Foundations — Energy in SHM — KE + PE = ½kA² (constant)

1.6.5 · D1 · Physics › Oscillations & Waves › Energy in SHM — KE + PE = ½kA² (constant)

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0 · Woh scene jo hum describe kar rahe hain

Is chapter mein sab kuch ek block ke saath hota hai jo ek smooth table par spring se bandha hai. Picture dekho aur use apne dimag mein rakho — neeche har symbol us picture ke kisi hisse ka label hai.

• Block sirf left–right slide kar sakta hai.
• Ek special spot hai jahan spring na stretched hai na squeezed — equilibrium point. Hum us spot ko "zero" kahenge.
• Block ko bahar khicho, choddo, aur woh hamesha aage-peechhe jhoolti rehti hai (ideal, frictionless duniya mein).

Ab hum parts ke naam rakhte hain.

1 · $x$ — displacement (block abhi kahan hai)

Picture: $x$ woh chota sa arrow hai jo centre line se block tak jaata hai. Yeh ek signed number hai — sign direction batata hai. Figure s01 mein wapas dekho: $x$ marked arrow bilkul yahi hai.

Topic ko iski zaroorat kyun hai: spring ka push aur uski stored energy dono depend karte hain ki tum centre se kitna door ho. "Kitna door" ke liye koi number ke bina, hum koi formula nahi likh sakte.

2 · $A$ — amplitude (block jitna door kabhi bhi ja sakta hai)

Picture: block $x=+A$ (far right) aur $x=-A$ (far left) ke beech jhoolti hai. Woh do extreme spots turning points hain: block wahan ek split second ke liye rukta hai aur reverse ho jaata hai.

Topic ko iski zaroorat kyun hai: $A$ ek baar, shuru mein, set ho jaata hai — jitna tune khicha utne se. Kyunki motion ke dauran yeh kabhi nahi badlta, yeh ek constant total energy banane ke liye perfect cheez hai — isliye $\frac{1}{2}k{A}^2$ kaam karta hai.

3 · $k$ — spring constant (spring kitna stiff hai)

Picture: socho ki block ko distance $x$ par khiincha. Spring wapas pull karta hai. Stretch double karo → pull double ho jaata hai. "Pull $=$ (number) $\times$ stretch" mein woh proportionality number $k$ hai.

Yeh Hooke's Law hai: $F=-kx.$

• $k$ wahi stiffness hai jo humne abhi describe ki.
• $x$ section 1 ka displacement hai.
• Minus sign ka matlab hai force displacement ke opposite direction mein point karta hai: right khiincho ($x>0$) toh spring left pull karta hai ($F<0$). Yeh "hamesha centre ki taraf vaapis aimed" behaviour hi ise restoring force banata hai.

Topic ko iski zaroorat kyun hai: $k$ position ko force se link karta hai, aur force woh cheez hai jo energy store karti hai. Poora energy formula $k$ par build hota hai.

4 · $m$ — mass (block kitna heavy hai)

Picture: ek heavy block usi spring par sluggishly respond karta hai; ek light block quickly snap karta hai. Usi spring ke saath, alag weights → alag rhythms.

Topic ko iski zaroorat kyun hai: kinetic energy (motion energy) mass par depend karti hai. Saath hi, $m$ aur $k$ milkar decide karte hain ki oscillation kitni fast hai (agla symbol).

5 · $\omega$ — angular frequency (kitni fast cycle karta hai)

Yeh un do cheezoon se banta hai jo humne already dekhi hain: $\omega =\sqrt{\frac{k}{m}}.$

Is formula ko words mein padhna: stiffer spring (bada $k$) → faster wobble; heavier block (bada $m$) → slower wobble. Square root isliye hai kyunki ${\omega }^2=k/m$ naturally nikal ke aata hai jab tum motion equation solve karte ho (yeh SHM definition note mein hai).

Topic ko iski zaroorat kyun hai: $\omega$ woh bridge hai jo parent ko energy do equivalent tareekon se likhne deta hai, $\frac{1}{2}k{A}^2$ aur $\frac{1}{2}m{\omega }^2{A}^2$. Woh ek hi cheez hain kyunki $m{\omega }^2=k$.

6 · $t$, $\cos$, $\varphi$ — time ke function ke roop mein motion

Parent yahan se shuru karta hai: $x(t)=A\cos (\omega t+\varphi ).$

Chalte hain har piece earn karte hain.

• $t$ = time seconds mein. "$x(t)$" bas matlab hai "time $t$ par $x$ ki value" — position time ka function hai.
• $\cos$ (cosine) = ek smooth wave jo hamesha $+1$ aur $-1$ ke beech rock karti rehti hai, kabhi badi nahi hoti. Ise $A$ se multiply karo aur tumhe $+A$ aur $-A$ ke beech smooth rock milti hai — bilkul block ki motion. Isliye cosine aur koi aur shape nahi: yeh repeat karti hai, bounded hai, aur iski wiggle spring se match karti hai.
• $\varphi$ (phi) = phase, ek starting-offset. Yeh bas batata hai "block $t=0$ par cycle mein kahan tha." Agar far right se release karo, $\varphi =0$; kahin aur se release karo aur $\varphi$ wave ko sideways shift kar deta hai.

Topic ko iski zaroorat kyun hai: yeh prove karne ke liye ki energy constant hai, parent is $x(t)$ ko energy formulas mein plug karta hai. Phir magic identity ${\sin }^2+{\cos }^2=1$ time ko wipe out kar deti hai — yahi pura point hai.

7 · $v$ — velocity (block kitna fast aur kis direction mein move kar raha hai)

Picture: kinaron ke paas block creep karta hai, rukta hai, reverse ho jaata hai ($v\to 0$). Beech se guzarte waqt woh bhaagte jaata hai ($v$ sabse bada). Yeh SHM mein velocity hai.

Parent likhta hai: $v(t)=\frac{dx}{dt}=-A\omega \sin (\omega t+\varphi ).$

Symbol $\frac{dx}{dt}$ derivative hai — padho "jis rate se $x$ change hota hai jab time tick karta hai." Hum yahan derivative use karte hain, sirf "distance over time" nahi, kyunki block ki speed har instant alag hoti hai; derivative woh tool hai jo instantaneous speed deta hai.

Topic ko iski zaroorat kyun hai: kinetic energy $\frac{1}{2}m{v}^2$ hai, isliye motion-energy ke baare mein baat karne se pehle humein $v$ chahiye.

8 · Energy, $\frac{1}{2}$, aur "$\propto$"

• Energy = ek single number jo "kaam karne ki capacity" measure karta hai. Joules (J) mein measure hoti hai. Yahan do flavours hain: kinetic (kyunki move kar raha hai) aur potential (kyunki stretched hai).
• $\frac{1}{2}$ jo $\frac{1}{2}m{v}^2$ aur $\frac{1}{2}k{x}^2$ mein hai woh guess nahi hai — yeh force ko $0$ se $x$ tak stretch build hone par add up (integrate) karne se nikalta hai. Maximum ka half, kyunki force linearly zero se badhi. (Parent Step 1 mein yeh integral dikhata hai.)
• $\propto$ ka matlab hai "proportional to" — "ke saath step mein badhna." $E\propto A^2$ likhne ka matlab hai: agar $A$ double karo, $E$ $2^2=4$ se badh jaati hai. Equals sign ki zaroorat nahi kyunki hum sirf scaling ki parwah karte hain, exact constant ki nahi.

Topic ko iski zaroorat kyun hai: ye dono energies, jod do, constant total deti hain — headline result. "Coins nahi khoye" wali idea conservation of mechanical energy hai; uss assumption ko drop karo aur tumhe damped oscillations milta hai jahan total slowly leak ho jaata hai.

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