Foundations — Work-energy theorem — derivation from Newton's second law

1.3.3 · D1 · Physics › Work, Energy & Power › Work-energy theorem — derivation from Newton's second law

Yeh page assume karta hai ki tumhe kuch nahi aata siwaaye arithmetic ke aur arrow-picture ka matlab. Hum parent note (parent topic) mein har letter aur squiggle ko ground up se, ek-ek karke banate hain, har ek apni jagah justify karta hai pehle agla aane se.

0. Stage: ek line par ek particle

Kisi bhi symbol se pehle, scene picture karo. Ek chhota sa object — hum ise particle kehte hain kyunki hum iska size ignore karte hain aur ise ek dot treat karte hain — ek seedhe track par baitha hai. Woh track par alag-alag jagahon par ho sakta hai, aur move kar sakta hai.

Is topic mein sab kuch pehle is ek line par hota hai (1-D). Baad mein hum 3-D ke liye direction-arrows add karte hain.

1. Position $x$ — dot kahan hai

• Picture: track par rakha hua ek ruler. Dot number $x$ ke upar baitha hai.
• Topic ko yeh kyun chahiye: work hai "force spread over distance". Distance yani change in position, isliye hum work ke baare mein baat nahi kar sakte jab tak dot kahan hai yeh pin down na kar lein.

2. Ek tiny step $dx$ — distance ka almost-nothing wala tukda

• Picture: ruler ko zoom in karo jab tak ek baal-jitna patla slice bache; woh width hai $dx$.
• Topic ko yeh kyun chahiye: agar force har position par alag hai, toh hum force ko poori distance se ek baar mein multiply nahi kar sakte. Isliye hum distance ko countless slivers $dx$ mein kaatate hain, har tiny piece ko handle karte hain jahan force almost constant hai, phir sab ko wapas jodtey hain. Woh "slivers jodna" integral hai (§9).

3. Velocity $v$ aur derivative $\frac{dv}{dt}$

• Picture: dot do nearby instants par; $dx$ woh gap hai jo woh slide hua, $dt$ snapshots ke beech ka time. Unka ratio speed hai.

• Topic ko yeh kyun chahiye: Newton's law (§6) literally $F=m\frac{dv}{dt}$ ke roop mein likha jaata hai. Aur derivation mein famous "$vdv$ trick" kuch nahi hai siwaaye in do ratios $\frac{dx}{dt}$ aur $\frac{dv}{dt}$ ko juggle karne ke. Agar yeh symbols samajh aaye, toh poori derivation sirf bookkeeping hai.

4. Mass $m$ — speed up karna kitna mushkil hai

• Picture: do dots, ek bhaari wala bada drawn, ek halka wala chhota; same push chhote wale ko zyada move karta hai.
• Topic ko yeh kyun chahiye: $F=ma$ mein $m$ force aur motion ke beech ka bridge hai, aur kinetic energy $K=\frac{1}{2}m{v}^2$ ke andar bhi hai.

5. Force $F$ — push ya pull

• Picture: dot par ek horizontal arrow. Lamba arrow = badi force; peeche pointing arrow = negative force.

• Topic ko yeh kyun chahiye: theorem $W_{net}$ ke baare mein hai, jo net force se bana hai — sirf woh force nahi jo tum lagate ho. Yeh akela word "net" parent note mein sabse common mistake ka source hai.

6. Newton's Second Law $F_{net}=ma$

• Picture: ek bada arrow (force) ek dot ke upar jo consequently ek bada velocity-change arrow pata hai, heavy mass se scale down karke.
• Topic ko yeh kyun chahiye: yeh sirf ek fundamental input hai. Work–Energy Theorem nature ka koi naya law nahi hai — yeh $F=ma$ re-organised hai. Dekho Newton's Second Law.

7. Work $W$ — force distance par add up hoti hai

• Picture: force (upar-axis) versus position (across-axis) ke graph ke neeche ka area shade karo. Woh shaded area hai work.

• Topic ko yeh kyun chahiye: $W$ theorem ka ek aadha hissa hai. Ise "force-vs-position graph ke neeche area" ki tarah padhna hi variable-force case (spring) ko natural banata hai — dekho Work done by a variable force.

8. Kinetic energy $K=\frac{1}{2}m{v}^2$ — motion-energy

• Picture: ek speedometer jo ek energy-tank se wired hai; tank speed se tezi bhar ta hai kyunki $v$ squared hai.
• Kyun $v^2$ use hota hai (ek scalar): square karna velocity ka sign throw away kar deta hai, isliye left ya right same speed par move karne par same $K$ milti hai. Energy ki koi direction nahi hoti — isliye $K$ ek scalar hai aur hamesha $\ge 0$. Zyada detail Kinetic Energy mein.

9. Integral sign ${\int }_a^b$ — saare slivers jodo

• Picture: §7 ka shaded area, ab $dx$ width aur $F$ height ke patले rectangles mein kaata gaya; integral unke areas jodta hai.
• Topic ko yeh kyun chahiye: theorem ka left side, "net work", yahi integral hai. Right side, $vdv$ trick ke baad, ek aur integral $\int mvdv$ hai jo $\frac{1}{2}m{v}^2$ evaluate karta hai.

10. Vectors $\vec{F},\vec{v},d\vec{r}$ aur dot product $\cdot$

• Picture: displacement arrow par seedha neeche light daalo; force arrow ka us par shadow "aligned part" hai jo dot product rakhta hai.
• Topic ko yeh kyun chahiye: 3-D mein, work hai $\int \vec{F}\cdot d\vec{r}$. Dot product woh tool hai jo answer karta hai "is push ka kitna hissa actually motion mein help karta hai?" — yeh automatically angles aur signs handle karta hai, isliye hum unhe haath se track nahi karte. Isliye normal force (hamesha sideways) zero work karta hai.

Prerequisite map

Ise top-down padho: position se velocity, velocity se acceleration; mass + force + acceleration se Newton's law; force + distance se work; work integral se sum hoti hai, kinetic energy mein change ke barabar set hoti hai, yahi theorem hai.

Equipment checklist

Right side cover karo aur reveal karne se pehle har ek ka answer dene ki koshish karo.

Connections

• Newton's Second Law — woh single fundamental input jiske liye yahan har symbol kaam karta hai.
• Kinetic Energy — jahan $K=\frac{1}{2}m{v}^2$ fully explore hota hai.
• Work done by a variable force — kyun integral (§9) unavoidable hai.
• Power — time-rate cousin, $P=\vec{F}\cdot \vec{v}$, §10 ka dot product reuse karta hai.
• Impulse–Momentum Theorem — wahi $F=ma$ distance ki jagah time par integrate hua.
• Conservation of Mechanical Energy — theorem kya ban jaata hai jab forces conservative hain.