Coefficients of friction — measurement, material dependence

WHAT is a coefficient of friction?

WHY a ratio? Experiment shows friction scales linearly with how hard the surfaces are squeezed together ($N$). Dividing it out gives a number that's the same regardless of weight — so we can predict friction for any load.

HOW do we measure $\mu$? (Two classic methods — derive both)

Method 1 — The inclined plane (angle of repose)

Place the block on a ramp and slowly tilt until it just begins to slip. At that critical angle $\theta_c$:

Set up coordinates along/perpendicular to the incline. Forces on the block:

• Weight $mg$ splits into: along-slope $mg\sin\theta$ (drives sliding), into-slope $mg\cos\theta$ (presses surfaces).
• Normal force $N$.
• Friction $f$ up the slope.

Perpendicular balance (no motion off the surface): $N = mg\cos\theta \quad\text{— Why? No acceleration perpendicular to ramp.}$

At the verge of slipping, friction is maxed out: $f = \mu_s N$, and along-slope balance still holds: $mg\sin\theta_c = \mu_s N = \mu_s\, mg\cos\theta_c$

Cancel $mg$ (Why? mass-independence — the same trick that makes Galileo's ramps work): $\boxed{\mu_s = \tan\theta_c}$

Beautiful result: the mass cancels, so you only need a protractor — no force meter, no scale!

For $\mu_k$: tilt slightly past $\theta_c$ and find the angle where the block slides at constant velocity (zero acceleration). Same algebra gives ==$\mu_k = \tan\theta_{\text{slide}}$==.

Method 2 — Horizontal pull

Pull horizontally with a spring scale until the block just moves; read the force $F_{\max}$. $N = mg \quad\Rightarrow\quad \mu_s = \frac{F_{\max}}{mg}$ Then keep it moving steadily (constant velocity, $a=0$): the reading $F_{\text{slide}}$ gives $\mu_k = \frac{F_{\text{slide}}}{mg}.$ Why constant velocity? If $a=0$ then applied force exactly equals kinetic friction — no leftover net force to confuse the reading.

Material dependence — what changes $\mu$?

Factors: material hardness/softness, surface roughness, contamination/lubricants, and surface cleanliness. NOT (to first order): apparent contact area, sliding speed, weight.

Worked examples

Flashcards

Connections

• Newton's Second Law — friction enters as a force term in $\sum F = ma$.
• Static vs Kinetic Friction — the two regimes summarized here.
• Inclined Plane Dynamics — same coordinate decomposition.
• Normal Force — the $N$ that $\mu$ multiplies.
• Lubrication & Tribology — how films lower $\mu$.
• Free Body Diagrams — tool used in both measurement methods.

Concept Map

Hinglish (regional understanding)