Equilibrium with Friction

In equilibrium problems with friction, the ordinary equations of statics are supplemented by dry-friction conditions. Unlike a smooth contact, a rough-surface reaction has both normal and tangential components.

Free-body diagram

It is convenient to resolve the rough-contact reaction into a normal reaction $N$ and a friction force $F_{fr}$. The normal component is perpendicular to the surface, while friction acts tangentially and opposes possible relative sliding.

Equilibrium check procedure

1. Draw the free-body diagram and show the normal reaction $N$ and friction force $F_{fr}$.
2. Direct friction opposite to the possible or impending relative sliding.
3. First write the ordinary equilibrium equations and determine the friction force required for rest.
4. Check the static-friction condition $|F_{fr}|\le\mu N$.
5. If the inequality is satisfied, equilibrium is possible within the adopted model. If the required friction exceeds $\mu N$, static equilibrium is impossible.
6. Use $|F_{fr}|=\mu N$ only for impending sliding.

Inclined plane

For a body on an inclined plane, its weight resolves into $G\sin\alpha$ along the plane and $G\cos\alpha$ normal to it. With no other forces, rest requires $G\sin\alpha\le\mu G\cos\alpha$, or $\tan\alpha\le\mu$. The limiting inclination corresponds to the angle of friction.

Direction of friction

In an equilibrium problem, friction direction is determined from the tendency of relative motion that would occur without friction, not from actual motion. If an assumed friction direction is opposite to the required one, the calculated friction value will be negative.

Impending equilibrium

When a problem states that a body is on the verge of motion, is about to slide, or asks for a minimum or maximum force immediately before motion, the limiting condition $|F_{fr}|=\mu N$ is generally used. Ordinary static equilibrium requires the inequality instead.

Example

A 1 kN block rests on a horizontal surface with $\mu=0.25$. A horizontal force of 180 N requires 180 N of friction. Since $F_{fr,max}=0.25\cdot1000=250$ N, equilibrium is possible. If the horizontal force rises to 300 N, static friction can no longer maintain rest.

Common mistakes

• immediately setting $F_{fr}=\mu N$ without an impending-motion condition;
• failing to check the calculated friction against its limiting value;
• directing friction with, rather than against, the tendency to slide;
• ignoring changes in the normal reaction caused by inclined external forces;
• using the dry-friction model when the problem specifies a different contact model.