Newtonian mechanics

Friction force

Friction opposes relative motion between contacting surfaces; kinetic friction is often modeled as μN.

A book at rest does not slide on its own; when pushed, the table exerts a resisting force. Friction is a contact force parallel to the surfaces that opposes impending or actual relative sliding.

0\le f_s\le\mu_sN,\qquad f_k=\mu_kN

Definition: Static and kinetic friction

Static friction adjusts to balance a tangential force up to $f_{s,\max}=\mu_sN$ . Once sliding begins, kinetic friction is approximated by $f_k=\mu_kN$ . $N$ is the normal force (N); the dimensionless coefficients $\mu_s,\mu_k$ depend on the surfaces. Often $\mu_k<\mu_s$ .

Change the slope and friction coefficient; inspect weight, normal force, friction, and motion.

What determines friction?

On a horizontal surface with no vertical acceleration, $N=mg$ , so kinetic friction is about $\mu_kmg$ ; adding mass usually increases it. On a slope of angle $\theta$ , $N=mg\cos\theta$ and kinetic friction is $\mu_kmg\cos\theta$ . Friction helps us walk but also heats and wears surfaces.

Example: Pulling a box across a floor

A 5 kg box slides on a horizontal floor with $\mu_k=0.20$ . Take $g=10\,m/s^2$ . Find the kinetic friction.

Solution

$N=mg=50\,N$ , so $f_k=\mu_kN=0.20\times50=10\,N$ , opposite the sliding direction.

Static friction exists even before sliding: it adjusts to balance a horizontal pull up to its limit $\mu_sN$ . A box with $N=50 N$ and $\mu_s=0.4$ can have up to 20 N of static friction, so a 12 N pull does not yet make it slide. Once sliding with $\mu_k=0.3$ , kinetic friction is about 15 N.

Example: Comparing the two kinds of friction

A stationary box is pulled with 8 N; its maximum static friction is 18 N. Does it slide?

Solution

No. Static friction adjusts to 8 N and balances the pull.

To increase useful friction, tire treads and shoe soles grip the ground; to reduce friction, bearings can be lubricated. Static friction lets a foot push against the ground without slipping, while sliding friction heats contacting surfaces. The model $f=mu N$ approximates many dry surfaces.

Quick check

What is the magnitude of static friction while an object remains at rest?

A block slides on a horizontal floor. If the kinetic coefficient increases while N stays fixed, what happens to kinetic friction?

References

Young, Freedman (2019). University Physics