**Branch :**First Year-Engineering Syllabus

**Subject :**Elements of Mechanical Engineering

## Coulomb Theory of Friction

**Coulomb Theory of Friction:**** **Let us first consider static friction, using the example in Fig. 1b. As long as F is smaller than a certain limit F0, the box stays at rest and equilibrium yields H = F. The tangential force H attains its maximum value H = H_{0} for F = F_{0}. Charles Augustine de Coulomb (1736–1806) showed in his experiments that this limit force H_{0} is in a first approximation proportional to the normal force N:

The proportionality factor μ0 is commonly referred to as the coefficient of static friction. It depends solely on the roughness of surfaces in contact, irrespective of their size. Table 1 shows several numerical values for different configurations. Note that coefficients derived from experiments can only be given within certain tolerance limits; the coefficient for “wood on wood”, for example, strongly depends on the type of wood and the treatment of the surfaces. It should also be noted that relates the tangential force and the normal force only in the limit case when slip is impending; it is not an equation for the static friction force H. A body adheres to its base as long as the condition of static friction

eq...(1) table..1

is fulfilled. The orientation of the friction force H always opposes the direction of the motion that would occur in the absence of friction. For complex systems, this orientation is often not easily identifiable, and must therefore be assumed arbitrarily. The algebraic sign of the result shows if this assumption was correct. In view of a possible negative algebraic sign of H, we generalise condition (1) as follows:

..................eq.(2)

The normal force N and the static friction force H can be assembled into a resultant forceW (Fig. 1a). Its direction is defined by the angle ϕ, which can be derived from

fig.....1

Referring to the limit angle ϕ_{l} as e_{0} (in the case H = H_{0}) yields

This so-called “angle of static friction” e_{0} is related to the coefficient of static friction:

A “static friction wedge” for a plane problem (Fig. 1b) is obtained by drawing the static friction angle e_{0} on both sides of the normal n. If W is located within this wedge, H <H_{0} is valid and the body stays at rest.