1.3.7
Kuch forces honest bankers hote hain: block ko hill pe push karo, wapis slide karo, aur gravity har joule wapas de deti hai jo tumne kharch ki. Ye conservative forces hain. Friction aur air drag thieves with a one-way door hain: ye hamesha mechanical energy lete hain aur use heat/sound mein convert karte hain, kabhi wapas nahi dete, chahe tum kisi bhi direction mein move karo. Defining feature: jo work ye karte hain wo liye gaye PATH pe depend karta hai, sirf start aur end points pe nahi .
Definition Conservative vs Non-conservativeEk force conservative hai agar wo kisi bhi closed loop mein jo work kare wo zero ho:
∮ F ⋅ d r = 0 W potential energy U
Ek force non-conservative hai agar ∮ F ⋅ d r = 0 koi potential energy function exist nahi karta . Friction aur air drag iske headline examples hain.
Friction/drag loop test kyun fail karte hain? Kyunki ye hamesha motion ko oppose karte hain (F v ghata deta hai. Loop integral strictly negative hota hai, kabhi zero nahi.
Ye path pe kyun depend karta hai: L total distance travelled hai, displacement nahi. A se B tak lambe winding route se jao → bada L hai non-conservative signature.
Worked example Block sliding phir wapas aana — loop testEk block ko rough floor pe distance d N = m g
Forward: W 1 = − μ k m g d
Backward: W 2 = − μ k m g d
Closed loop mein total: W = − 2 μ k m g d = 0
Ye step kyun? Ek conservative force return pe + μ m g d velocity ke saath reverse karti hai — toh ye kabhi cancel nahi ho sakti.
Intuition Terminal velocity = drag ki bookkeepingGirte waqt, drag speed ke saath badhta hai jab tak gravity balance na ho jaye. Net force = 0 m g = 2 1 C d ρ A v t 2 ⇒ v t = C d ρ A 2 m g saara gravitational PE seedha air ko heat karne mein chala jaata hai — KE aur nahi badhti.
Worked example Block rough incline se neechaBlock mass m L θ μ k
Δ U = − m g L sin θ L sin θ W n c = − μ k N L = − μ k ( m g cos θ ) L W n c = Δ K E + Δ U − μ k m g cos θ L = 2 1 m v 2 − 0 + ( − m g L sin θ ) v = 2 g L ( sin θ − μ k cos θ ) Ye step kyun? Gravity se release hua energy (m g L sin θ μ k ≥ tan θ ≤ 0
Worked example Bullet ek block mein (friction + sticking)Ek 10 g bullet 400 m/s pe 2 kg block mein embed hoti hai, jo phir rough floor pe 0.5 m slide karta hai (μ k = 0.4 μ
Step 1 (momentum, NOT energy): 0.01 ( 400 ) = ( 2.01 ) v ⇒ v = 1.99 Kyun? Collision bhi non-conservative hai — KE lost hoti hai; yahan sirf momentum conserved hai.Step 2 (friction isko rokti hai): W n c = Δ K E − μ k ( 2.01 ) ( 9.8 ) ( 0.5 ) = 0 − 2 1 ( 2.01 ) ( 1.99 ) 2 μ k ( 2.01 ) ( 9.8 ) ( 0.5 ) = 3.96 2 1 ( 2.01 ) ( 1.99 ) 2 = 3.98 Friction se generate hua heat Q = 3.98
Common mistake Classic errors ko steel-man karnaMistake 1: "Friction work displacement use karta hai." Kyun sahi lagta hai: baaki har work formula displacement d Fix: friction path ke along direction reverse karta hai, toh tumhe f k total path length L pe integrate karna hoga, net displacement pe nahi. Round trip mein zero displacement hota hai lekin nonzero friction work hota hai.
Mistake 2: "Collision mein energy conservation use karo." Kyun sahi lagta hai: energy "hamesha conserved" hoti hai. Fix: mechanical energy inelastic collisions/friction mein conserved nahi hoti — collisions ke liye momentum use karo, phir sliding phase ke liye energy use karo.
Mistake 3: "Drag friction jaisa constant hota hai." Kyun sahi lagta hai: dono motion ko oppose karte hain. Fix: drag speed ke saath badhta hai (∝ v v 2 F d − F d L
Mistake 4: "Heat lost = Δ U Fix: heat = − W n c = Δ K E + Δ U
Recall Feynman: ek 12-saal ke bachche ko explain karoSocho ek toy car carpet pe slide karo. Gravity ek fair friend hai: agar tum car ko ramp pe lift karo aur wapas roll karne do, tumhe saari mehnat wapas mil jaati hai. Lekin carpet ek sneaky pickpocket hai — har centimetre pe wo thodi energy chura leta hai aur use warmth mein badal deta hai (apne haath tez tez ragdo — wo garam ho jaate hain!). Koi baat nahi agar tum aage jao ya peeche; carpet hamesha churaata hai. Toh agar tum car ko bade circle mein push karo aur start pe wapas aao, gravity sab kuch wapas deti hai (zero net), lekin carpet ne trip ke har part pe thoda liya — wo wapas nahi milega. Wo stolen energy gayab nahi hua; wo heat ban gaya. Air ek girte skydiver ke saath yahi karta hai: jitna tez wo girte hain, utna hi air pushback karta hai, jab tak wo speed up karna band nahi kar dete — yahi terminal velocity hai.
"Friction Pays a Path-Toll; Conservatives Pay only the Endpoints."
Non-conservative ⇒ N ever returns, C for C losed loop ≠ 0. Sign rule: W f r i c − hota hai.
Non-conservative force mathematically kya define karta hai? Closed loop mein work nonzero hota hai,
∮ F ⋅ d r = 0 ; koi potential energy define nahi ki ja sakti; work path-dependent hota hai.
Path length L pe kinetic friction dwara kiya gaya work? W f r i c = − μ k N L
Friction ki potential energy kyun nahi ho sakti? Iska work path/total distance pe depend karta hai, sirf endpoints pe nahi, toh U
Non-conservative forces ke saath modified energy equation? W n c = Δ K E + Δ U = Δ E m ec h
"Lost" mechanical energy kahan jaata hai? Heat (aur sound) mein: Q = − W n c = μ k N L
Do drag-force models aur unke regimes? Low speed F d = b v F d = 2 1 C d ρ A v 2
Terminal velocity formula (quadratic drag)? v t = 2 m g / ( C d ρ A ) , set
m g = drag.
Rest se rough incline se L neecha slide karne wale block ki final speed? Incline pe block ke slide karne ki condition? tan θ > μ k
Collisions mein momentum (energy nahi) kyun use karte hain? Collisions non-conservative hote hain — KE lost hoti hai — lekin momentum hamesha conserved hota hai agar koi external impulse na ho.
Closed loop work not zero
No potential energy exists
Low speed drag linear in v
High speed drag v squared
