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

1.3.3 · Physics › Work, Energy & Power

WHAT hai yeh theorem?

WHAT yeh relate karta hai: force-over-distance ( $W$ ) ↔ speed change ( $Δ K$ ).
WHY yeh useful hai: yeh time aur acceleration ko skip kar deta hai. Agar tumhe force vs. position pata hai, tum directly final speed paa sakte ho — $a (t)$ solve karne ki zaroorat nahi.

WHY Newton's second law se shuru karein?

Newton ka law batata hai ki force velocity ko time mein kaise change karta hai: $F = m \frac{d v}{d t}$ .

Lekin hum jaanna chahte hain ki force velocity ko distance par kaise change karta hai. Toh poora game yeh hai: time derivative ko position derivative mein convert karo. Bas yahi ek trick poora theorem produce karti hai.

HOW derive karein (1-D, scratch se)

Formula Derivation step-by-step

Step 1 — Newton's 2nd law se shuru karo. $F_{n e t} = ma = m \frac{d v}{d t}$ Yeh step kyun? Yeh hamara ek hi fundamental input hai — sab kuch yahan se aana chahiye.

Step 2 — Dono sides ko $d x$ se multiply karo (hum work chahte hain, $\int F d x$ ). $F_{n e t} d x = m \frac{d v}{d t} d x$ Kyun? Work force times displacement hoti hai, isliye $d x$ introduce karna hume $W = \int F d x$ ki taraf le jaata hai.

Step 3 — Chain-rule trick. Note karo $\frac{d x}{d t} = v$ , toh rearrange karo: $m \frac{d v}{d t} d x = m d v \frac{d x}{d t} = m v d v$ Kyun? Yahi iska dil hai: humne awkward $\frac{d v}{d t}$ ko clean $v d v$ se replace kar diya, time ko poori tarah eliminate karke.

Step 4 — Initial se final state tak integrate karo. $\int_{x_{i}}^{x_{f}} F_{n e t} d x = \int_{v_{i}}^{v_{f}} m v d v$ Kyun? Left side net work ki definition hai; right side ab ek aasaan integral hai.

Step 5 — Evaluate karo. $W_{n e t} = m [\frac{v ^{2}}{2}]_{v_{i}}^{v_{f}} = \frac{1}{2} m v_{f}^{2} - \frac{1}{2} m v_{i}^{2} = Δ K ■$

Vector / general form

3-D motion ke liye jahan $F = m \frac{d v}{d t}$ : $F \cdot d r = m \frac{d v}{d t} \cdot v d t = m v \cdot d v = d (\frac{1}{2} m v \cdot v) = d (\frac{1}{2} m v^{2})$ Integrate karo: $\int F \cdot d r = Δ K$ . Same theorem, dot product direction ko automatically handle kar leta hai.

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

Worked Examples

Common Mistakes (Steel-manned)

Recall checkpoints

Recall Feynman: ek 12-saal ke bachche ko explain karo

Socho tum ek toy car push kar rahe ho. Jitna zyada aur jitni der push karo (woh hai "work"), utna fast woh end mein jaayega (woh hai "motion ki energy"). Agar tum usse pakad ke ulta push karo, tum use slow kar dete ho — tum "negative work" kar rahe ho aur uski speed chura rahe ho. Theorem bas yahi kehta hai: tumne poori distance mein jo bhi pushing ki, woh sab add karke exactly barabar hai car ki go-fast energy mein kitna change aaya. Kuch bhi nahi kho jaata; yeh bas motion ke liye bookkeeping hai.

Flashcards

Work–Energy Theorem ko words mein state karo.

Ek particle par kiya gaya net work uski kinetic energy mein change ke barabar hota hai:

W_{n e t} = Δ K

Kaun sa single calculus trick

F = ma

ko work–energy theorem mein convert karta hai?

\frac{d v}{d t} d x = \frac{d x}{d t} d v = v d v

likhna, time ko eliminate karna.

Kinetic energy kya hai aur yeh scalar kyun hai?

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

; yeh speed squared par depend karta hai, velocity ki direction par nahi.

W_{n e t} = Δ K

mein kaun si forces count hoti hain?

SAARI forces — net force (applied, friction, gravity, normal, etc.).

W = F d

kab valid hai

W = \int F \cdot d r

ki jagah?

Sirf jab force constant aur displacement ke parallel ho.

Ek block constant velocity par drag kiya jaata hai.

W_{n e t}

kya hai?

Zero, kyunki

Δ K = 0

(tumhara work friction ke negative work se cancel ho jaata hai).

Work negative kyun ho sakta hai?

Jab force ka displacement ke opposite component ho,

F \cdot d r < 0

, kinetic energy remove hoti hai.

Force

F (x) = 6 x

ek mass par 0 se 2 m tak act karta hai. Net work?

\int_{0}^{2} 6 x d x = 12 J

Theorem yahan

F = ma

se zyada convenient kyun hai?

Yeh force-over-distance ko directly speed se link karta hai, acceleration aur time ko skip karke.

Connections

Newton's Second Law — derivation ka starting point.
Kinetic Energy — woh quantity jo change hoti hai.
Work done by a variable force — kyun integral form ki zaroorat hai.
Conservation of Mechanical Energy — woh jo milta hai jab forces conservative hain.
Power — work karne ki rate, $P = \frac{d W}{d t} = F \cdot v$ .
Impulse–Momentum Theorem — time-integrated sibling ( $\int F d t = Δ p$ ) vs. yeh distance-integrated wala.