Limiting Friction

What limiting friction is

As an applied force increases, static friction increases to match it. But static friction cannot increase indefinitely. At some point it reaches its maximum value — this maximum is called limiting friction:

$f_{s,max} = \mu_s N$

At this point, the body is on the verge of sliding. Any additional force will cause motion.

Significance

Limiting friction is the threshold between rest and motion.

• Applied force $< f_{s,max}$: body stays at rest, static friction $=$ applied force
• Applied force $= f_{s,max}$: body is just about to move — the critical point
• Applied force $> f_{s,max}$: body moves, static friction is replaced by kinetic friction

Experimental basis — Coulomb's Laws of Friction

Limiting friction was established experimentally by Charles-Augustin de Coulomb (the same Coulomb of electrostatics). His observations:

1. Limiting friction is proportional to the normal force: $f_{s,max} \propto N$
2. Limiting friction is independent of the area of contact
3. Limiting friction depends on the nature and condition of the surfaces in contact

From observation 1: $f_{s,max} = \mu_s N$, where $\mu_s$ is the constant of proportionality — the coefficient of static friction.

$\mu_s > \mu_k$ always

The coefficient of static friction is always greater than the coefficient of kinetic friction for the same surfaces. It takes more force to start an object sliding than to keep it sliding.

This is why:

• A car wheel that locks up (slides) has less braking force than one that rolls (static friction at contact patch)
• Anti-lock braking systems (ABS) prevent wheel lockup to maintain maximum braking force