Non-conservative forces — friction, air drag

WHAT is a non-conservative force?

WHY do friction/drag fail the loop test? Because they always oppose motion ( $\vec{F}$ points opposite to $\vec{v}$ ). On a closed loop you move in every direction, but the force flips to oppose you each time — so every segment subtracts energy. The loop integral is strictly negative, never zero.

HOW friction removes energy (derivation from scratch)

Why this depends on path: $L$ is the total distance travelled, not displacement. Go from A to B by a long winding route → larger $L$ → more energy lost. That path-dependence is the non-conservative signature.

HOW air drag works

Figure — Non-conservative forces — friction, air drag

The Work–Energy theorem with non-conservative forces

Formula Modified energy conservation — derived

Start from the full work–energy theorem (always true): $W_{total} = \Delta KE$ Split total work into conservative + non-conservative parts: $W_{cons} + W_{nc} = \Delta KE$ By definition $W_{cons} = -\Delta U$ . Substitute: $-\Delta U + W_{nc} = \Delta KE \;\Rightarrow\; \boxed{W_{nc} = \Delta KE + \Delta U = \Delta E_{mech}}$ The non-conservative work equals the change in total mechanical energy. Since friction/drag give $W_{nc}<0$ , mechanical energy steadily decreases — the "lost" amount becomes heat $Q = -W_{nc} = \mu_k N L$ .

Worked example Block down a rough incline

Block mass $m$ , slides distance $L$ down incline angle $\theta$ , $\mu_k$ . Find final speed from rest.

$\Delta U = -mgL\sin\theta$ (drops height $L\sin\theta$ )
$W_{nc} = -\mu_k N L = -\mu_k (mg\cos\theta) L$
Apply $W_{nc} = \Delta KE + \Delta U$ : $-\mu_k mg\cos\theta\, L = \tfrac{1}{2}mv^2 - 0 + (-mgL\sin\theta)$ $v = \sqrt{2gL(\sin\theta - \mu_k\cos\theta)}$ Why this step? Energy released by gravity ( $mgL\sin\theta$ ) is split: part becomes KE, part stolen by friction. If $\mu_k \geq \tan\theta$ , the bracket $\leq 0$ → block never starts/keeps moving. Beautifully consistent with statics!

Worked example Bullet into a block (friction + sticking)

A 10 g bullet at 400 m/s embeds in a 2 kg block, which then slides 0.5 m on rough floor ( $\mu_k = 0.4$ ) before stopping. Find $\mu$ consistency / heat.

Step 1 (momentum, NOT energy): $0.01(400) = (2.01)v \Rightarrow v = 1.99$ m/s. Why? Collision is non-conservative too — KE is lost; only momentum is conserved here.
Step 2 (friction stops it): $W_{nc} = \Delta KE$ : $-\mu_k(2.01)(9.8)(0.5) = 0 - \tfrac12(2.01)(1.99)^2$ . Check: $\mu_k(2.01)(9.8)(0.5)=3.96$ J vs $\tfrac12(2.01)(1.99)^2 = 3.98$ J ✔.
Heat generated by friction $Q = 3.98$ J (plus huge heat in the embedding collision).

Common mistake Steel-manning the classic errors

Mistake 1: "Friction work uses displacement." Why it feels right: every other work formula uses displacement $\vec{d}$ . The fix: friction reverses direction along the path, so you must integrate $f_k$ over total path length $L$ , not net displacement. A round trip has zero displacement but nonzero friction work.

Mistake 2: "Use energy conservation in a collision." Why it feels right: energy is "always conserved." The fix: mechanical energy is not conserved in inelastic collisions/with friction — use momentum for collisions, then energy for the sliding phase.

Mistake 3: "Drag is constant like friction." Why it feels right: both oppose motion. The fix: drag grows with speed ( $\propto v$ or $v^2$ ), so you can't write $-F_d L$ with constant $F_d$ unless speed is constant (e.g. at terminal velocity).

Mistake 4: "Heat lost = $\Delta U$ ." Fix: heat $= -W_{nc} = \Delta KE + \Delta U$ , accounting for KE too.

Recall Feynman: explain to a 12-year-old

Imagine sliding a toy car across a carpet. Gravity is a fair friend: if you lift the car up a ramp and let it roll back, you get all your effort back. But the carpet is a sneaky pickpocket — every centimetre it rubs, it steals a little energy and turns it into warmth (rub your hands fast — they get hot!). It doesn't matter if you go forward or back; the carpet always steals. So if you push the car in a big circle and return to start, gravity gives you everything back (zero net), but the carpet took a bit on every part of the trip — you can't get that back. That stolen energy didn't vanish; it became heat. Air does the same to a falling skydiver: the faster they fall, the harder air pushes back, until they stop speeding up — that's terminal velocity.

Flashcards

What defines a non-conservative force mathematically?

Work around a closed loop is nonzero,

\oint \vec{F}\cdot d\vec{r}\neq 0

; no potential energy can be defined; work is path-dependent.

Work done by kinetic friction over path length L?

W_{fric}=-\mu_k N L

(negative, uses total path length not displacement).

Why can't friction have a potential energy?

Its work depends on the path/total distance, not just endpoints, so

U

is undefined.

Modified energy equation with non-conservative forces?

W_{nc}=\Delta KE+\Delta U=\Delta E_{mech}

Where does the "lost" mechanical energy go?

Into heat (and sound):

Q=-W_{nc}=\mu_k N L

Two drag-force models and their regimes?

Low speed

F_d=bv

(viscous/laminar); high speed

F_d=\tfrac12 C_d\rho A v^2

(turbulent).

Terminal velocity formula (quadratic drag)?

v_t=\sqrt{2mg/(C_d\rho A)}

, set

mg=

drag.

Final speed of block sliding L down rough incline from rest?

v=\sqrt{2gL(\sin\theta-\mu_k\cos\theta)}

Condition for block to slide at all on incline?

\tan\theta>\mu_k

(else friction bracket non-positive).

Why use momentum (not energy) in collisions?

Collisions are non-conservative — KE is lost — but momentum is always conserved if no external impulse.

Connections

Conservative forces & potential energy — the mirror concept; loop integral = 0.
Work–Energy theorem — the parent law $W_{total}=\Delta KE$ used here.
Mechanical energy conservation — the special case when $W_{nc}=0$ .
Friction — static & kinetic — source of $f_k=\mu_k N$ .
Terminal velocity & projectile with drag — application of drag models.
Inelastic collisions — another non-conservative energy sink.
Heat & first law of thermodynamics — where the lost energy ends up.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho yaar, forces do tarah ke hote hain. Gravity jaisa conservative force ek imandar dost hai — block ko upar le jao aur wapas niche aane do, jitni energy lagayi thi utni poori wapas mil jaati hai. Lekin friction aur air drag thodi sneaky cheezein hain — ye non-conservative hain. Inka kaam hamesha motion ke against hota hai, isliye jis bhi raaste se chalo, ye energy chura kar heat mein badal dete hain, aur ye energy kabhi wapas nahi milti. Sabse important baat: friction ka kaam path par depend karta hai, sirf start aur end point par nahi.

Iska sabse mast proof hai "closed loop test". Block ko aage dhakelo $d$ distance aur wapas le aao same jagah. Displacement to zero ho gaya, par friction ne aage bhi energy churaayi aur peeche bhi — total loss $-2\mu_k mg d$ , zero nahi! Yahi non-conservative ki pehchaan hai. Formula yaad rakho: $W_{fric}=-\mu_k N L$ , jahan $L$ total distance hai, displacement nahi. Yeh galti sabse common hai exams mein.

Energy ka master equation banta hai: $W_{nc}=\Delta KE+\Delta U=\Delta E_{mech}$ . Matlab friction/drag ka kaam directly bata deta hai ki mechanical energy kitni ghati. Jo energy "kho gayi" woh actually heat ban gayi: $Q=-W_{nc}=\mu_k N L$ . Energy destroy nahi hoti, sirf form change karti hai.

Air drag mein ek aur cheez special hai — yeh speed ke saath badhta hai ( $F_d=bv$ slow speed par, $\tfrac12 C_d\rho A v^2$ fast speed par). Isliye girte hue object ki speed ek point par constant ho jaati hai — terminal velocity — jab drag gravity ko balance kar deta hai. Skydiver ki yahi kahani hai. Aur haan, collisions mein hamesha momentum use karo energy nahi, kyunki collision bhi ek non-conservative process hai jahan KE lost ho jaati hai.

Non-conservative forces — friction, air drag

WHAT is a non-conservative force?

HOW friction removes energy (derivation from scratch)

HOW air drag works

The Work–Energy theorem with non-conservative forces

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Work, Energy & Power

Connections