Spring-mass system — horizontal, vertical: Foundations

1.6.8 · D1 · Physics › Oscillations & Waves › Spring-mass system — horizontal, vertical

Parent note ko aaram se padhne se pehle, tumhe usme aane wale har symbol ko mehnat karke earn karna hoga. Neeche, har item deta hai: simple words → picture → yeh topic ko kyun chahiye. Yeh is order mein hain ki har cheez sirf upar wali cheezein use karti hai — koi symbol tab tak nahi aata jab tak uska block define na kar de.

Building blocks, ek ek karke

1. Natural length aur coordinate axis

Spring ko waise hi baithe socho, coils evenly spaced, na stretched na squished. Har "kitna door bhage" measurement isi relaxed shape ke baare mein sochne se shuru hoti hai.

Axis convention (ek baar fix karo, hamesha ke liye): hum ek axis ko spring ki line of action ke along point karte hain aur usi direction ko positive kehte hain.
- Horizontal case: table par ek spring. Hum axis ko right ki taraf point karte hain aur rightward ko positive kehte hain.
- Vertical case: ek hanging spring. Hum axis ko seedha neeche point karte hain aur downward ko positive kehte hain. Har case mein sirf ek axis hoti hai, motion ke along chosen — toh "positive" kabhi ambiguous nahi hai, bas matlab hai "jis taraf axis point kar rahi hai."
Yeh topic ko kyun chahiye: natural length reference hai yeh jaanne ke liye ki spring kitni stretch hui hai, aur single-axis convention se hum ek clean equation likh sakte hain left/right/up/down ko haath se juggle karne ki jagah.

2. Position aur coordinate $x$

Socho ek mass ek smooth table par baith raha hai, ek ruler axis ke along neeche rakha hua hai. Ruler ka zero mark wahan rakha hai jahan mass bina kisi disturbance ke rest karta hai. Toh $x$ bas mass ke neeche ruler kya read kar raha hai yeh hai.

Figure — Spring-mass system — horizontal, vertical

Figure 1 — mass ek smooth table par; ruler ka zero resting spot par baith raha hai, toh $x$ mass ke neeche reading hai. Dashed magenta line $x = 0$ mark karti hai aur magenta arrow spring ko wapas uski taraf kheenchta dikhata hai. Navy axis arrow positive direction (rightward) dikhata hai.

Yeh topic ko kyun chahiye: page ka har law — force, energy, wave-jaisi motion — ek rule ki tarah likha hai ki $x$ time ke saath kaise behave karta hai. Ek clean zero point ke bina, koi bhi formula simplify nahi hoga.

3. Equilibrium — "ghar ka base"

Socho mass bilkul still hai, spring na zyada kheench rahi hai na zyada dhake maar rahi hai doosri forces se. Woh balance point ghar hai.

Yeh topic ko kyun chahiye: oscillation is point ke around swing karna hai. Horizontal case mein ghar spring ki natural length hai; vertical case mein gravity pehle spring ko stretch karti hai, toh ghar neeche shift ho jaata hai — lekin woh phir bhi bas "balance point" hai.

4. Spring constant $k$

Do springs socho: ek flimsy slinky ( $k$ chhota) aur ek stiff car spring ( $k$ bada). Dono ko same distance se khicho — stiff wala bahut zyada waapis kheechega. Woh "zyada per metre" exactly $k$ hai.

Figure 2 — same stretch $x$ ek soft spring (top, chhota pull-back) aur ek stiff spring (bottom, bada pull-back) par apply ki gayi. Magenta arrow pull-back force $k x$ dikhata hai; kyunki stretch $x$ dono ke liye identical hai, lamba arrow badi stiffness $k$ reveal karta hai.

Yeh topic ko kyun chahiye: $k$ us "pull-back" ki strength hai jo oscillation cause karta hai. Yeh har period aur frequency formula mein aata hai. Yeh seedha Hooke's Law se aata hai.

5. Force $F$ aur $F = - k x$ mein minus sign

Minus kyun? Picture dekho: spring ko right mein stretch karo ( $x > 0$ ) aur woh left kheechti hai (force negative). Use left mein push karo ( $x < 0$ ) aur woh right push karti hai (force positive). $F$ ka sign hamesha $x$ ke sign ka opposite hota hai — woh opposite-ness hi " $-$ " encode karta hai.

Yeh topic ko kyun chahiye: yeh minus sign puri wajah hai ki motion oscillate karta hai bhaag jaane ki jagah. Hooke's Law dekho.

6. Mass $m$ aur inertia

Socho same finger-push se ek marble ko thappad maaro versus bowling ball ko: marble uchhal jaata hai, bowling ball muskil se hilta hai. Woh reluctance inertia hai, $m$ se measure hoti hai.

Yeh topic ko kyun chahiye: inertia isliye hai ki mass ghar par overshoot karta hai rukne ki jagah. Spring use ghar kheenchti hai → inertia use aage le jaati hai → spring use doosri taraf wapas kheenchti hai. Pull-back plus overshoot equals oscillation.

7. Velocity $v$ aur dot notation $\overset{x}{˙}$

Socho mass mid-swing mein hai: us instant par $v$ woh speed hai jo ek imaginary speedometer par dikhti hai jo mass se chipki hai, direction ke liye sign ke saath. Jahan position graph steeply rise karta hai, $v$ bada hai.

Hum "dot" kyun use karte hain aur sirf " $v$ " kyun nahi: dot ek compact tarika hai kehne ka ki "jo bhi neeche hai uska time-derivative." Yeh stack karne deta hai: ek dot = velocity, do dot = acceleration. Yeh equation of motion ko chhota rakhta hai.

8. Acceleration $\overset{x}{¨}$ (double dot)

Socho mass apne swing ke door wale end par hai, momentarily ruka hua ( $v = 0$ ) lekin spring se sabse zyada kheencha ja raha hai. Woh abhi move nahi kar raha, lekin uski motion sabse tezi se change hone wali hai — woh badi acceleration hai.

Yeh topic ko kyun chahiye: Newton's second law $F = m \overset{x}{¨}$ spring ki pull ko motion ke change se link karta hai. Ise $F = - k x$ ke saath combine karo aur master equation milti hai.

9. Amplitude $A$

Swing ke do turning points socho, ghar ke left aur right. Ghar se unki distance har taraf $A$ hai (equal, kyunki spring symmetric hai).

Yeh topic ko kyun chahiye: $A$ total energy budget set karta hai aur maximum speed ( $v_{ma x} = A ω$ ), jaisa neeche energy block precise karta hai.

10. Cosine function aur general solution

Socho ek wheel steady rate par ghoom raha hai, uski rim par ek dot hai. Us dot ki shadow ek wall par cast hoti hai aur bahar jaati hai, slow hoti hai, reverse karti hai, aur wapas aati hai — wheel ka ek full turn shadow ka ek full back-and-forth hai. Woh shadow ki motion cosine hai. Isliye cosine oscillation ki natural language hai: yeh hai steady circular turning side se kaisa dikhta hai.

11. Greek letter $ϕ$ (phase)

Socho same wiggle time axis ke left ya right mein slide hui. $ϕ$ woh slide measure karta hai. Agar far end se release karo, $ϕ = 0$ ; ghar se moving release karo, toh $ϕ$ shift hota hai.

Yeh topic ko kyun chahiye: general solution $x (t) = A cos (ω t + ϕ)$ kisi bhi starting condition fit karni chahiye, aur $ϕ$ woh dial hai jo start adjust karta hai.

12. Period $T$ , frequency $f$ , angular frequency $ω$

Yeh teeno sab describe karte hain ki motion time mein kaise spaced out hai. Yeh ek clock ke teen views hain.

Figure 3 — position-versus-time wiggle $x (t) = A cos (ω t)$ . Orange arrow amplitude $A$ mark karta hai; magenta span ek full period $T$ mark karta hai (ek complete hump-and-dip).

Socho position-versus-time graph ki ek full cosine wiggle: $T$ ek complete hump-and-dip ki horizontal width hai. $f$ count karta hai ki ek second mein kitne humps fit hote hain. $ω$ usi rhythm ko measure karta hai ek spinning wheel ke angle use karke jo exactly yeh wiggle trace karega.

$ω$ kyun aur sirf $f$ kyun nahi? Equation of motion naturally ek $cos (ω t)$ produce karta hai, jiska argument ek angle hona chahiye. $ω$ woh tool hai jo elapsed time ko woh angle mein convert karta hai — exactly isliye woh $2 π$ carry karta hai. Yeh Simple Harmonic Motion se link hota hai.

13. Energy words: KE, PE, aur $E$

Socho ek bank account: swing ke ends par sara paisa "spring" account mein hai (PE full, KE zero); ghar par sara "motion" account mein hai (KE full, PE zero). Total kabhi nahi badlata — bas dono ke beech slosh karta hai. Energy in SHM dekho.

Yeh topic ko kyun chahiye: $E$ ka conservation usi $ω$ tak pahunchne ka doosra, independent raasta deta hai, aur turant max speed deta hai.

Yeh foundations topic ko kaise feed karte hain

Map ko top-down ek story ki tarah padho: coordinate $x$ (equilibrium se measure kiya, jo khud natural length se referenced hai) aur stiffness $k$ milke Hooke's law produce karte hain. Hooke's law plus Newton's law — jise mass $m$ aur rates ke liye dot notation chahiye — milke equation of motion banate hain. Woh ek equation, cosine se solve hoke, angular frequency $ω = k / m$ deliver karta hai, jo timing words $T = 2 π m / k$ aur $f$ set karta hai. Alag se, amplitude $A$ $ω$ ke saath milke energy picture fix karta hai. Equation of motion aur energy view dono ek hi cheez ke do faces hain: Simple Harmonic Motion.

Equipment checklist

Right side cover karo aur khud ko test karo. Agar koi answer surprising lage, woh block upar se dobara padho.

Spring ki natural length kya hoti hai?

Uski length jab kuch bhi use kheench, dhake, ya hang nahi karta — relaxed length.

x

ke liye positive direction kaise choose ki jaati hai?

Motion ki single axis ke along — horizontal ke liye rightward, vertical ke liye downward; "positive" bas matlab hai "jis taraf axis point kar rahi hai."

x

kya measure karta hai, aur kis point se?

Mass kitna door bhaga hai, axis ke along equilibrium (ghar) position se measure kiya.

Equilibrium kya hai?

Woh spot jahan net force zero hai; mass wahan rest karta hai aur uske around oscillate karta hai.

Spring constant

k

kya batata hai?

Spring kitni stiff hai — newtons of pull per metre of stretch (N/m).

F = - k x

mein minus sign ka matlab kya hai, aur force ki magnitude kya hai?

Force displacement ke opposite direction mein point karti hai (ghar ki taraf wapas); uski size

k x

hai.

Mass

m

kya represent karta hai?

Inertia — motion change karne ki resistance; yeh woh overshoot cause karta hai jo oscillation jaari rakhta hai.

Ek dot

\overset{x}{˙}

ka matlab kya hai? Do dot

\overset{x}{¨}

ka?

Ek dot = velocity (position ka rate of change); do dot = acceleration (velocity ka rate of change).

Spring-mass system ki equation of motion likho.

m \overset{x}{¨} = - k x

, yaani

\overset{x}{¨} + (k / m) x = 0

, yaani

\overset{x}{¨} = - (k / m) x

Amplitude

A

kya hai?

Ghar se sabse badi distance jo mass kabhi pahunchta hai.

Solution cosine kyun hai, aur woh kya hai?

x (t) = A cos (ω t + ϕ)

— cosine hi ek akela shape hai jiska second derivative khud ka negative multiple hai,

\overset{x}{¨} = - ω^{2} x

match karta hai.

k

aur

m

ke terms mein

ω

kya hai, aur use kaise paate hain?

ω = k / m

;

x = A cos (ω t + ϕ)

ko master equation mein substitute karo aur

ω^{2} = k / m

match karo.

m

aur

k

ke terms mein

T

aur

f

likho.

T = 2 π m / k

aur

f = \frac{1}{2 π} k / m

Phase

ϕ

kya karta hai?

Set karta hai ki mass

t = 0

par swing mein kahan start karta hai — wave ki time shift.

Static extension

y_{0}

kya hai, aur ise kya fix karta hai?

Hanging spring ka downward stretch balance par;

k y_{0} = m g

se fix hota hai, toh

y_{0} = m g / k

Teen energy terms likho.

KE = \frac{1}{2} m v^{2}

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

, aur total

E = KE + PE

Spring PE mein

\frac{1}{2}

factor kyun hota hai?

Force

0

k x

tak linearly ramp karta hai; kaam triangular area hai

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

Total energy kiske equal hoti hai, aur woh kab sabse aasaan dikhti hai?

E = \frac{1}{2} k A^{2}

— turning point par sabse clean jahan

v = 0

aur

x = A

hai, toh sari energy spring PE hai.

Connections

Spring-mass system — horizontal, vertical — parent topic jo yeh foundations unlock karti hain.
Hooke's Law — $F = - k x$ aur $k$ ki meaning ka source.
Simple Harmonic Motion — jahan $ω$ , $T$ , $ϕ$ sab milke aate hain.
Energy in SHM — KE/PE/total-energy trio action mein.
Springs in Series and Parallel — stiffnesses $k$ combine karna.
Simple Pendulum — ek cousin oscillator jisse timing words compare karo.