite point karta hai; sign us taraf ke hisaab se flip hota hai jis par tum ho, isliye force hamesha restoring hoti hai.

Plain words mein

\frac{d x}{d t}

kya hai?

Position–time graph ka instantaneous slope — velocity

v

\frac{d ^{2} x}{d t ^{2}}

kya hai?

Velocity ka rate of change — acceleration

a

Newton ka law

F = ma

kya contribute karta hai?

Yeh force rule aur acceleration ko motion ki ek equation,

ma = - k x

, mein bridge karta hai.

ω

kaise define hota hai aur yeh kya control karta hai?

ω = k / m

; yeh rhythm set karta hai — bada

k

tez, bada

m

dheema.

ω

T

aur

f

kaise aate hain?

T = 2 π / ω

(seconds per cycle),

f = 1/ T = ω /2 π

(cycles per second).

SHM sine/cosine hi kyun hona chahiye?

Sirf sine/cosine do derivatives ke baad minus themselves mein return karte hain,

a = - ω^{2} x

ko match karte hue.

K E

U

E

kya hain aur kaunsa constant hai?

K E = \frac{1}{2} m v^{2}

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

E = K E + U

; total

E

constant rehta hai.

Connections

Parent topic — SHM definition & $F=-kx$
Hooke's Law — jahan linear spring force $F = - k x$ aati hai
Uniform Circular Motion — woh circle jiska shadow hamara block hai ( $ω$ ka source)
Energy in SHM — bowl $U = \frac{1}{2} k x^{2}$ aur energy sloshing
Taylor Series — har valley bottom parabola kyun hoti hai
Simple Pendulum — disguise mein ek aur restoring force
Damped Oscillations — friction add karne par kya badalta hai