Friction — static (maximum), kinetic, rolling

1. The three regimes

2. Deriving the friction laws from first principles

We don't prove friction from deeper physics in this course — instead we build the standard Amontons–Coulomb model from two experimental observations and reason out the consequences.

Observation 1 — Proportionality to normal force. Press surfaces harder ⇒ more real contact area at the asperities ⇒ more bonds ⇒ more force to break them. Experiment shows the friction force scales linearly with $N$ : $f \propto N \quad\Longrightarrow\quad f = \mu N$ The constant $\mu$ (mu) is the coefficient of friction — a pure number capturing surface roughness/stickiness.

Observation 2 — Independence of apparent area. Doubling the apparent contact area halves the pressure, so each patch carries less load and forms fewer bonds. The two effects cancel, so $f$ doesn't depend on apparent area. (That's why $\mu$ alone suffices.)

Figure — Friction — static (maximum), kinetic, rolling

The graph above is the single most important diagram: friction force vs applied force. Note the diagonal static region, the sharp peak at $f_s^{max}$ , and the drop to the lower constant kinetic plateau.

3. Angle of friction & friction on an incline

4. Worked examples

Worked example Example 1 — Will the box move?

A $10\ \text{kg}$ box on the floor, $\mu_s=0.5$ , $\mu_k=0.3$ , $g=10$ . You push horizontally with $40\ \text{N}$ . Find friction & acceleration.

Step 1 — find the ceiling. $f_s^{max}=\mu_s N=\mu_s mg=0.5\times100=50\ \text{N}$ . Why this step? We must know the max static force before deciding if it slips.

Step 2 — compare push to ceiling. Push $=40\ \text{N} < 50\ \text{N}$ . So it does not slide. Why? Static friction is self-adjusting; it can supply up to 50 N.

Step 3 — friction value. Since static, $f_s =$ applied $=40\ \text{N}$ (Newton's 2nd law, $a=0$ ). Why? No acceleration ⇒ net force zero ⇒ friction balances the push exactly. Not 50 N!

Step 4 — acceleration: $a=0$ .

5. Common mistakes (Steel-man them)

6. Flashcards

Static friction obeys an equality or inequality?

Inequality:

f_s \le \mu_s N

(self-adjusting up to a max).

What is

f_s^{max}

\mu_s N

— the maximum static friction, reached just before sliding.

Formula for kinetic friction?

f_k=\mu_k N

, roughly constant and independent of speed.

Which is larger,

\mu_s

\mu_k

\mu_s>\mu_k

— harder to start sliding than to keep it sliding.

Does friction depend on apparent contact area?

No — it depends on the normal force

N

(and

\mu

Relation between angle of repose and

\mu_s

\tan\theta_r=\mu_s

What is the angle of friction

\lambda

Angle of the resultant contact force from the normal;

\tan\lambda=\mu_s

Why is rolling friction tiny?

\mu_r=a/R

where

a

(deformation offset)

\ll R

, so

f_r\ll f_k

A 10 kg box,

\mu_s=0.5

, pushed with 40 N — friction value?

40 N (static, balances push exactly; not 50 N).

Walking forward — which direction does static friction on your foot point?

Forward (your foot tends to slide backward).

Recall Feynman: explain to a 12-year-old

Imagine two pieces of sandpaper. Their tiny bumps lock together like Velcro. Static friction is the grip while nothing moves — it pushes back exactly as hard as you push, but only up to a limit. Push past the limit and the bumps snap apart: now it's sliding (kinetic) friction, which is a bit weaker, like Velcro that's already torn. A wheel barely touches and just peels off the tiny dip it makes, so rolling friction is teeny — that's why rolling a suitcase is way easier than dragging it.

Connections

Newton's Second Law — friction is just one force in $\sum F = ma$ .
Free Body Diagrams — always draw $N$ and $f$ correctly.
Inclined Plane Problems — angle of repose lives here.
Rolling without Slipping — condition $v=\omega R$ ; rolling friction.
Circular Motion — friction provides centripetal force for cornering cars.
Work-Energy Theorem — kinetic friction dissipates energy as heat.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, friction ko samajhne ka sabse important point yeh hai: static friction ek fixed number nahi hai, woh adjust karta hai. Jab tum ek bhaari box ko halka sa push karte ho aur woh nahi hilta, toh friction utna hi force laga raha hai jitna tumhara push — na zyada na kam. Tum jor lagao, friction bhi badhta jaata hai, par ek maximum tak — woh maximum hai $f_s^{max}=\mu_s N$ . Jaise hi tumhara push us ceiling ko cross karta hai, box slip ho jaata hai aur ab kinetic friction lagta hai jo thoda kam hota hai ( $\mu_k < \mu_s$ ). Isiliye box ko start karna mushkil, par chalu rakhna easy lagta hai.

Doosra key idea: friction area pe depend nahi karta, sirf normal force $N$ pe karta hai. Log sochte hain bada surface matlab zyada grip, lekin bada area matlab pressure kam, dono effect cancel ho jaate hain. Toh formula simple rehta hai: $f=\mu N$ .

Incline pe ek pyaara result hai — angle of repose. Plane ko tilt karte jao jab tak block just slip na karne lage; us angle ka tangent hi $\mu_s$ hota hai: $\tan\theta_r=\mu_s$ . Exam mein yeh seedha marks dilata hai.

Aur rolling friction ( $\mu_r$ ) bahut chhota hota hai kyunki wheel sirf zameen ko thoda sa deform karta hai aur uss chhote se gaddhe se nikalta rehta hai. Isiliye suitcase ghaseet-ne se behtar hai use pahiyon pe rrol karna. Yaad rakho: $\mu_s > \mu_k \gg \mu_r$ .

Friction — static (maximum), kinetic, rolling

1. The three regimes

2. Deriving the friction laws from first principles

3. Angle of friction & friction on an incline

4. Worked examples

5. Common mistakes (Steel-man them)

6. Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Newton's Laws & Dynamics

Connections