Static, Dynamic and Rolling Friction
Table of contents
The friction plays a fundamental role in various areas of our lives:
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When walking: Friction between the shoes and the ground allows us to move by pushing against the surface and pushing off.
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When braking a vehicle: Friction between the brake pads and the brake rotors generates the force required to bring the vehicle to a halt.
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When making a fire: Friction between two materials, such as a lighter and a matchbox, generates heat to ignite the match.
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When playing musical instruments: Friction between the strings of an instrument and the fingers produces a sound.
These are just a few examples of friction. Friction occurs almost everywhere in nature. Knowing about friction allows you to design more efficient systems, reduce wear, and achieve optimized motion.
In the following, we will look at three important types of friction: Static friction, kinetic friction and rolling friction. In all three cases, an important force will play a role: the normal force \( F_{\text N} \).
Normal force is the force exerted on the body by the surface on which the body is placed. In this way, the body does not simply fall through the surface. The normal force acts perpendicular to the surface on which the body is placed. "Normal" is meant in the geometric sense and means perpendicular.
Static friction
Let's look at a box standing on an inclined plane and not moving (see illustration 1). Why doesn't it just slide down? Static friction is responsible for this. It acts at the contact surface between the box and the inclined plane and prevents the box from sliding down. Static friction occurs whenever two surfaces are in contact and attempt to move relative to each other, but do NOT move.
The static friction generates a static friction force \( \boldsymbol{F}_{\text s} \), which always acts AGAINST the motion. Since we are considering an inclined plane as an example here, a certain force \( \boldsymbol{F} \) acts to try to move the box downward. However, the box does not move yet because the static friction force is as large as the force \( \boldsymbol{F} \), that is: \( \boldsymbol{F}_{\text s} = \boldsymbol{F} \). We can increase this force \( \boldsymbol{F} \) by pushing the box. The harder we push, the greater the force \( \boldsymbol{F} \). Also the static friction force \( F_{\text s} \) increases by the same amount and cancels out the force \( \boldsymbol{F} \), that is: \( \boldsymbol{F}_{\text s} = \boldsymbol{F} \). This happens up to a point where the force \( \boldsymbol{F} \) can no longer be compensated by the static friction force. There is no longer an equilibrium of forces and the box slides down. In this state, \( \boldsymbol{F} \) is greater than \( \boldsymbol{F}_{\text s} \).
The maximum possible static frictional force \( \boldsymbol{F}_{\text s} \), which can just balance the force \( \boldsymbol{F} \), is proportional to the normal force \( \boldsymbol{F}_{\text N} \). The proportionality constant between the static friction force and the normal force is called static friction coefficient \( \class{green}{\mu_{\text s}} \):
This dimensionless number \( \class{green}{\mu_{\text s}} \) indicates how hard it is to set a body in motion.
The static friction coefficient depends on the material of the box and the surface on which it is placed. The static friction force therefore varies depending on whether it is a wooden or metal box on an inclined plane. The material of the inclined plane, whether it is made of wood or metal, for example, also affects the static friction. The table below gives some examples of different materials of the box (or other body) and the surface on which it stands:
| Surfaces | Static friction coefficient \( \class{green}{ \mu_{\text s}} \) |
|---|---|
| Steel on steel | 0.2 |
| Wood on wood | 0.5 |
| Stone on wood | 0.9 |
| Stone on stone | 1.0 |
A wooden box with a mass of 10 kg is standing on a horizontal surface made of wood. What force must be applied to set the box in motion?
To calculate the static friction force \( F_{\text s} \), we multiply the static friction coefficient \( \class{green}{ \mu_{\text s}} \) by the normal force \( F_{\text N} \). The static friction coefficient here corresponds to 0.5, since both the box and the floor on which it stands are made of wood. The normal force is equal to the weight force \( F_{\text g} = \class{brown}{m} \, g \) of the box, that is the mass \( \class{brown}{m} \) times the gravitational acceleration: \( g = 9.8 \, \mathrm{m}/\mathrm{s}^2 \).
~&=~ \class{green}{\mu_{\text s}} \, F_{\text g} \\\\
~&=~ \class{green}{\mu_{\text s}} \, \class{brown}{m} \, g \\\\
~&=~ \class{green}{0.5} \cdot \class{brown}{10 \, \mathrm{kg}} \cdot 9.8 \, \frac{\mathrm{m}}{\mathrm{s}^2} \\\\
~&=~ 49 \, \mathrm{N} \end{align} $$
So you have to apply \( 49 \, \mathrm{N} \) to the box to make it move.
Kinetic (sliding) friction
Sliding friction occurs whenever two surfaces slide past each other. For example, when a box slides down an inclined plane (see illustration 2), kinetic friction acts on the contact surface between the box and the inclined plane. There is no sliding friction if the box is not moving.
The kinetic friction generates a kinetic friction force \( \boldsymbol{F}_{\text k} \), which always acts AGAINST the motion. Experimentally it can be found that the kinetic friction force \( \boldsymbol{F}_{\text k} \) is proportional to the normal force \( \boldsymbol{F}_{\text N} \):
The ratio of the kinetic friction force to the normal force is called kinetic friction coefficient \( \class{blue}{\mu_{\text k}} \). This dimensionless number indicates how hard it is to keep a body in motion on a given surface.
Just as with the static friction coefficient, the kinetic friction coefficient depends on the material of the box and the surface on which it is placed. The table below gives some examples of different materials of the box (or other body) and the surface on which it stands and the resulting coefficient of kinetic friction:
| Surfaces | Kinetic friction coefficient \( \class{blue}{\mu_{\text k}} \) |
|---|---|
| Steel on steel | 0.1 |
| Wood on wood | 0.4 |
| Stone on wood | 0.7 |
| Stone on stone | 0.9 |
A stone box weighing 20 kilograms slides over a horizontal surface made of wood. What kinetic friction force acts against the sliding motion?
To calculate the kinetic friction force \( F_{\text k} \), we multiply the kinetic friction coefficient \( \class{blue}{ \mu_{\text k}} \) by the normal force \( F_{\text N} \). The coefficient of kinetic friction here is 0.7, since the box is made of stone and the floor on which it stands is made of wood. The normal force is equal to the gravitational force \( F_{\text g} = \class{brown}{m} \, g \) of the box, that is, the mass \( \class{brown}{m} \) times the gravitational acceleration: \( g = 9.8 \, \mathrm{m}/\mathrm{s}^2 \).
~&=~ \class{blue}{\mu_{\text k}} \, F_{\text g} \\\\
~&=~ \class{blue}{\mu_{\text k}} \, \class{brown}{m} \, g \\\\
~&=~ \class{blue}{0.7} \cdot \class{brown}{20 \, \mathrm{kg}} \cdot 9.8 \, \frac{\mathrm{m}}{\mathrm{s}^2} \\\\
~&=~ 137.2 \, \mathrm{N} \end{align} $$
During the sliding movement \( 137.2 \, \mathrm{N} \) act against the direction of movement.
Rolling friction
Rolling friction occurs when a body (for example, a ball or a wheel) rolls on a surface. For example, when a ball rolls down an inclined plane (see Illustration 3), rolling friction acts on the contact surface between the ball and the inclined plane. Rolling friction does not exist when a round body does not roll.
Rolling friction generates a rolling friction force \( \boldsymbol{F}_{\text r} \), which always acts AGAINST the motion. Experimentally it can be stated that the rolling friction force \( \boldsymbol{F}_{\text r} \) is proportional to the normal force \( \boldsymbol{F}_{\text N} \):
The ratio of the rolling friction force to the normal force is called rolling friction coefficient \( \class{red}{\mu_{\text r}} \). This dimensionless number indicates how difficult it is to roll a round body on a given surface.
The rolling friction coefficient depends on the material of the round body and the surface on which it rolls. The table below gives some examples of different materials of the round body and the surface on which it rolls and the resulting rolling friction coefficient:
| Surfaces | Rolling friction coefficient \(\mu_{\text r}\) |
|---|---|
| Tire on asphalt | 0.011 bis 0.015 |
| Railroad wheel on rail | 0.001 bis 0.002 |
| Tire on concrete | 0.01 bis 0.02 |
| Tire on sand | 0.2 bis 0.4 |
A tire with a mass of 10 kg rolls over a flat asphalt road. What is the rolling friction force acting against the movement? (Rolling friction coefficient is 0.015).
To calculate the rolling friction force \( F_{\text r} \), we multiply the rolling friction coefficient \( \class{red}{ \mu_{\text r}} \) by the normal force \( F_{\text N} \). The normal force is equal to the weight force \( F_{\text g} = \class{brown}{m} \, g \).
~&=~ \class{red}{\mu_{\text r}} \, F_{\text g} \\\\
~&=~ \class{red}{\mu_{\text r}} \, \class{brown}{m} \, g \\\\
~&=~ \class{red}{0.015} \cdot \class{brown}{10 \, \mathrm{kg}} \cdot 9.8 \, \frac{\mathrm{m}}{\mathrm{s}^2} \\\\
~&=~ 1.47 \, \mathrm{N} \end{align} $$
During the rolling motion, 1.47 N act against the direction of motion. As you can see, the rolling friction force is negligible compared to the kinetic friction.