Coefficient of Static Friction Formula: μs = fs,max / N

The coefficient of static friction formula is $\mu_s = \frac{f_{s,\max}}{N}$ : the biggest grip force available before slipping, divided by the normal force. If a 12 kg cabinet just starts to move when pushed with 47 N, its $\mu_s = \frac{47}{117.6} \approx 0.40$ .

Static friction is the grip between surfaces that are not sliding. It is a self adjusting force: it matches any push up to a limit, then gives way. The coefficient measures that limit. The same number also comes from the angle of repose, $\mu_s = \tan\theta$ . It is part of the friction family in the laws of motion and works with the general friction force formula.

Quick facts

Formula
$\mu_s = \frac{f_{s,\max}}{N}$
Inequality
$f_s \le \mu_s N$
Angle of repose
$\mu_s = \tan\theta_r$
Unit
None, dimensionless
Flat curve, no banking
$v_{\max} = \sqrt{\mu_s g r}$
Usual size
Larger than $\mu_k$ for the same surfaces

Static coefficient of friction formula: what limiting friction means

Push gently on a heavy cabinet. It does not move, so static friction equals your push. Push harder and friction rises to match. At one particular push the cabinet just begins to slide. Friction at that instant is limiting friction, $f_{s,\max}$ , and the coefficient is that limit divided by N.

Static friction as the push grows
Your push	Static friction	Motion
0 N	0 N	At rest
20 N	20 N	At rest
47 N (limit)	47 N	About to slide
60 N	Kinetic friction takes over	Sliding

The table uses the 12 kg cabinet from the first example. Before the limit, the formula $\mu_s N$ gives only the maximum, not the actual friction.

Coefficient of static friction formula from the angle of repose

Place a block on a plank and lift one end slowly. At the angle $\theta_r$ where the block just starts to slip, the component of weight down the slope equals limiting friction: $mg\sin\theta_r = \mu_s mg\cos\theta_r$ . So

$\mu_s = \tan\theta_r$

Mass and g both cancel, so the method works for any block. Sand and gravel piles settle at their angle of repose for the same reason.

Worked examples on the formula for coefficient of static friction

Example 1: starting a cabinet

A 12 kg cabinet starts to slide when pushed horizontally with 47 N. Find μs, using g = 9.8 m/s².

N = 12 × 9.8 = 117.6 N.

$\mu_s = \frac{47}{117.6} = 0.400 \approx 0.40$

Example 2: from the angle of repose

A matchbox on a tilted board begins to slide at 31°. Find μs.

$\mu_s = \tan 31^\circ = 0.601 \approx 0.60$

Example 3: car on a flat bend, with km/h

A car takes an unbanked bend of radius 50 m. Between the tyres and the road μs = 0.70. Find the greatest safe speed in km/h.

Static friction supplies the centripetal force: $\mu_s mg = \frac{mv^2}{r} \Rightarrow v = \sqrt{\mu_s g r}$ .

$v = \sqrt{0.70 \times 9.8 \times 50} = \sqrt{343} = 18.5\ \text{m/s}$

In km/h: 18.5 × 3.6 = 66.7 km/h, so about 67 km/h. The tyres do not slide on the road, which is why the static coefficient is the one to use.

Pressing a book against a wall

Hold a book against a wall by pressing horizontally with force F. The wall pushes back with N = F, and static friction holds the book up. It stays put while $mg \le \mu_s F$ , so the smallest press that works is $F_{\min} = \frac{mg}{\mu_s}$ . For a 1.5 kg book with $\mu_s = 0.50$ , $F_{\min} = \frac{1.5 \times 9.8}{0.50} = 29.4\ \text{N}$ . Pressing harder than that does not raise the friction above mg; it only raises the limit.

Static against kinetic friction

For the same surfaces, $\mu_s$ is normally larger than $\mu_k$ . That gap is why anti lock brakes help: a wheel that keeps rolling grips with static friction, while a locked, skidding wheel only gets the smaller kinetic friction. Compare the two on the coefficient of kinetic friction formula page, and see the banking of road formula for curves where the road is tilted to help.

Slips with μs

Writing $f_s = \mu_s N$ for an object that is not on the verge of slipping.
Using the slope angle of a moving block in tan θ; only the angle at which slipping starts counts.
Giving μs a unit.
Forgetting that wet or oily surfaces lower μs a lot.

Frequently asked questions

How do you calculate the coefficient of static friction?

μs = fs,max / N, the limiting static friction divided by the normal force. It also equals tan θ at the angle of repose.

What is limiting friction?

It is the greatest value static friction can reach, just before the surfaces start to slide. It equals μs N.

Can the coefficient of static friction be greater than 1?

Yes. It only means the limiting friction exceeds the normal force. Soft rubber on rough dry surfaces can do this.

Why is the static coefficient used for a car turning a bend?

A rolling tyre does not slide over the road at the contact patch. The grip that keeps the car on its curve is therefore static friction.