Non-conservative forces — friction, air drag: Question bank

Before the questions, three anchors so every symbol below is already earned:

Work $W = F \cdot d r$ means "force times the distance moved along the force." Opposing force ⇒ negative work.
Mechanical energy $E_{m ec h} = K E + U$ : motion energy plus stored (height/spring) energy.
Non-conservative = a force whose total work depends on the path length travelled, not just where you start and end. See Work–Energy theorem and Conservative forces & potential energy.

True or false — justify

Friction always does negative work on a moving object.

False. On the surface doing the sliding it is negative, but friction can do positive work on a passenger carried forward by a decelerating truck, or on the block under your hand — it's the relative-motion direction that decides the sign, not the word "friction."

A round trip on a rough floor has zero net friction work because displacement is zero.

False. Friction work uses ==total path length

L

==, not net displacement. A round trip has

L = 2 d \neq = 0

, so

W_{f r i c} = - μ_{k} N (2 d) < 0

If a force does zero work on one particular closed loop, it must be conservative.

False. Conservative requires

\oint F \cdot d r = 0

for every loop. One accidental zero (e.g. friction on a loop where you never actually move) proves nothing.

Air drag is non-conservative for the same reason friction is.

True. Drag always points opposite to velocity, so on any closed loop it subtracts on every segment — the loop integral is strictly negative, exactly like friction.

At terminal velocity a skydiver's mechanical energy stays constant.

False. Height keeps dropping so

U

falls, while

K E

is constant — total

E_{m ec h}

decreases, all of it going into heating the air. See Terminal velocity & projectile with drag.

Since energy is always conserved, kinetic energy is conserved in a bullet-into-block collision.

False. Total energy is conserved, but kinetic energy is not — much becomes heat and deformation. Only momentum is conserved, so use momentum first. See Inelastic collisions.

Doubling the speed doubles the quadratic drag force.

False. Quadratic drag is

F_{d} = \frac{1}{2} C_{d} ρ A v^{2}

, so doubling

v

quadruples the force. Only linear (Stokes) drag

F_{d} = b v

doubles.

Heat generated by friction equals the loss in potential energy.

False. Heat

= - W_{n c} = Δ K E + Δ U

. It accounts for the change in both KE and U, not U alone. See Heat & first law of thermodynamics.

μ_{k} > tan θ

, a block already sliding down a rough incline will speed up.

False. The bracket

(sin θ - μ_{k} cos θ) < 0

, so gravity's pull along the slope loses to friction — the block decelerates (and would never start from rest).

Static friction can do work but never removes mechanical energy as heat.

True (in the ideal no-slip case). With no relative sliding, no surfaces rub, so no heat is generated; static friction can still transfer energy (e.g. a rolling wheel) without dissipating it. See Friction — static & kinetic.

Spot the error

"Block slides distance

d

and returns; friction did

- μ m g d

forward and

+ μ m g d

back, so net friction work is zero."

Error: friction does not flip sign with displacement direction — it flips to oppose velocity, so the return trip is also

- μ m g d

. Net

= - 2 μ m g d

, the loop-test signature of non-conservative.

"Terminal velocity: I set drag times distance equal to

m g

Error: mixing a force with an energy statement. Terminal velocity comes from balancing forces

m g = \frac{1}{2} C_{d} ρ A v_{t}^{2}

, not from a work equation.

"On the incline example the block loses

m g L sin θ

to friction."

Error: gravity releases

m g L sin θ

; friction steals only

μ_{k} m g cos θ L

. The rest becomes KE. Confusing the released PE with the friction loss.

"I'll use

W = - F_{d} L

for a skydiver falling through the atmosphere."

Error:

F_{d}

is not constant during the fall — it grows as

v

grows. Only at terminal velocity (constant

v

) can you use a constant-force product.

"Since friction is non-conservative, I can't use the work–energy theorem here."

Error: the work–energy theorem

W_{t o t a l} = Δ K E

is always true. Non-conservative forces just mean you can't fold their work into a potential energy

U

"A conservative force must always be zero around a loop, so gravity does no work when I lift a book."

Error: confusing net over a loop with over one leg. Gravity does negative work lifting up and positive coming down; the round-trip sum is zero, but a single leg is not.

Why questions

Why does friction's work depend on path but gravity's doesn't?

Gravity depends only on the height difference (endpoints), so a potential energy

U

exists. Friction opposes motion at every step, so longer/windier paths rub more — total distance, not endpoints, sets the loss.

Why is a potential energy function impossible for friction?

A potential energy would let you assign one number

U

to each position independent of history. But friction's work depends on how you got there (the path length), so no single position-only function can reproduce it.

Why must we use momentum, not energy, in an inelastic collision?

Kinetic energy vanishes into heat/deformation by an unknown amount, so energy bookkeeping has a missing term. Momentum has no such loss (no external impulse), giving a clean, solvable equation.

Why does drag change from linear to quadratic as speed rises?

At low speed viscous shear dominates (momentum loss

\propto v

). At high speed you knock aside a mass of air per second

\propto v

, each parcel carrying momentum

\propto v

, giving

\propto v^{2}

Why is

Q = - W_{n c}

rather than

Q = + W_{n c}

Non-conservative forces do negative work on the system (

W_{n c} < 0

). The heat produced is a positive quantity equal to the energy removed, so

Q = - W_{n c} > 0

Why does a block on an incline need

tan θ > μ_{k}

just to start moving?

The gravity component along the slope is

m g sin θ

and maximum friction is

μ_{s} m g cos θ

. Motion begins only when

sin θ > μ_{s} cos θ

, i.e.

tan θ > μ_{s}

Edge cases

What is friction's work when a block sits still on a rough floor?

Zero — no sliding means no path length is traversed, so

W_{f r i c} = - μ_{k} N L

with

L = 0

. Static friction with no motion dissipates nothing.

What happens to the drag work at

v = 0

(dropped from rest)?

Drag is exactly zero at that instant (

F_{d} \propto v

v^{2}

), so initially the fall is pure free-fall; drag only bites once speed builds. See Terminal velocity & projectile with drag.

On an incline with

μ_{k} = tan θ

exactly, what final speed does the block reach?

The bracket

(sin θ - μ_{k} cos θ) = 0

, so

v = 0

— if already moving it coasts at constant velocity (net force zero); from rest it never starts. A perfect knife-edge.

For a perfectly horizontal round trip (

N = m g

constant), how does the friction loss compare to a hilly round trip of the same displacement?

The hilly trip usually has greater path length and varying

N

, so generally more loss — the point is that displacement alone can't tell you the friction work; you need the actual traversed path.

In the limit of a frictionless, drag-free world, does the modified equation

W_{n c} = Δ E_{m ec h}

still hold?

Yes, trivially:

W_{n c} = 0

, so

Δ E_{m ec h} = 0

— mechanical energy is conserved, recovering Mechanical energy conservation as the special case.

If drag opposed velocity but grew smaller with speed, could it still be non-conservative?

Yes — non-conservativeness comes from the force always opposing motion (loop integral negative), not from how the magnitude scales with

v

. The direction rule alone forbids a potential energy.

Recall One-line reflex checks

Friction work uses path length not displacement ::: because it re-opposes velocity on every leg. Collision first move ::: conserve momentum, never kinetic energy. Heat produced ::: $Q = - W_{n c} = Δ K E + Δ U$ . Block starts sliding when ::: $tan θ > μ_{s}$ .

Connections

Parent topic — full derivations these traps test.
Work–Energy theorem · Mechanical energy conservation · Conservative forces & potential energy — the honest counterparts.
Friction — static & kinetic · Terminal velocity & projectile with drag · Inelastic collisions · Heat & first law of thermodynamics — where these traps recur.