Frictional Force Between Solids

Figure 22.5: The kinetic frictional force $F_k$ is exerted on the upper body by the stationary lower body. The upper body is moving with velocity $v$ and is pressed together with the lower body by a normal force $N$. It may also be acted upon by an additional non-normal external force $F_{ext}$.

The frictional force $F_k$ between two solid objects in contact obeys an empirical law.^23.1 If the two objects are sliding over each other, the frictional force on each object acts so as to oppose the relative motion of the two objects. (See figure 22.5.) The frictional force is proportional to the normal force $N$ pressing the objects together:

$$F_k = \mu_k N ~~~ \mbox{(kinetic friction)} .$$ (23.13)

The dimensionless quantity $\mu_k$ is called the coefficient of kinetic friction. This quantity is different for different pairs of materials rubbing together. It is typically of order one, but may be much less for particularly slippery materials.

Equation (22.13) is only valid if the two objects are moving relative to each other. If they are not in relative motion, but if some other force is being exerted on one of them, a static frictional force $F_s$ will precisely counteract this force so as to result in zero net force on the object. However, the static frictional force will keep the bodies from slipping only up to some limit defined by

$$\vert F_s \vert \le \mu_s N ~~~ \mbox{(static friction)} ,$$ (23.14)

where $\mu_s$ is the coefficient of static friction. Generally we find that $\mu_s > \mu_k$, so gradually increasing the external force on an object in static frictional contact with another object will cause it to suddenly break loose and accelerate when the maximum sustainable static frictional force is exceeded. Once the object is in motion, a lesser external force is needed to keep it moving at a constant velocity.