Friction Notes

Last reviewed 16 Sept 2026 · 7 min read

What friction is

When one surface slides, or tends to slide, over another, a resisting force acts tangentially at the contact and opposite to the motion or intended motion. This is friction. It arises from surface roughness and molecular adhesion.

Static friction acts while the body is at rest; it grows with the applied force up to a maximum, the limiting friction.
Kinetic (dynamic) friction acts once sliding starts; it is slightly less than limiting friction.
Rolling friction (rolling resistance) is much smaller than sliding friction.

Laws of dry (Coulomb) friction

Friction acts opposite to the direction in which the body moves or tends to move.
Limiting friction is proportional to the normal reaction: $F = μ R_{N}$ .
Limiting friction does not depend on the area of contact (for the same normal force).
Friction depends on the nature of the surfaces in contact.
Kinetic friction is somewhat less than limiting friction and is nearly independent of sliding speed (at moderate speeds).

Coefficient of friction, angle of friction and repose

FormulaKey definitions

Coefficient of friction: $μ = \frac{F _{l im}}{R _{N}}$
Angle of friction $ϕ$ : angle between the normal reaction and the resultant of normal reaction and limiting friction: $tan ϕ = μ$ .
Angle of repose $α$ : maximum inclination of a plane at which a body rests without sliding under its own weight. At this angle $W sin α = μ W cos α$ → $tan α = μ$ → angle of repose = angle of friction.
Cone of friction: the cone generated by revolving the limiting resultant reaction about the normal; semi-vertex angle $ϕ$ . A body stays at rest if the resultant reaction lies within the cone.

Body on a horizontal plane

Weight $W$ , force $P$ at angle $θ$ above the horizontal, limiting condition:

R_{N} = W - P sin θ, P cos θ = μ (W - P sin θ) \Rightarrow P = \frac{μ W}{cos θ + μ sin θ}

RememberLeast force to move a body on a horizontal plane

$P$ is least when $θ = ϕ$ (the pull is inclined at the angle of friction): $P_{min} = W sin ϕ$ . Pulling at a slight upward angle is easier than pushing horizontally, which is easier than pushing downward at an angle.

Body on an inclined plane

Plane at angle $α$ , weight $W$ , $μ = tan ϕ$ .

Case (force $P$ parallel to the plane)	Force required
To move the body up	$P = W (sin α + μ cos α) = W \frac{sin ( α + ϕ )}{cos ϕ}$
To just prevent it sliding down ( $α > ϕ$ )	$P = W (sin α - μ cos α) = W \frac{sin ( α - ϕ )}{cos ϕ}$
Horizontal force to move up	$P = W tan (α + ϕ)$
Least force to move up (pull inclined at $ϕ$ to the plane)	$P_{min} = W sin (α + ϕ)$

If $α < ϕ$ the body stays at rest without any force (self-locking).

Ladder friction

A ladder of length $L$ and weight $W$ rests against a wall at angle $θ$ with the floor. At the point of slipping, friction acts at the foot (away from the wall, preventing slip) and — if the wall is rough — upward at the top.

Equations: $\sum F_{x} = 0$ , $\sum F_{y} = 0$ , $\sum M = 0$ about the foot. With a smooth wall and floor coefficient $μ$ , a uniform ladder with no other load is about to slip when

tan θ = \frac{1}{2 μ}

Adding a person higher up the ladder makes slipping more likely.

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