Conservation of mechanical energy — derivation

1.3.8 · Physics › Work, Energy & Power

KYA prove kar rahe hain hum?

Claim yeh hai: Agar sirf conservative forces hi work karti hain, tab $E = K + U$ time mein constant rehti hai.

YEH sach kyun hona chahiye? (proof ka engine)

Do facts saara kaam karte hain:

Work–energy theorem (Newton ke 2nd law se): net work = kinetic energy mein change.
Potential energy ki definition: conservative force dwara kiya gaya work minus uski potential energy ka change hota hai.

Inhe side by side rakho aur result khud nikal aata hai. Chalte hain, har piece ko scratch se derive karte hain.

KAISE — First principles se Derivation

Step 1 — Work–energy theorem

Newton ke second law se ek dimension mein shuru karo, $F_{n e t} = ma = m \frac{d v}{d t}$ .

Object jab $d x$ displacement karta hai to net work: $d W_{n e t} = F_{n e t} d x = m \frac{d v}{d t} d x$

Toh: $d W_{n e t} = m v d v$

State 1 ( $v_{1}$ ) se State 2 ( $v_{2}$ ) tak integrate karo: $W_{n e t} = \int_{v_{1}}^{v_{2}} m v d v = \frac{1}{2} m v_{2}^{2} - \frac{1}{2} m v_{1}^{2} = Δ K$

Step 2 — Conservative force se Potential energy

Ek conservative force ke liye, hum potential energy is tarah define karte hain ki uska kiya hua work potential energy ki drop ke barabar ho: $W_{co n s} = - Δ U = - (U_{2} - U_{1})$

Earth ke paas gravity ke liye, $F = - m g$ (neeche). Height $h_{1}$ se $h_{2}$ tak move karne par uska work: $W_{g r a v} = \int_{h_{1}}^{h_{2}} (- m g) d h = - m g (h_{2} - h_{1}) = - (m g h_{2} - m g h_{1})$ $W_{co n s} = - Δ U$ se compare karne par jaana-maana $U = m g h$ milta hai. Humne ise derive kiya, assume nahi kiya.

Step 3 — Combine karo

Agar work karne wali sirf conservative force hai, tab $W_{n e t} = W_{co n s}$ . Step 1 ko Step 2 ke barabar rakho: $Δ K = W_{n e t} = W_{co n s} = - Δ U$

$Δ K + Δ U = 0 ⟹ Δ (K + U) = 0$

Figure — Conservation of mechanical energy — derivation

Forecast-then-Verify

Recall Answer padhne se pehle forecast karo

Ek ball height $h$ se rest se drop ki jaati hai. Energy conservation use karke zameen par uski speed predict karo, phir kinematics se verify karo.

Energy: $K_{1} + U_{1} = K_{2} + U_{2} \Rightarrow 0 + m g h = \frac{1}{2} m v^{2} + 0 \Rightarrow v = 2 g h$ . Kinematics: $v^{2} = u^{2} + 2 g h = 0 + 2 g h \Rightarrow v = 2 g h$ . ✓ Same answer — energy method ne time ko bilkul skip kar diya.

Worked examples

Common mistakes (Steel-manned)

Recall Feynman: 12-saal ke bacche ko explain karo

Energy ko pocket money ki tarah socho jo tum do pockets mein rakh sakte ho: ek "moving" pocket (kinetic) aur ek "stored-up" pocket (potential — jaise upar hona ya dabaa hua spring). Jab ball girती hai, paisa "high-up" pocket se "moving" pocket mein slide hota hai — lekin tumhari dono pockets mein total paisa kabhi nahi badlta. Paisa kho sirf tab sakta hai jab ek chipku chor jiska naam friction hai kuch pakad le aur use heat mein badal de. Friction nahi, chor nahi → tumhara total hamesha exactly same rehta hai.

Flashcards

Mechanical energy kya hoti hai?

Kinetic aur potential energy ka sum,

E = K + U

Work–energy theorem state karo.

Kisi object par kiya gaya net work uski kinetic energy ke change ke barabar hota hai,

W_{n e t} = Δ K

Conservative force se potential energy kaise define hoti hai?

Us force dwara kiye gaye work ke negative ke roop mein:

W_{co n s} = - Δ U

Mechanical energy kab conserved hoti hai?

Jab sirf conservative forces hi work karti hain (friction jaise non-conservative forces se koi net work nahi).

Energy use karke height

h

se girae gaye object ki

v

derive karo.

m g h = \frac{1}{2} m v^{2} \Rightarrow v = 2 g h

W_{co n s} = - Δ U

mein minus sign kyun aata hai?

Taaki force dwara kiye gaye work aur stored PE ke change ke opposite signs hon (force +work kare ⇒ PE gire).

Yahan energy conservation kinematics se aasaan kyun hai?

Yeh time aur force direction ko eliminate kar deta hai; sirf do states compare karte hain.

Friction hone par

Δ E = 0

ki jagah kya aata hai?

Δ (K + U) = W_{n c}

, non-conservative forces dwara kiya gaya work (friction ke liye negative).

String tension pendulum bob par koi work kyun nahi karti?

Yeh hamesha bob ki velocity ke perpendicular hoti hai, toh

F \cdot d = 0

Spring potential energy formula aur uska derivation source?

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

- \int_{0}^{x} (- k x^{'}) d x^{'}

se.

Connections

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