# 5.3: Friction Forces

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Whenever a solid comes into contact with another solid or a fluid, the force at the interface has both a normal and a shear component. The focus of this section is on the shear component of the force. These shear forces are produced by shear stress at the interface and are called friction forces.

Traditionally friction forces are classified into two categories: dry friction (Coulomb friction) or fluid friction. Dry friction is a shear force produced at a solid-solid interface and occurs regardless of whether or not there is relative motion (sliding) at the interface. Fluid friction, by contrast, is a shear force produced at a solid-fluid interface by the relative motion between the bulk fluid and the solid. The major characteristics of these friction models are presented in the table below:

Dry Friction Model Fluid Friction Model

Magnitude of the force acting on a system assumed to depend on

• the magnitude of the normal force $$N$$ at the boundary and
• the relative motion (sliding) between the surfaces.

If the surfaces do not slide (no relative motion),

$F_f \leq F_{s, \text{ max}} = \mu_s N \nonumber$ \begin{align*} \text{where} \quad F_f &= \text{the friction force} \\ F_{s, \text{ max}} &= \ maximum \text{ static–friction force possible,} \\ \mu_s &= \text{coefficient of static friction} \\ N &= \text{the normal force at the interface} \end{align*} \nonumber

If the surfaces do slide (relative motion),

$F_f = F_k = \mu_k N \nonumber$ \begin{align*} \text{where} \quad F_k &= \text{the kinetic-friction force} \\ \mu_k &= \text{coefficient of kinetic friction} \end{align*} \nonumber

Direction of the force acting on a system is in the plane of contact and in the direction of relative motion of the external object.

Magnitude of the force acting on a system is assumed to depend on the relative velocity between the surface and the fluid free stream.

Models for fluid friction:

Viscous friction forces occur between close-fitting surfaces with a gas or liquid lubricant at the interface and its magnitude is modeled as being proportional to the relative velocity between the surfaces (the sliding velocity), $$V_{\text{slide}}$$:

$F_{\text{viscous}} = k_1 V_{\text{slide}} \nonumber$

Fluid dynamic drag forces are exerted on any object immersed in a moving fluid, and its magnitude is modeled as being proportional to the fluid velocity squared: $F_{\text{drag}} = k_2 V_{\text{fluid}}^2 \nonumber$ where $$V_{\text{fluid}}$$ is the velocity of the bulk fluid measured relative to the solid surface.

Direction of the force acting on a system is in the direction of the fluid velocity $$V_{\text{fluid}}$$ (or sliding velocity $$V_{\text{slide}}$$) measured relative to the surface.

Note the significant differences between fluid friction and dry friction. A dry friction force is proportional to the normal force at the interface between between the two solids and occurs with or without relative motion. With no sliding, we have static friction; with sliding, we have kinetic (or sliding) friction. In contrast, a fluid friction force is independent of the normal force at the interface and is zero when there is no relative motion between the bulk fluid and the solid surface. Detailed information about the motion of the fluid is required to evaluate the proportionality constants in the fluid friction models. In this section we will concentrate on dry friction. A detailed discussion of fluid friction, especially how to find the proportionality constants, will be reserved for a later course, e.g. ES 202—Fluid and Thermal Systems.

To explore dry friction in more detail, consider a block resting on a surface as shown in Figure $$\PageIndex{1}$$. Under what conditions will the block move and what is the value of the friction force at the interface? For our study, select a closed system that encloses just the block. Note that the lower boundary of our system is placed at the interface between the block and the horizontal surface that it rests on. A linear momentum system diagram (free body diagram) showing all linear momentum interactions with the surroundings is also shown in the figure.

Figure $$\PageIndex{1}$$: Block moving on a horizontal surface with friction.

Writing the rate-form of conservation of linear momentum in vector form we have for the closed system: $\frac{d \mathbf{P}_{\mathrm{sys}}}{dt} = \mathbf{F} + \mathbf{P} + \mathbf{F}_{\mathrm{f}} + \mathbf{N} \nonumber$ where $$\mathbf{F}_{\mathbf{f}}$$ is the dry friction force at the interface between the block and the surface. We can simplify this equation by recalling that for a closed system $$\mathbf{P}_{\mathrm{sys}} = m_{\mathrm{sys }} \mathbf{V}_{G}$$. To write this equation in terms of scalar components use the $$(x, y)$$ coordinate system shown in the figure. In the $$x$$-direction we have the following: $\frac{d\left(m V_{x}\right)}{d t} = F \cos \theta-F_{\mathrm{f}} \quad \rightarrow \quad m \ \frac{d V_{x}}{d t} = F \cos \theta - F_{\mathrm{f}} \nonumber$ In the $$y$$-direction, we have the following result:

\begin{aligned} \frac{d\left(m V_{y}\right)}{d t} &= N-P-W-F \sin \theta \\ \underbrace{m \cancel{ \frac{d V_y}{d t} }^{=0} }_{\begin{array}{c} \text { No motion in } \\ y \text{ direction } \end{array}} &= N-P-W-F \sin \theta \quad \rightarrow \quad N=P+W+F \sin \theta \end{aligned} \nonumber Note that the normal force $$N$$ does not just equal the weight of the block. Please be careful in determining the normal force. Students often assume that the friction force is proportional to the weight of the object. This mistake is the result of carelessly assuming that the normal force always equals the weight.

To investigate how the friction force influences the system behavior, we apply the dry friction model: \begin{align*} \text { No sliding: } \quad V_{x}=0 \quad \rightarrow \quad F_{\mathrm{f}} = F \cos \theta \leq \mu_{\mathrm{s}} N \\ \text { Surfaces sliding: } \quad V_{x} \neq 0 \quad \rightarrow \quad F_{\mathrm{f}} = \mu_{\mathrm{k}} N \end{align*} \nonumber where $$V_x$$ is the relative (sliding) velocity at the interface between the surfaces in contact. Typically the coefficient of kinetic friction is approximately $$75 \%$$ of the coefficient of static friction.

If there is no sliding, i.e. relative velocity between the surfaces is zero, the friction force can assume a range of values less than or equal to the value of the maximum possible static-friction force. Students frequently assume that the static-friction force is single valued and always equals the maximum possible static-friction force. This is incorrect and a major cause of errors. Only when sliding is impending does the value of the friction force equal that of the maximum static-friction force.

By contrast, if the surfaces are sliding, the friction force has a single value and equals the kinetic-friction force. Note that the kinetic-friction force only depends on the coefficient of kinetic friction and the normal force.

In many problems with dry friction it is unclear whether or not the system slides due to the applied loads. In these problems, the system (block) can behave in one of three ways:

Case Relative motion at the interface Friction Force
I No motion — The block is not sliding. $$F_{\mathrm{f}} < \mu_{\mathrm{s}} N$$
II Impending motion — The block is not sliding but is on the verge of sliding. $$F_{\mathrm{f}} = \mu_{\mathrm{s}} N$$
III Motion — The block is sliding. $$F_{\mathrm{f}} = \mu_{\mathrm{k}} N$$

When the motion is indeterminate do the following:

• First assume there is no sliding and determine the required friction force to maintain this condition.
• Next, compare the required friction force against the maximum possible static friction force, $$F_{\mathrm{s, max} }=\mu_{\mathrm{s}} N$$.
• If the required friction force is less than or equals the maximum possible static friction force, your assumption of no sliding was correct and the actual friction force at the interface equals the required friction force calculated previously.
• If the required friction force exceeds the maximum possible static friction force, your assumption of no sliding was incorrect. Under these conditions, the system slides and the actual friction force equals the kinetic-friction force.

Alternatively, instead of assuming something about the motion and then solving for the friction force required, you can determine the value of the external forces that are required for impending motion. To do this, assume that sliding is impending and the friction force equals the maximum possible static-friction force. Then determine the external forces required for this condition. Actual external forces with values that exceed this required value will produce sliding, and the friction force will equal the kinetic-friction force value. Smaller values will result in no motion, and the actual friction force can be predicted using the actual external forces.

To test your understanding, reconsider the problem discussed above. Using the graph shown below plot a graph showing how the friction force $$F_{\text {f }}$$ generated at the block interface changes as the applied force $$F$$ increases from zero with $$\theta=0$$.
Figure $$\PageIndex{2}$$: Coordinate system for graphing $$F_{\text{f}}$$ vs $$F$$.