Sliding Friction. Coulomb's Laws

Dry sliding friction arises at the contact between two rough bodies and opposes their relative sliding or tendency to slide. In the simplest Coulomb model, the tangential friction force is related to the normal contact reaction.

Friction during static equilibrium

As long as the body does not slide, the friction force adjusts to the external loading within the range required for equilibrium. Therefore, $F_{fr}=\mu N$ must not automatically be used for every static condition.

Limiting friction

At impending sliding, the maximum dry-friction force is proportional to the normal reaction:

$$F_{fr,max}=\mu N.$$

During static equilibrium, friction is not necessarily equal to $\mu N$: it takes the value required for equilibrium within $|F_{fr}|\le\mu N$.

The equality applies at impending sliding. The friction force acts opposite to the direction of the impending relative motion.

Coefficient of friction

The dry-friction coefficient $\mu$ is a dimensionless property of the contacting pair within the adopted model. Its value depends on the materials and surface condition. In the elementary Coulomb model, the limiting friction force is proportional to the normal reaction.

Angle of friction

The total reaction of a rough surface is the vector sum of the normal reaction $N$ and the friction force. At impending sliding it is inclined from the normal by the angle of friction $\varphi$, for which $\tan\varphi=\mu$.

Cone of friction

In a three-dimensional problem, the possible direction of the limiting total reaction forms a cone around the contact normal. If the resultant contact reaction lies inside the friction cone, sticking equilibrium may be possible; a reaction on the cone surface corresponds to impending sliding.

Example

A block is pressed against a horizontal surface with normal reaction $N=500$ N and coefficient of friction $\mu=0.30$. The maximum static-friction force is $F_{fr,max}=0.30\cdot500=150$ N. If the applied horizontal force is only 80 N and there are no other horizontal forces, the equilibrium friction force is 80 N, not 150 N.

Common mistakes

• always setting $F_{fr}=\mu N$ instead of using the static-friction inequality;
• assigning the friction direction without considering impending motion;
• confusing the normal reaction with the total rough-contact reaction;
• treating the coefficient of friction as a dimensional quantity;
• failing to check whether the friction force found from equilibrium exceeds its limiting value.