Laws of friction

Push a heavy box across the floor and, for a while, nothing happens — the box just sits there resisting you. Push harder and suddenly it lurches into motion, and once moving it takes noticeably less force to keep it sliding than it took to start it. That everyday jolt is the whole story of friction in miniature: a force that adjusts itself to oppose you up to a limit, then settles into a smaller, steadier value once sliding begins. The laws of friction turn this behaviour into two clean, testable relations — one for the moment just before slipping, one for while it slides — plus the geometric idea of an angle at which a surface alone can hold a block in place without any push at all. This article builds those relations from the definitions up, applies them to a block on a floor and a block on an incline, and flags the traps ISC examiners repeatedly probe.

The core insight

Static friction self-adjusts up to a maximum $f_{s(max)} = \mu_s N$ , beyond which the body slides under a smaller, roughly constant kinetic friction $f_k = \mu_k N$ , and the incline angle at which sliding just begins — the angle of repose — satisfies $\tan\theta = \mu_s$ .

The core idea

Friction opposes relative sliding, or the tendency toward it, between two surfaces in contact, acting tangentially along the surface. It splits into two regimes that behave very differently and must never be treated as the same formula.

Static friction is self-adjusting. When a body is pushed but does not move, static friction $f_s$ is not a fixed number — it automatically takes on whatever value, from zero up to a ceiling, is needed to balance the applied force exactly. That ceiling is the limiting friction:

f_{s(max)} = \mu_s N

where: $N$ is the normal reaction between the surfaces, and $\mu_s$ is the coefficient of static friction, a dimensionless number set only by the nature of the two surfaces in contact (not by area, and not by the applied force). Below this limit, the actual static friction equals whatever the applied force requires for equilibrium — using $\mu_s N$ as "the" friction force whenever the body is at rest is the single most common ISC mistake, and it is wrong unless the body is on the verge of slipping.

Kinetic friction takes over once sliding starts. Once the applied force exceeds $f_{s(max)}$ , the body accelerates and a smaller, roughly speed-independent resistance acts:

f_k = \mu_k N

where: $\mu_k$ is the coefficient of kinetic friction, generally slightly less than $\mu_s$ for the same pair of surfaces — which is exactly why a box "breaks free" with a jerk and then slides more easily. The four experimental laws of friction summarise this picture: limiting friction is proportional to the normal reaction; it is independent of the apparent contact area; friction opposes the direction of relative motion (or its tendency); and kinetic friction is approximately independent of sliding speed over ordinary ranges.

Rolling friction resists a body that rolls rather than slides, and is generally much smaller than sliding friction for the same surfaces — the reason wheels and ball bearings are used wherever possible to cut frictional losses.

Angle of friction and angle of repose. The angle of friction $\lambda$ is the angle the resultant of $N$ and $f_{s(max)}$ makes with the normal, so $\tan\lambda = \mu_s$ . The angle of repose $\theta$ is the steepest a rough incline can be tilted while a resting body just stays put without sliding; as the derivation below shows, $\tan\theta = \mu_s$ too, so the two angles are numerically equal even though one is defined at a single contact point and the other from a whole block on a slope. Lubrication — a thin film of oil or grease between two surfaces — replaces solid-on-solid sliding friction with the much smaller viscous drag of the fluid film; at this level it is understood only qualitatively.

Figure

A graph of friction force against applied force showing static friction rising along a line up to a peak at the limiting value then dropping to a lower constant kinetic friction as the block starts sliding

Derivation AAngle of repose from force balance on an incline

The goal is to show, using nothing beyond resolving a weight into components along and perpendicular to a rough incline and applying the two laws $N$ balances the perpendicular pull while limiting friction balances the parallel pull at the verge of slipping, that the incline angle at which a resting block just begins to slide satisfies the single clean relation $\tan\theta = \mu_s$ , tying the everyday tilt-test experiment directly to the coefficient of static friction without ever having to measure a force by hand.

Step 1. Set up the incline. A block of mass $m$ rests on a plane inclined at angle $\theta$ to the horizontal, with coefficient of static friction $\mu_s$ between block and surface. Its weight $mg$ acts vertically downward; resolve it into a component $mg\cos\theta$ perpendicular to the incline and a component $mg\sin\theta$ along the incline, pulling the block down the slope.

Step 2. Balance forces perpendicular to the incline. There is no acceleration perpendicular to the surface, so the normal reaction exactly cancels the perpendicular weight component: $N = mg\cos\theta$ .

Step 3. Balance forces along the incline at the verge of slipping. As $\theta$ is increased slowly, static friction acts up the slope and grows to match $mg\sin\theta$ , right up until the block is on the point of sliding — at that instant friction has reached its ceiling, $f_{s(max)} = \mu_s N$ , and equilibrium along the slope reads $mg\sin\theta = f_{s(max)} = \mu_s\,mg\cos\theta$ .

Step 4. Solve for the angle. The mass $m$ and gravity $g$ cancel from both sides, leaving $\sin\theta = \mu_s\cos\theta$ , so $\tan\theta = \mu_s$ . This defines the angle of repose: tilt any rough incline slowly and note the angle at which a resting block just starts to slip, and that angle's tangent is the coefficient of static friction — a simple, force-meter-free way to measure $\mu_s$ experimentally.

Worked example

A block of mass $10\ \text{kg}$ rests on a horizontal floor with $\mu_s = 0.4$ and $\mu_k = 0.3$ between block and floor. Take $g = 9.8\ \text{m/s}^2$ .

Step 1. Find the normal reaction. The floor is horizontal and the block does not accelerate vertically, so $N = mg = 10 \times 9.8 = 98\ \text{N}$ .

Step 2. Find the maximum static friction available. $f_{s(max)} = \mu_s N = 0.4 \times 98 = 39.2\ \text{N}$ .

Step 3. Check a horizontal push of $35\ \text{N}$ . Since $35\ \text{N} < 39.2\ \text{N}$ , static friction is able to self-adjust to exactly $f_s = 35\ \text{N}$ and balance the push. The block does not move; it stays in static equilibrium, and it would be wrong to use $\mu_s N$ as "the" friction here — the actual static friction is only as large as needed, $35\ \text{N}$ , not $39.2\ \text{N}$ .

Step 4. Now push with $45\ \text{N}$ instead. This exceeds $f_{s(max)} = 39.2\ \text{N}$ , so the block slides, and kinetic friction takes over: $f_k = \mu_k N = 0.3 \times 98 = 29.4\ \text{N}$ . The net force is $45 - 29.4 = 15.6\ \text{N}$ , giving acceleration $a = 15.6/10 = 1.56\ \text{m/s}^2$ . The block slides with acceleration $1.56\ \text{m/s}^2$ .

Competitive stretch

A block just begins to slide down a rough incline at $22°$ . By the derivation above, this angle is the angle of repose, so $\mu_s = \tan 22° \approx 0.404$ — a value close to the $\mu_s = 0.4$ used in the floor example, showing the same physical scale of "moderately rough" contact appearing in two different experiments.

Common traps

Using $\mu_s N$ as the friction force whenever a body is at rest. It is only the maximum available value; actual static friction equals whatever equilibrium demands, up to that ceiling.
Assuming friction depends on the contact area. By the laws of friction it depends only on the normal reaction and the nature of the surfaces, never on how large the contact patch is.
Mixing up the angle of friction ( $\tan\lambda = \mu_s$ , defined at the contact point) with the angle of repose (the incline tilt at which sliding just begins) — numerically equal but conceptually distinct definitions.
Assuming $\mu_k \ge \mu_s$ . Kinetic friction is normally slightly smaller than the static limit, which is why a stuck object "breaks free" abruptly rather than sliding smoothly from rest.

Ask the tutor: want to work through a few more angle-of-repose or self-adjusting-friction problems together?

If static friction can take any value up to the maximum, how do I know what value to actually use in a problem?

You never assume it — you find it from equilibrium. Draw the free body diagram, write the balance of forces along the surface, and whatever value of friction makes that balance work out is the static friction, provided it does not exceed $f_{s(max)} = \mu_s N$ . If the required value would exceed that ceiling, equilibrium is impossible and the body slides, at which point you switch to $f_k = \mu_k N$ instead. The $10\ \text{kg}$ block example above shows exactly this: a $35\ \text{N}$ push needs only $35\ \text{N}$ of static friction, not $39.2\ \text{N}$ .

Why is kinetic friction usually smaller than the maximum static friction?

At a microscopic level, surfaces in contact are never perfectly smooth — they touch at scattered high points called asperities, which interlock and slightly weld together when at rest, giving static friction its larger limiting value. Once sliding begins, these contacts do not have time to fully re-form before they are sheared apart and new ones made, so on average the resistance is a little lower. This is why $\mu_k$ is generally somewhat less than $\mu_s$ , and why an object often "jerks" free the instant it starts moving.

Why does the angle of repose experiment give the same answer as measuring mu s with a spring balance?

Because both are measuring the identical physical ratio, just via different routes. Pulling the block horizontally with a spring balance until it just slips gives $\mu_s = f_{s(max)}/N$ directly as a ratio of two forces. Tilting the same block on an incline until it just slips instead lets gravity itself supply and vary that ratio through $\sin\theta$ and $\cos\theta$ , and the force balance reduces to $\tan\theta = \mu_s$ once mass cancels out — so the incline method reads off $\mu_s$ purely from an angle, with no force measurement needed at all.

Laws of friction

The core idea

Worked example

Competitive stretch

Common traps

Before this

Where this leads