Equations Introduced and Used for this Topic:

• [latex]a=\dfrac{\Delta v}{t}[/latex]
• [latex]\vec{a}=\dfrac{\Delta \vec{v}}{t}[/latex]
• [latex]\Delta v=v_f-v_i[/latex]
• [latex]v=\dfrac{(v_1+v_f)}{2}[/latex]
• [latex]v_f=v_i+at[/latex]
• [latex]2ad={v_f}^2-{v_i}^2[/latex]
• [latex]d=v_it+\dfrac{1}{2}at^2[/latex]
• [latex]d=\dfrac{(v_i+v_f)t}{2}[/latex]

• [latex]F_f=\mu N \text{ or }\mu mg[/latex] (simplest case on horizontal surface)
• [latex]F_f=\mu N \text{ where }N=mg\text{ cos ø}[/latex] (on an inclined surface)

• [latex]F_{fs}=\mu _sN[/latex]
• [latex]F_{fk}=µ_kN[/latex]
• [latex]F_{fr}=µ_rN[/latex]
• [latex]F=ma[/latex]
• [latex]w=mg[/latex]

Where…

• [latex]F[/latex] is Force, measured in newtons (N)
• [latex]w[/latex] is Weight (it is a Force), measured in newtons (N)
• [latex]m[/latex] is Mass, commonly measured in kilograms (Kg), grams (g) or tonnes (t)
• [latex]a[/latex] is Acceleration (deceleration is negative), measured in metres per second squared (m/s²)
• [latex]g[/latex] or [latex]a_g[/latex] is the Acceleration due to Gravity and varies from 9.78 m/s² (equator) to 9.83 m/s² (North or South poles) … The average value of gravity is taken to be either 9.80 m/s²  or 9.81 m/s²
• [latex]F_f[/latex] is the Force of Friction, measured in newtons (N)
• [latex]µ_s, µ_k\text{ & }µ_r[/latex] are Coefficients of Friction and are magnitudes without units
• [latex]N[/latex] is the Normal Force, measured in newtons (N)
• [latex]F_{fs}[/latex] is the Force of Static Friction, measured in newtons (N)
• [latex]F_{fk}[/latex] is the Force of Kinetic Friction, measured in newtons (N)
• [latex]F_{fr}[/latex] is the Force of Rolling Friction, measured in newtons (N)
• ø is the measure of the Incline of the Surface the object is resting on

And…

• [latex]v[/latex] is Speed, commonly measured in metres/second (m/s) or kilometres/hour (km/h)
• [latex]\vec{v}[/latex] is Velocity, commonly measured in metres/second (m/s) or kilometres/hour (km/h) and includes a direction
• [latex]v_i[/latex] is Initial Speed, commonly measured in metres/second (m/s) or kilometres/hour (km/h)
• [latex]\vec{v_i}[/latex] is Initial Velocity, commonly measured in metres/second (m/s) or kilometres/hour (km/h) and includes a direction
• [latex]v_f[/latex] is Final Speed, commonly measured in metres/second (m/s) or kilometres/hour (km/h)
• [latex]\vec{v_f}[/latex] is Final Velocity, commonly measured in metres/second (m/s) or kilometres/hour (km/h) and includes a direction
• [latex]∆ d[/latex] is Distance travelled, commonly measured in metres (m), kilometres (km)
• [latex]\Delta \vec{d}[/latex] is Change in Displacement, commonly measured in metres (m), kilometres (km)
  and includes a direction
• [latex]a[/latex] is Acceleration (deceleration is negative), measured in metres per second squared (m/s2)
• [latex]\vec{a}[/latex] is Vector Acceleration, measured in metres per second squared (m/s2) and includes a direction
• [latex]t[/latex] is time, commonly measured in seconds (s) or hours (h)

13.1 Friction

Equations Introduced or Used for this Section:

• [latex]F_{fs}=\mu _sN[/latex]
• [latex]F_{fk}=µ_kN[/latex]
• [latex]F_{fr}=µ_rN[/latex]
• [latex]F=ma[/latex]
• [latex]w=mg[/latex]

The by product of friction is generally heat.

Friction is a resistive force that acts to oppose the relative motion between two surfaces (objects, fluids, gases) that are moving past one another or are attempting to move. Studies of friction find their origin in various Greek philosophers as Aristotle (384-322 BCE), Victruvius (80 or 70 BCE- 15 BCE) and Pliny the Elder (23-79 CE), who were interested in both the causes of friction and how to reduce the amount of friction. The difference in frictional types of static and kinetic was known as early as Themistus (317-390 CE) who stated “it is easier to further the motion of a moving body than to move a body at rest“.

The first to develop the classic laws of friction was Leonardo da Vinci (1452-1519), who recorded his investigations using sketches of sliding blocks (shown above) on inclined surfaces. His ideas to use sliding blocks are still used today in demonstrating the fundamentals of friction that exists between solid bodies. Da Vinci’s works were never published and remained unknown until over 150 years later they were discovered again by Guillaume Amontons (1663-1705) and published as the Laws of Static Friction:

• 1st Law: The force of friction is directly proportional to the applied load.
• 2nd Law: The force of friction is independent of the apparent area of contact.
• 3rd Law: Kinetic friction is independent of the sliding velocity.

Amonton’s laws can be demonstrated by objects such as a brick resting on an inclined plane. Expanding the concept of friction from static to kinetic was accomplished by Bernard Forest de Bélidor (1698-1761) and Leonhard Euler (1707-1783).  Charles-Augustin de Coulomb (1736-1806) established that friction depended on several factors, such as the nature of the materials and coatings of the surfaces in contact, the weight or force of the load on the surface (Normal Force) and the length of time these surfaces were to remain in contact. The classical model of friction was completed by Arthur Morin (1795-1880) in his work on differences between sliding and rolling frictions, and that of Osborne Reynolds (1842-1912), who added the concept of friction that exists in the flow of fluids.

Current research is exploring the role of friction in earthquakes, in artificial limbs and hip replacements, tiny sensors that control air bags, nanotech investigations of polymer molecules to create fluids to reduce friction, and even exploring the role of rupture waves similar to sound waves in altering friction depending on the direction of motion.  Some of the main motivations for this and other work are to increase the longevity and to reduce energy loss as we become more aware of the importance in not losing energy in devices.  Studies of friction are of importance.

In this section, we will quantify static, kinetic and rolling frictions. The equations for these follow a pattern, that of the frictional force being a coefficient given by a standard symbol µ (value or ratio) of the Normal Force. The equations for all three of these frictions are shown below.

• [latex]F_{fs}=\mu _sN[/latex]
• [latex]F_{fk}=µ_kN[/latex]
• [latex]F_{fr}=µ_rN[/latex]

In all cases the symbol for friction is given by F_f with a subscript to distinguish which type of friction it is: s for static, k for kinetic and r for rolling.

The Normal Force is often described as the support force of a surface or other stable object that is in contact with an object being supported.  For an object on a desk or a chair, the Normal Force is the supporting force that acts to balance whatever force (generally gravity) that is holding the object to the surface.  If a person pushes a book against a wall, then the Normal Force will be balancing this force.

Finding the coefficients of friction between two bodies is a complicated activity, since it is an attempt to quantify the relationship between the Frictional Force and the Normal Force.  Technically, it is done through multiple experiments of two objects under similar conditions that can be affected by temperature, humidity, air pressure, dust, the types of solids, whether any fluids are present between the two surfaces, if the surfaces are flat and uniform … This list goes on. In theory, the coefficient of friction also takes into account both mechanical interactions and actions at the atomic level.

The following examples explore introductory friction questions on a flat horizontal surface.

13.2 Friction & Newton’s Second Law

Equations Introduced or Used for this Section:

• [latex]F_f=\mu N\text{ or “}\mu mg\text{" (simplest case)}[/latex]
• [latex]F_f=\mu N\text{ where }N=mg\text{ cos ø}[/latex]

• [latex]F_{fs}=\mu _sN[/latex]
• [latex]F_{fk}=µ_kN[/latex]
• [latex]F_{fr}=µ_rN[/latex]
• [latex]F=ma[/latex]
• [latex]w=mg[/latex]

In this section of questions we explore the relationship between Net Force, Applied Force, Frictional Forces and Newton’s second law. Central to this is the concept that friction resists the vector sum of applied forces, and if there is an unbalanced force (Net Force) that results from the blend of applied forces and friction, then acceleration results. Quantified, this relationship looks as follows:

[latex]F_\text{net}=F_\text{applied}-F_f\text{ where }F_\text{net}=ma[/latex]

The results of these calculations mean:

For an object at rest:

• If [latex]F_\text{net}=0[/latex], then [latex]F_\text{applied}=F_f[/latex]

For an object in motion:

• If [latex]F_\text{net}>0[/latex], then [latex]F_\text{net}=ma[/latex] and the object or system is accelerating.
• If [latex]F_\text{net}<0[/latex], then [latex]F_\text{net}=ma[/latex] and the object or system is decelerating or remains in a static situation.

One feature of interest is the role of friction in accelerating and decelerating. Friction will dictate the maximum value that some objects can accelerate or decelerate. An examples of this is the friction of tires on a road. The maximum rate of braking when decelerating depends on the friction that exists between the tires and the road. Likewise, the maximum rate that a vehicle can accelerate on a surface is dictated by the maximum friction that exists between the tires and the road. One can always add extra forces to complicate accelerating and decelerating but at the simplest:

acceleration & deceleration < maximum friction

If we are to summarize and put all the above on a sketch showing a change of applied force compared to friction as a function of time, then we would have something as follows.

There are two distinct regions shown with five possibilities:

1. If we are looking at how much force we are applying to an object at rest, the object will remain at rest until we are able to apply a force that is greater than the static force that resists it.
   • When the [latex]F_\text{applied}= F_{fs}[/latex] and the object remains at rest or the [latex]F_\text{net} = 0[/latex]
   • When the [latex]F_\text{applied} > F_{fs}[/latex] and the object will accelerate and the [latex]F_\text{net}= ma[/latex]
2. If we are looking at how much force we are applying to an object in motion, the object will react differently depending on how the applied force compares to the frictional force.
   •  When the [latex]F_\text{applied}= F_{fs}[/latex] and the object remains at constant speed or the [latex]F_\text{net}=0[/latex]
   •  When the [latex]F_\text{applied}>F_{fs}[/latex] and the object will accelerate and the [latex]F_\text{net}=ma[/latex]
   •  When the [latex]F_\text{applied}< F_{fs}[/latex] and the object will decelerate and the [latex]F_\text{net}=ma[/latex]

13.3 Friction & Accelerated Motion

Equations Introduced or Used for this Section:

• [latex]a=\dfrac{\Delta v}{t}[/latex]
• [latex]\vec{a}=\dfrac{\Delta \vec{v}}{t}[/latex]
• [latex]\Delta v=v_f-v_i[/latex]
• [latex]v=\dfrac{(v_1+v_f)}{2}[/latex]
• [latex]v_f=v_i+at[/latex]
• [latex]2ad={v_f}^2-{v_i}^2[/latex]
• [latex]d=v_it+\dfrac{1}{2}at^2[/latex]
• [latex]d=\dfrac{(v_i+v_f)t}{2}[/latex]

• [latex]F_f=\mu N \text{ or }\mu mg[/latex] (simplest case on horizontal surface)
• [latex]F_f=\mu N \text{ where }N=mg\text{ cos ø}[/latex] (on an inclined surface)

• [latex]F_{fs}=\mu _sN[/latex]
• [latex]F_{fk}=µ_kN[/latex]
• [latex]F_{fr}=µ_rN[/latex]
• [latex]F=ma[/latex]
• [latex]w=mg[/latex]

13.4 Friction on an Inclined Surface

Equations Introduced or Used for this Section:

Basic Trigonometric Ratios

• [latex]\text{Sin}=\dfrac{\text{Opposite}}{\text{Hypotenuse}}[/latex]
• [latex]\text{cos}=\dfrac{\text{Adjacent}}{\text{Hypotenuse}}[/latex]
• [latex]\text{Tan}=\dfrac{\text{Opposite}}{\text{Adjacent}}[/latex]

[latex]F_f=\mu N \text{ where }N=mg\text{ cos ø}[/latex]

• [latex]F_{fs}=\mu _sN[/latex]
• [latex]F_{fk}=µ_kN[/latex]
• [latex]F_{fr}=µ_rN[/latex]
• [latex]F=ma[/latex]
• [latex]w=mg[/latex]

When we consider multiple forces acting on a body or a system we are drawing what is known as Free Body Diagrams. Free body diagrams represent a picture of the forces that are acting on a body or a system. In the examples below, the forces illustrated are friction, applied forces, the Normal force and weight.

You can see for these situations that the angle of the incline is 0° and as such cos 0° = 1. Because of this, introductory physics students often remember the Normal force as equaling [latex]N=mg[/latex] instead of the proper [latex]N=mg\text{ cos ø}[/latex], where the only force holding the object to the surface is gravity. This is a request for you as students to use the equation for Normal Force as given by the equation: [latex]N=mg\text{ cos ø}[/latex]

These diagrams become a little more complicated when used to analyze objects on inclined surfaces.

Both of these examples are nearly identical to the previous Free Body Diagrams, except that they now take the weight vector and break it up into the force that is pulling the object directly into the surface and the part of the force that is now acting to pull the object down the plane.

These Free Body diagrams can be also drawn showing Forces that act in either direction, both up and down the inclined surface.  As above, the Free Body Diagrams are identical to the ones above with one addition, the applied force and the direction of the frictional force.  The frictional force will always act opposite to the net force between applied force and the force of gravity pulling the object down the incline. In the example to the left, the frictional force is shown as acting either up or down the incline. Friction in this case acts opposite to the direction of the vector sum of the applied force and the force of gravity pulling down the incline.

Exercise Answers

13.1 Friction

1. ≥ 196 N
2. 0.306
3. 0.041
4. ≤ 7200 N
5. 19.6 N
6. 0.85

13.2 Friction & Newton’s Second Law

1. 0.44 m/s²
2. 1. 0.41
   2. 1.1 m/s² 
3. 39.2 m/s²

13.3 Friction & Accelerated Motion

1. 0.031
   1. 104 m
   2. 151 m
2. 48 m
3. 0.19
4. 331 m
5. 0.023
6. 10 s
7. The person will move
8. − 1.72 N

13.4 Friction on an Inclined Surface

1. 0.54 m/s²
2. 793 N
3. 22.8°

Media Attributions

• “yellow and gray haul truck carrying sand and soil, Dumper, Coal Mining” from Wallpaper Flare is in the public domain.
• “Tribometre Vinci 1” by Léonard de Vinci is in the public domain.