Engineering Statics: Open and Interactive

Friction has two distinct regions as shown in Figure 9.1.6, and the value of \(\mu\) is different in each region. The region from point one to point two, where the force of friction increases linearly with load is called the static friction region. Here you must use the coefficient of static friction\(\mu_s\text{.}\) The region from point two to point three, where the friction remains roughly constant is called the kinetic friction region. In this region you must use the coefficient of kinetic friction\(\mu_k\text{.}\)

The coefficient of friction suddenly drops at point two, causing the friction force \(F_f\) to drop as well. Point two is called the point of impending motion, because here the situation is unstable. If the applied force increases ever so slightly, the opposing friction force suddenly decreases and the object begins to move.

To better understand the behavior of Coulomb friction imagine an object resting on a rough surface as shown in Figure 9.1.5.(b). When force \(P\) is gradually increased from zero, the normal force \(N\) and the frictional force \(F_f\) both change in response. Initially both \(P\) and \(F_f\) are zero and the object is in equilibrium. The interaction between the two surfaces in contact means that friction is available but it is not engaged \(F_f=P=0\text{.}\)

As \(P\) increases, the opposing friction force \(F_f\) increases as well to match and hold the object in equilibrium. In this static-but-not-impending phase \(F_f= P\text{.}\)

When \(P\) reaches point two, motion is impending because friction has reached its maximum value. \(F_{f_\text{max}}=\mu_\text{s} N = P\text{.}\) If force \(P\) increases slightly beyond \(F_{f_\text{max}}\text{,}\) the friction force suddenly drops to the kinetic value \(F_f= \mu_k N\text{.}\) The applied force exceeds the frictional force breaking equilibrium and causing the object to accelerate, and accelerating bodies are beyond the scope of Statics!

Notice that static friction at impending motion is greater than kinetic friction, because the coefficient \(\mu_\text{s} \gt \mu_k\) for most materials. Practically, this tells us that once a material starts to move it is easier to keep moving than it was to get it started from rest.

If you wonder why we include kinetic friction in a statics course, remember that a sliding body moving at constant velocity is in equilibrium.

The friction angle\(\phi_s\) is defined as the angle between the friction resultant and the normal force. At impending motion, the friction angle reaches its maximum value. The friction resultant and friction angle are used for screw, flexible belt, and journal bearing type problems.

In Figure 9.1.7 a block of weight \(W\) is pushed sideways by force \(P\text{.}\) The reaction forces can be represented as separate friction and normal forces, or as combined friction force \(R\) acting at friction angle \(\phi_s\text{,}\) measured from the normal direction.

The normal force supporting the object is distributed over the entire contact surface, however it is common on two dimensional problems to replace the distributed force with an equivalent concentrated force acting at a particular spot on the contacting surface, as we did in Chapter 7. This point is rarely at the exact center of the contact surface.

This is illustrated in Figure 9.1.8 where we see in that an object at rest has a uniformly distributed normal force along the bottom surface and the resultant normal force is located directly below the weight. The friction force is not engaged. When a pushing force \(F_\text{push}\) is applied, the distributed normal force changes shape, and the resultant normal force \(N\) shifts to the right to maintain equilibrium. The resultant normal force continues to shift to the right the harder you push.

This can be understood from the principle of a two-force body. As we discussed previously, the combined push and weight can be represented a single load acting down and to the right. The friction and normal forces can be combined into a single reaction force acting up and to the left at a point on the bottom surface. These two forces must share the same line of action to maintain equilibrium, so as the pushing force increases, the friction angle changes and the point shifts to the right. If the point shifts off the physical object then the required friction is greater than the friction available and motion begins.