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Friction & Drag

Source lecture(s): SC133 Lec 7

Intuition

Real surfaces grip and real fluids resist. Friction is the sideways force between touching surfaces — microscopic welds forming and breaking. Drag is the fluid's resistance to a body pushing through it. Both oppose relative sliding, both convert orderly kinetic energy into heat, and both are why perpetual motion looks so plausible on ice and so absurd in honey.

Friction: static vs kinetic

Static friction holds surfaces still. It is adjustable: it matches whatever force tries to slide the object, up to a ceiling —

\[f_s \leq \mu_s N\]

Kinetic friction acts once sliding begins, roughly constant:

\[f_k = \mu_k N\]

with \(\mu_k < \mu_s\) (breaking free is harder than keeping going — the reason wheels lock and ABS exists). Both are proportional to the normal force \(N\), not to weight and not to contact area.

Worked example: block on an incline

A block rests on a slope of angle \(\theta\). When does it slip?

Along the incline: gravity component \(mg\sin\theta\) vs static friction \(\leq \mu_s N = \mu_s mg\cos\theta\). Slipping starts when

\[\tan\theta_c = \mu_s\]

— measuring the critical angle is measuring \(\mu_s\). For \(\mu_s = 0.6\): \(\theta_c \approx 31°\).

Drag and terminal velocity

At everyday speeds in air, drag grows with the square of speed:

\[D = \tfrac12 C \rho A v^2\]

(\(\rho\) air density, \(A\) cross-section, \(C\sim0.5\)–1 drag coefficient). A falling body accelerates until drag balances weight — terminal velocity:

\[v_t = \sqrt{\frac{2mg}{C\rho A}}\]

Skydiver spread-eagle: \(\sim 60\,\text{m/s}\); head-down: \(\sim 90\,\text{m/s}\); raindrop: \(\sim 7\,\text{m/s}\) (a mercy — pea-sized hail at free-fall-from-cloud speed would be lethal). Because \(v_t \propto \sqrt{m/A}\), small creatures fall slowly: an ant reaches harmless terminal velocity almost instantly.

Physical interpretation

Friction converts kinetic energy to thermal energy at rate \(f_k v\) — the energy bookkeeping still balances, but mechanical energy alone does not. Drag at high speeds is the first taste of fluid mechanics; the quadratic law comes from the momentum the body must give the fluid it shoves aside, and the dimensionless story behind \(C\) is the Reynolds number.

Common mistakes

  • Using \(f_s = \mu_s N\) always. That's the maximum; static friction is usually smaller — exactly what's needed to prevent sliding, no more.
  • Friction always opposes motion? It opposes relative slipping. Friction is what pushes a car forward (tires grip road) and what makes walking possible.
  • \(N = mg\) reflexively — false on inclines and whenever other vertical forces act.
  • Using constant-acceleration kinematics with drag. Drag depends on \(v\), so \(a\) changes continuously; you need calculus or numerics (PHY653 knows how).

Knowledge graph position

Prerequisites: Newton's laws. Leads to: Work & energy (dissipation), fluid mechanics (PC316).

Quiz

Q1 (computational). A 10 kg crate needs 49 N to start moving and 39 N to keep moving at constant velocity. Find \(\mu_s\) and \(\mu_k\).

Answer

\(N = mg = 98\,\text{N}\). \(\mu_s = 49/98 = 0.5\); \(\mu_k = 39/98 = 0.4\) (constant velocity ⇒ applied force balances kinetic friction).

Q2 (conceptual). Why do heavy and light skydivers not fall together, when Galileo says free fall is universal?

Answer

With drag, terminal velocity \(v_t = \sqrt{2mg/C\rho A}\) depends on \(m/A\). Free fall is universal only in vacuum; drag breaks the equivalence by caring about size and mass separately.

Q3 (multiple choice). A car brakes hardest without skidding when its wheels: (a) lock completely (b) keep rolling at the verge of slipping (c) spin freely

Answer

(b). Rolling contact uses static friction (\(\mu_s > \mu_k\)); a locked wheel slides on smaller kinetic friction. Hence anti-lock brakes.