Foundations — Conservation of mechanical energy — derivation

1.3.8 · D1 · Physics › Work, Energy & Power › Conservation of mechanical energy — derivation

Isse pehle ki hum prove karein ki total frozen rehta hai, humein exactly pata hona chahiye ki parent note mein har scribble ka kya matlab hai. Neeche, har symbol ko zero se banaya gaya hai: plain words → ek picture → kyun is topic ko yeh chahiye. Upar se neeche padho; har item uske upar wale par depend karta hai. Chhota "§" sign simply "is page ka numbered section" matlab hai — e.g. "§8" tumhe neeche section 8 par point karta hai.

1. Mass — $m$

Picture: ek solid block table par rakha hua, "$m$" tag ke saath. Yeh kyun chahiye: har energy formula ($\frac{1}{2}m{v}^2$, $mgh$) $m$ se scale hota hai — mass double karo, same speed ya height par energy double ho jaati hai.

2. Time — $t$

Picture: chalte block ke paas ek stopwatch tick kar rahi hai. Yeh kyun chahiye: velocity aur acceleration dono "per second" hain, isliye rate of change ka har measure $t$ ke against measure hota hai. Clock ki ek chhoti si tick $dt$ likhi jaati hai (time ka ek sliver, §3).

3. Position aur displacement — $x$, $h$, $dx$

Picture: ek number line; $x$ par ek dot, ek $dx$ length ka arrow use right nudge karta hua — neeche figure dekho. Yeh kyun chahiye: work force hai jo ek distance par act karta hai, isliye humein "movement ka ek chhota sa hissa" $dx$ ke baare mein baat karni aani chahiye aur saare chhote hisson ko add up karna chahiye.

Neeche ka figure inn teenon ideas ko — position $x$, ek sliver $dx$, aur kaise ek distance ka sliver time ke sliver par divide karne se velocity milti hai — ek picture mein fix karta hai. Yellow step $dx$ ko notice karo jo blue dot ko rightward nudge kar raha hai, aur green label $v=dx/dt$ padh raha hai.

4. Velocity aur acceleration — $\frac{dx}{dt}$, $\frac{dv}{dt}$

Picture: number line par wahi dot, ab ek speedometer needle attached ke saath.

Yeh kyun chahiye: poori derivation Newton's law $F=m\frac{dv}{dt}$ se shuru hoti hai, isliye line one se pehle hum "rate of change" mein fluent hone chahiye. $\frac{dv}{dt}$ fraction do numbers divided nahi hai — yeh ek idea hai: speed-vs-time graph ki steepness.

5. Force — $\vec{F}$, $F_{net}$

Picture: ek block par kaafi saare arrows (neeche gravity, right taraf hand pushing); unka sum ek bold ${\vec{F}}_{net}$ arrow hai. Notation par note: jab bhi kisi quantity ki direction hoti hai (force $\vec{F}$, displacement $\vec{d}$, acceleration $\vec{a}$, velocity $\vec{v}$) to hum uske upar arrow daal sakte hain. Ek single seedhi line par hum aksar arrow drop kar dete hain aur sirf ek sign ($+$ ya $-$) rakhte hain direction record karne ke liye — yeh woh sab "direction" hai jo ek 1-D line ke paas hoti hai. Yeh kyun chahiye: Newton's second law aur work dono force se start hote hain. Hum kaun sa force act kar raha hai isse link karte hain yeh decide karne ke liye ki energy conserved hai ya nahi.

6. Newton's second law — ${\vec{F}}_{net}=m\vec{a}$

7. Work — $W=\vec{F}\cdot \vec{d}$

Picture: ek force arrow aur ek displacement arrow. Work sirf force ka woh part count karta hai jo motion ke saath aligned hai.

• Motion ke along force → positive work (speeds up).
• Motion ke against force → negative work (slows down).
• Sideways force (90°) → zero work, chahe kitna bhi strong ho.

8. Kinetic energy aur Work–energy theorem — $K=\frac{1}{2}m{v}^2$

Picture: chalti hui block, ek glowing halo jiska brightness speed ke saath badhta hai.

Yeh kyun chahiye: yeh un do pillars mein se ek hai jis par poora proof khada hai. Yeh abstract "work" (§7) ko motion-pocket $K$ mein ek concrete change mein convert karta hai. Poora statement aur derivation Work-energy theorem mein hai.

9. Integration — $\int$

Picture: ek curve ke neeche kaafi saare thin rectangles, unke areas ek shaded region mein sum ho rahe hain. Yeh kyun chahiye: ek real distance par work, aur potential energy, dono infinitely many tiny contributions ke sums hain. Integration ke bina hum sirf ek waqt mein ek sliver handle kar sakte.

10. Potential energy — $U$ — aur force se uska link

Picture: do "batteries" — ek ball shelf par upar rakhi hui hai, doosri ek compressed coil hai.

• $g$ = gravitational field strength, Earth ke paas lagbhag $9.8{\text{m/s}}^2$ — ground har kilogram ko kitni strongly pull karta hai. Gravitational potential energy dekho.
• $k$ = spring constant — ek spring kitni stiff hai; bada $k$ = dabana mushkil. Spring potential energy (Hooke's law) dekho.

11. Change symbol — $\Delta$

Yeh kyun chahiye: poora conclusion isi se likha gaya hai — $\Delta (K+U)=0$ literally kehta hai "total start se finish tak nahi badla."

12. Non-conservative force (e.g. friction)

Picture: rough ground par slide karta ek block, uske peeche tiny heat-squiggles upar uth rahe hain. Yeh kyun chahiye: conservation law sirf tab hold karta hai jab aisi forces koi net work nahi karti. Is "sticky" energy-thief ko precisely naam dena hi woh cheez hai jo hume proof ke exact condition state karne deti hai.

13. Mechanical energy — $E=K+U$

Picture: do pockets jinke contents aage-peeche shift hote hain, lekin dono ko saath taulne wala scale kabhi nahi hilta.

Foundations topic ko kaise feed karte hain

Ise upar se neeche padho: time, mass aur rate-of-change Newton's law banate hain; force, displacement aur integration work banate hain; work work–energy theorem deta hai; force + integration $\vec{F}=-dU/dx$ ke through potential energy bhi banate hain; aur do "change" boxes (friction absent hone par) conservation dene ke liye collide karte hain.

Equipment checklist

Khud test karo — right side cover karo.

Connections

• 1.3.08 Conservation of mechanical energy — derivation (Hinglish)
• Work-energy theorem
• Conservative and non-conservative forces
• Potential energy
• Gravitational potential energy
• Spring potential energy (Hooke's law)
• Energy lost to friction
• Simple harmonic motion energy
• Power