You’ve probably heard the term “friction” before. It may bring up images of slipping around on ice or of figuring out how high you can angle your binder before the pencil rolls off. As it turns out, those are actually two different types of friction – the first being kinetic friction and the second being static friction. In this piece, we’ll be learning about the types of friction, the role of the coefficient of friction, and how to use the friction force formula.

What is Friction?

Friction is something you are likely already familiar with and you may have some working definition of it in your mind. In physics, friction is defined as a contact force that resists movement. A contact force is one that requires objects to be touching. Resisting movement can mean one of two things – preventing something from starting to move or slowing something down if it’s already moving. These are two separate types of friction.

Static and Kinetic Friction

Static friction is the type of friction that will prevent an object from moving. Kinetic friction is the type of friction that will slow down an already-moving object. The difference between static and kinetic friction is that kinetic friction happens when something is sliding. You can probably think of some different examples in your life that involved these different types of friction – you may even be starting to realize how important they are. Without friction, you couldn’t hold onto your phone, the slightest touch would send any item sliding across the table to the floor, and you wouldn’t even be able to walk. 

Coefficient of Friction

Now we have a working definition for the two types of friction and we know a bit about when to apply them. But there are still two things we need to know before we can start calculating frictional forces. We need to know what a coefficient of friction is and we need to know precisely when to utilize each type of friction.

What is the Coefficient of Friction?

The coefficient of friction is a constant that helps us understand the amount of frictional force two surfaces apply when moving or attempting to move across one another. This is usually some value between zero and one (though it can be higher in extreme cases) that essentially tells us how large the frictional force will be relative to the normal force. Higher coefficients of friction correspond to higher frictional forces. The coefficient of friction changes with each situation and with each type of friction. Depending on the situation you’re in, you’ll be looking at a coefficient of static friction or a coefficient of kinetic friction. 

This may seem strange and confusing, but you’ve got some experience with this as well. For example, many people keep cases on their phones because the glass backing of smartphones has a lower coefficient of static friction than a rubber case. This makes it easier for the phone to slip out of your hand than the case. If you’ve ever tried moving a large piece of furniture across the floor, then you’ve probably got some experience with why we need both coefficients. Have you ever noticed it’s usually easier to keep something moving than it is to get the same object to start moving? That’s because the coefficient of static friction (the force keeping the object in place) is generally higher than the coefficient of kinetic friction (the force working against a moving object).

Table of Common Coefficients of Friction

We’ll go more in-depth on how to actually find the coefficient of friction later. For now, here are some common examples of both the coefficient of static friction and the coefficient of kinetic friction and how these two values tend to differ.

Frictional System	Coefficient of Static Friction	Coefficient of Kinetic Friction
Rubber on dry concrete	1.0	0.7
Rubber on wet concrete	0.7	0.5
Metal on wood	0.5	0.3
Shoes on wood	0.9	0.7
Shoes on ice	0.1	0.03

When to Use Static Friction vs Kinetic Friction

The coefficient of friction depends of which of the two types of friction we are working with. In general, we use static friction when something is stationary and kinetic energy when something is moving. However, you can’t think simply of something moving because if an object is rolling along a surface, then no individual piece of that object is sliding across the ground. If an object is sliding, we are dealing with kinetic friction. You may think this is a slight distinction, but imagine if you were to use the wrong coefficient while designing parts of a car. In a situation like that, having your numbers correct could prevent major problems from occurring. So, we use kinetic friction when the molecules of one object would be sliding past the molecules of another and we use static friction in cases where individual molecules would be stationary relative to each other. 

Friction Force Formula

As with everything in physics, we have an equation that allows us to calculate the force of friction in any situation. Although there are two different types of friction, you really only need to remember one equation:

Friction Formula F_{friction}= \mu F_{N}

Here, F_{friction} simply informs us the force we’re calculating for is a frictional one, \mu is the coefficient of friction, and N represents the normal force. You may now be wondering, if there’s only one equation, how will you ever know which coefficient of friction to use? The answer is that it’s often up to you to decide.

Once you determine which type of friction you are using, you can add a subscript to the coefficient of friction to know which one you’re working with. Typically, the coefficient of static friction is denoted as \mu_{s} while the coefficient of kinetic friction is denoted as \mu_k. Now that we know the equation for friction and how to denote which type of friction we’re working with, let’s get into some problem-solving.

Example: How to Calculate the Force of Friction

A heavy box is pushed across a factory floor. The normal force on the box is 50\text{ N}, the coefficient of kinetic friction is 0.3, and the coefficient of static friction of 0.5. What is the force of friction on the box?

Step 1: Determine the Type of Friction Present

We were given the coefficients for both types of friction, but we only need one. In this case, our box is being pushed, which implies a sliding motion. So, we’ll need to use the coefficient of kinetic friction for this problem.

Step 2:  Identify What You Know

Now that we know what type of friction we’re working with, we can get started following our normal problem-solving steps by pulling the relevant information out of the problem.

• F_{N}=50\text{ N}
• \mu_{k}=0.3

Step 3: Identify the Goal

In this case, we’re looking for the force of friction, so the goal can be written as:

• F_{f}=\text{?}

Step 4: Gather Your Tools

To solve this problem, we’ll only need our friction equation. We’ll substitute the coefficient of kinetic friction to make the equation specific to the question we’re trying to answer.

• F_{f}=\mu_{k}F_{N}

Step 5: Put it all Together

F_{f}=\mu_{k}F_{N}

F_{f}=0.3 \cdot 50\text{ N}

F_{f}=15\text{ N}

Example: How to Calculate the Coefficient of Friction

The same box comes to rest on the same factory floor. The normal force on the box is still 50\text{ N} and the frictional force on the box is now 25\text{ N}. What is the coefficient of friction between the box and the factory floor?

Step 1: Determine the Type of Friction Present

The question is left open so we could be looking for the coefficient of kinetic or static friction. Because our box is at rest, there is no sliding motion, and so we know we’ll be looking for the coefficient of static friction.

Step 2:  Identify What You Know

• F_{N}=50\text{ N}
• F_{f}=25\text{ N}

Step 3: Identify the Goal

In this case, we’re looking for the coefficient of static friction, so the goal can be written as:

• \mu_{s}=\text{?}

Step 4: Gather Your Tools

To solve this problem, we’ll only need our friction equation. We’ll substitute the coefficient of static friction to make the equation specific to the question we’re trying to answer. 

• F_{f}=\mu_{s}F_{N}

We aren’t looking for the force of friction this time, but we’ll be rearranging this equation in the next step.

Step 5: Put it all Together

We’ll start by rearranging our equation to solve for the coefficient of static friction.

F_{f}=\mu_{s}F_{N}

\mu_{s}=\dfrac{F_{f}}{F_{N}}

Now, we’ll plug in our numbers and start solving.

\mu_{s}=\dfrac{25\text{ N}}{50\text{ N}}

\mu_{s}=0.5

Experiment Overview: How to Find the Coefficient of Friction

Now that you know a little more about friction and how to calculate it, you may be wondering how we measure frictional forces and where the coefficients of friction even come from. The answer, as with many things in science, is an experiment. During a friction lab experiment, you have some object or objects that you know the mass of (allowing you to calculate the normal force on that object). From there, you can figure out how to measure the force needed to set a stationary object in motion (measuring static friction) or to keep an object moving at a constant speed (measuring kinetic friction). Taking the measurement of your frictional force along with the normal force on your object allows you to create a graph such as the one below.

Frictional Force vs Normal Force Graph

The graph above is friction vs normal force, which shows the frictional force along the y-axis and the normal force along the x-axis. This is the graph you would create while completing a coefficient of friction lab. The really important and interesting part of this graph is the slope. If you have the correct software, it will automatically create a trendline and provide you with the slope of that line as we have here.

Finding the Coefficient of Friction with Slope

If you recall how to calculate slope, you’ll know it is effectively the change in y-value divided by the change in x-value. So, the slope of this line here is:

\text{slope}=\dfrac{\text{change in frictional force}}{\text{change in normal force}}

If we were to look out our equation for the force of friction it would be:

\text{coefficient of friction}=\dfrac{\text{frictional force}}{\text{normal force}}

Although the slope of the line is looking at the change in the forces, at any instantaneous point, these two equations will be effectively the same. This is even more obvious when looking at things from a slope-intercept format, as shown below.

y-value		slope	x-value
y	=	m	x
F_{friction}	=	\mu	F_{N}

Regardless of how you go about proving it, the slope of the line created by measuring the frictional force on an object relative to the normal force on that object is equivalent to the coefficient of friction between those two materials. This is true for finding both types of friction.

Friction on an Inclined Plane

Now that we have a firm understanding of friction on a flat surface, let’s look at something moving along or sitting on top of an inclined plane. You can actually run an experiment with this now. Grab any small object (sticky notes, a pen, an eraser, your keys, etc.) and something flat that you can lift on one end to create an inclined plane (a book, a note pan, a clipboard, a phone, etc.). Now place your smaller object on the flat surface and slowly raise one side of the flat surface to create an inclined plane with a greater and greater angle. Did the object start to slide off immediately? Probably not. You had to reach a certain height before it would, but why? The answer to that question, of course, lies in friction.

Free Body Diagram with Friction

Friction on an inclined plane works just like friction on a flat surface with one key difference. The angle of the inclined plane will affect the magnitude of the normal force which will in turn affect the magnitude of the frictional force. If you need a total refresh on how forces on an inclined plane work, you can check here, but the diagram below tells us everything we need to know right now.

The angle of the plane’s incline is equivalent to the angle between the gravitational force and the normal force. The significance of this is that if the incline changes then that angle changes, causing the normal force as well as the frictional force to change. The diagram above shows the force of kinetic friction, but this works exactly the same for both types of friction.

Generally, to calculate the normal force, you would use the equation F_{N}=mg. In this case, however, because we have this angle present, the normal force will be F_{N}=mg\cos(\theta). If you aren’t comfortable with trigonometry and want to review, this may be a good resource. With all of this in mind, we effectively think of the equation for the force of friction on an inclined plane as:

F_{f}=\mu F_{N}=\mu mg\cos{\theta}

Note: There may be times when you want to take the sine of an angle instead of the cosine of an angle. We’ll be sticking to situations where the cosine of the angle will always work, but be prepared to analyze problems you see in classes for which angle will work best.

Example: Finding the Force of Friction on an Inclined Plane

A box with a mass of 10\text{ kg} sits at rest on a plane inclined 30^{\circ} above the horizontal. The coefficient of kinetic friction between the box and the plane is 0.2 and the coefficient of static friction is 0.7. What is the force of friction on the box?

Step 1: Determine the Type of Friction Present

We were given the coefficients for both types of friction, but we only need one. In this case, our box is sitting at rest so there is no motion. So, we’ll need to use the coefficient of static friction for this problem.

Step 2:  Identify What You Know

We have two places we can draw information from in terms of what we know now. We have the information we can pull from our problem statement:

• m=10\text{ kg}
• \theta=30^{\circ}
• \mu_{s}=0.7

We can also go ahead and assume that we are on Earth and pull the constant value for gravitational acceleration on Earth’s surface:

• g=9.81\text{ m/s}^{2}

Step 3: Identify the Goal

• F_{f}=\text{?}

Step 4: Gather Your Tools

To solve this problem, we’ll only need our friction on an inclined plane equation. We’ll substitute in the coefficient of static friction to make the equation specific to the question we’re trying to answer.

• F_{f}=\mu_{s}mg\cos\theta

Step 5: Put it all Together

F_{f}=\mu_{s}mg\cos\theta

F_{f}=0.7\cdot 10\text{ kg}\cdot 9.81\text{ m/s}^{2} \cdot \cos(30^{\circ}) 

F_{f}=60\text{ N}

Example: Finding the Coefficient of Friction on an Inclined Plane

A person sleds down a snowy hill with an angle 45^{\circ} above the horizontal. The person has a mass of 100\text{ kg} and experiences a frictional force of 30\text{ N}. What is the coefficient of kinetic friction between the sled and the snow?

Step 1: Determine the Type of Friction Present

We were told in this case that the person is sledding down a snowy hill which implies a sliding motion. So, for this problem, we’ll be looking for the coefficient of kinetic friction.

Step 2:  Identify What You Know

Again, we’ll be pulling information from our problem statement as well as the known constant of gravitational acceleration on Earth’s surface.

• \theta=45^{\circ}
• m=100\text{ kg}
• F_{f}=30\text{ N}
• g=9.81\text{ m/s}^{2}

Step 3: Identify the Goal

• \mu_{k}=\text{?}

Step 4: Gather Your Tools

Although we are now looking for the coefficient of kinetic friction instead of the force of kinetic friction, we’ll be starting with the same equation and rearranging it in our next step.

• F_{f}=\mu_{k}mg\cos\theta

Step 5: Put it all Together

We’ll begin by rearranging the equation to solve for the coefficient of kinetic friction:

F_{f}=\mu_{k}mg\cos\theta

\mu_{k}=\dfrac{F_{f}}{mg\cos\theta}

Now we can plug in our known values to finish solving the problem:

\mu_{k}=\dfrac{30\text{ N}}{100\text{ kg}\cdot9.81\text{ m/s}^{2}\cdot \cos(45^{\circ})}

\mu_{k}=0.04

Conclusion

From this piece, you should have everything you need to know about friction to get through any high school physics course. Both types of friction, static friction and kinetic friction, shape the world we live in. As you move through your day, you’ll be able to find countless examples of this force in your life. Even being able to walk or move around in a wheelchair is dependent on the existence of friction. This force will continue to appear within your physics journey and will come up if you choose to pursue any engineering work later.