Non-conservative forces — friction, air drag: Visual walkthrough

1.3.7 · D2 · Physics › Work, Energy & Power › Non-conservative forces — friction, air drag

Step 1 — "Work" hota kya hai (do arrows ka dot)

Har symbol kya karta hai, theek wahan jahan woh baitha hai:

$∣ F ∣$ — force arrow ki lambai (newtons mein).
$∣ d r ∣$ — step ki lambai (metres mein).
$ϕ$ — do arrows ke beech ka angle. Yahi poori kahani hai.
$cos ϕ$ — poochh raha hai "force ka kitna hissa motion ki direction mein hai?"

Kyun dot product, plain multiplication kyun nahi? Kyunki force ka sirf woh hissa jo step ke saath hai woh energy transfer karta hai. Aapki motion ke sideways wala force (jaise string se ball ko ghoomana) zero work karta hai — aur $cos 9 0^{\circ} = 0$ exactly yahi deliver karta hai. Hume ek aisa tool chahiye jo automatically aligned hissa rakhe aur sideways hissa fenk de; dot product wahi tool hai.

PICTURE:

Figure — Non-conservative forces — friction, air drag

Teen cases dekho. Jab $ϕ = 0^{\circ}$ (force step ke saath) $cos ϕ = + 1$ : poora positive work. Jab $ϕ = 9 0^{\circ}$ : zero. Jab $ϕ = 18 0^{\circ}$ (force step ke khilaaf) $cos ϕ = - 1$ : poora negative work — energy nikal gayi. Woh aakhri case yaad rakhna; friction wahan hamesha rehta hai.

Step 2 — Friction hamesha $ϕ = 18 0^{\circ}$ par kyun rehta hai

Har symbol kya karta hai:

$μ_{k}$ — coefficient of kinetic friction, ek pure number (koi units nahi) jo batata hai surfaces kitni "grippy" hain.
$N$ — normal force, surface kitna zor se wapas dabaata hai (newtons mein); yeh friction ka size set karta hai.
$\overset{v}{^}$ — unit velocity vector: length $1$ ka ek arrow jo exactly usi direction mein point karta hai jis taraf object move kar raha hai.
Minus sign — friction $\overset{v}{^}$ ke opposite point karta hai. Hamesha. Yahi friction ki poori personality hai.

Work ke liye yeh kyun matter karta hai: tiny step $d r$ motion ki direction mein point karta hai, yaani $\overset{v}{^}$ ke along. Friction $- \overset{v}{^}$ ke along point karta hai. Toh friction aur har step ke beech angle $ϕ$ hamesha $18 0^{\circ}$ hota hai — chahe aap kisi bhi taraf jao, chahe path kitna bhi curve kare.

PICTURE:

Curved path follow karo. Har point par cyan velocity arrow aur amber friction arrow exactly opposite point karte hain. Path ko ulta karo, aur friction bhi flip kar leta hai opposite rehne ke liye. Yeh kabhi aapki help nahi kar sakta.

Step 3 — Saare tiny steps ko add karna (the integral)

Har naya piece kya karta hai:

$\int$ — "saare tiny steps ko add karo"; yeh bas ek bahut lamba sum hai.
$∣ d r ∣$ — ek tiny step ki lambai (hamesha ek positive number).
$L = \int ∣ d r ∣$ — total path length, odometer reading, saari step-lengths ka sum.

$L$ kyun, displacement kyun nahi? Kyunki humne $∣ d r ∣$ sum kiya — har step ki lambai — kabhi cancellation nahi hone diya. East ki taraf ek step aur baad mein west ki taraf ek step dono positive length add karte hain. Yahi exactly reason hai ki friction path-dependent kyun hai: zyada lamba, tedha-medha raasta zyada $L$ pile up karta hai, isliye zyada energy chori hoti hai.

PICTURE:

A se B tak do routes: ek chhoti seedhi line aur ek lamba detour. Same endpoints, lekin detour ka odometer $L$ bada hai — toh uska friction bill $- μ_{k} N L$ bhi bada hai. Endpoints cost decide nahi karte; path karta hai.

Step 4 — Closed-loop test (definition, visible banaake)

Kya ho raha hai: wapas aate waqt velocity reverse hoti hai, toh friction bhi reverse ho jaata hai — woh phir bhi motion ka oppose karta hai, toh woh phir bhi subtract karta hai. Do negatives, cancelling pair nahi.

Yeh definition kyun hai non-conservative ki: ek force conservative tab hota hai jab $\oint F \cdot d r = 0$ (kisi bhi closed loop mein zero net work). Friction strictly negative loop deta hai, isliye woh non-conservative hai aur ==isko koi potential energy $U$ assign nahi ki ja sakti==.

PICTURE:

Gravity se compare karo (upar): bahar jaane par $- m g h$ cost hai, wapas aane par $+ m g h$ milta hai, loop zero sum karta hai — ek imaandar banker. Friction (neeche): aage $- μ_{k} m g d$ , wapas $- μ_{k} m g d$ , loop $- 2 μ_{k} m g d$ sum karta hai — ek one-way chor.

Step 5 — Work ko energy statement mein badalna

Har symbol kya karta hai:

$K E$ — kinetic energy, motion ki energy, $\frac{1}{2} m v^{2}$ .
$Δ K E = K E_{f ina l} - K E_{ini t ia l}$ — motion-energy kitni badli.
$W_{t o t a l}$ — har force ka kaam milakar.

Yahan se kyun shuru karein: yeh theorem Newton's second law ka ek re-labelling hai; yeh kisi bhi force ke liye kaam karta hai, imaandar ho ya chor. Yeh woh neenv hai jis par hum khade hain. Isse hum friction result ko saaf saaf nikalenge.

PICTURE:

Bar chart: total work andar $\to$ kinetic energy utni hi badhti hai. Abhi tak kuch kho nahi gaya — humne sirf accountant ka pehla rule bataya hai.

Work–Energy theorem dekho is neenv ke fuller treatment ke liye.

Step 6 — Forces ko imaandar aur chor mein baantna

$U$ kya hai? Potential energy — stored energy jo ek conservative force (jaise gravity) poori wapas de deta hai. Gravity ke liye $U = m g h$ . Conservative forces & potential energy dekho.

$W_{co n s} = - Δ U$ mein minus sign kyun? Jab gravity positive work karta hai (object girta hai), stored height-energy girti hai — toh $Δ U$ negative hota hai. Positive work ↔ falling $U$ : signs opposite hone chahiye. Yahi exactly $U$ ko define karta hai.

PICTURE:

Total work do buckets mein split hota hai: conservative bucket ( $- Δ U$ se rename kiya, reversible) aur non-conservative bucket $W_{n c}$ (the leak).

Step 7 — Master equation saamne aati hai

Har term kya karta hai:

$Δ K E + Δ U = Δ E_{m ec h}$ — total mechanical energy ka change (motion + stored).
$W_{n c}$ — chors ne kya kiya; friction ke liye yeh $- μ_{k} N L$ hai, hamesha negative.

Yeh poora point kyun hai: kyunki $W_{n c} < 0$ hai, mechanical energy $E_{m ec h}$ zaroor ghateghi. Woh gayab nahi hui — woh heat ban gayi (dekho Heat & first law of thermodynamics): $Q = - W_{n c} = μ_{k} N L$

PICTURE:

Sankey-style flow: energy andar aati hai, ek slice $μ_{k} N L$ heat/sound ke roop mein nikal jaata hai, aur sirf baacha hua hissa usable mechanical energy rehta hai. Yeh Mechanical energy conservation hai jisme uska imaandaar leak draw kiya gaya hai.

Step 8 — Edge cases (reader ko koi na-dikha scenario nahi milna chahiye)

Us bracket $(sin θ - μ_{k} cos θ)$ mein chhupe teen cases:

Case A — $tan θ > μ_{k}$ : bracket positive → real speed, block accelerate karta hai. (Steep ya slippery.)
Case B — $tan θ = μ_{k}$ : bracket zero → $v = 0$ throughout; gravity ka pull exactly friction ki chori ke barabar hai, block constant speed par slide karta hai agar nudge diya jaye.
Case C — $tan θ < μ_{k}$ : bracket negative → negative ka square root → koi motion nahi. Block shuru hi nahi hota (consistent with Friction — static & kinetic).

Har door band karne ke liye degenerate limits:

$θ = 0$ (flat floor): $sin θ = 0$ , bracket $= - μ_{k} < 0$ → koi self-starting motion nahi. Sahi hai.
$θ = 9 0^{\circ}$ (vertical): $cos θ = 0$ , friction term gayab, $v = 2 g L$ — pure free-fall. Sahi hai.
$μ_{k} = 0$ (frictionless): $v = 2 g L sin θ$ — imaandar-banker ka jawaab, saari PE → KE.

PICTURE:

Teen incline diagrams side by side. Bracket ka sign dekho kaise flip hota hai jab $θ$ us critical angle se gujarta hai jahan $tan θ = μ_{k}$ — amber friction arrow, cyan gravity-along-slope arrow ko overtake kar leta hai.

Ek-picture summary

Sab kuch ek blueprint par: ek step $d r$ , friction $18 0^{\circ}$ par oppose karta hua, odometer $L$ pile up hota hua, closed loop jo zero hone se refuse karta hai, aur energy ledger heat $Q = μ_{k} N L$ mein drain hota hua.

Recall Feynman: poora walkthrough simple words mein

Work bas "ek push motion ki kitni help karta hai" — jab push aur motion agree karein toh full credit, jab ladte hain toh full debit. Friction duniya ka sabse zidd contrarian hai: woh hamesha backwards point karta hai, jahan bhi tum ja rahe ho uske ulta. Toh har tiny step par, woh tumse charge karta hai. Un saare chhote charges ko poore safar mein add karo aur bill hai odometer reading times grip: $μ_{k} N L$ . Kyunki woh har step par charge karta hai — wapas ki trip par bhi — ek round trip jo wahan khatam ho jahan shuru hua tha phir bhi tumhe poorer chhod jaata hai; yahi closed-loop test hai, aur isliye friction kabhi energy store nahi kar sakta wapas dene ke liye. Iske saath hi physics accountant ka sabse purana rule kehta hai total work motion-energy ke change ke barabar hai. Humne sirf "total work" ko imaandar hisse mein (jise hum minus-change-in-stored-energy rename karte hain) aur chor hisse mein split kiya. Terms ko shuffle karo aur headline saamne aa jaati hai: chor ka work tumhari total mechanical energy ke change ke barabar hai — aur kyunki chor sirf leta hai, woh total sirf gir sakta hai. Missing joules gaye nahi; apne haath ragdo aur feel karo woh kahan gaye: heat.

Connections

1.3.07 Non-conservative forces — friction, air drag (Hinglish) — parent topic, saath mein padho.
Work–Energy theorem — Step 5 ki neenv.
Conservative forces & potential energy — jahan se Step 6 ka $- Δ U$ aata hai.
Mechanical energy conservation — yeh page wahi law hai with a leak.
Friction — static & kinetic — $tan θ$ vs $μ_{k}$ starting condition.
Terminal velocity & projectile with drag — drag ka same energy leak wala version.
Inelastic collisions — ek aur non-conservative sink (energy nahi, momentum use karo).
Heat & first law of thermodynamics — chori ki $Q$ kahan jaati hai.