Rolling Friction: Definition, Coefficient, Formula (W/ Examples)

Friction is a part of everyday life. While in idealized physics problems you often ignore things like air resistance and the frictional force, if you want to accurately calculate the motion of objects across a surface, you have to account for the interactions at the point of contact between the object and the surface.

This usually means either working with sliding friction, static friction or rolling friction, depending on the specific situation. Although a rolling object like a ball or wheel clearly experiences less frictional force than an object you have to slide, you'll still need to learn to calculate rolling resistance to describe the motion of objects such as car tires on asphalt.

Rolling friction is a type of kinetic friction, also known as ​rolling resistance​, which applies to rolling motion (as opposed to sliding motion – the other type of kinetic friction) and opposes the rolling motion in essentially the same way as other forms of friction force.

Generally speaking, rolling doesn't involve as much resistance as sliding, so the ​coefficient of rolling friction​ on a surface is typically smaller than the coefficient of friction for sliding or static situations on the same surface.

The process of rolling (or pure rolling, i.e., with no slippage) is quite different from sliding, because rolling includes additional friction as each new point on the object comes into contact with the surface. As a result of this, at any given moment there is a new point of contact and the situation is instantaneously similar to static friction.

There are many other factors beyond the surface roughness that influence rolling friction, too; for instance, the amount the object and the surface for the rolling motion deform when they're in contact affects the strength of the force. For example, car or truck tires experience more rolling resistance when they're inflated to a lower pressure. As well as the direct forces pushing on a tire, some of the energy loss is due to heat, called ​hysteresis losses​.

The equation for rolling friction is basically the same as the equations for sliding friction and static friction, except with the rolling friction coefficient in place of the similar coefficient for other types of friction.

Using ​F​_k,r for the force of rolling friction (i.e., kinetic, rolling), ​F​_n for the normal force and ​μ​_k,r for the coefficient of rolling friction, the equation is:

\(F_{k,r} = μ_{k,r}F_n\)

Since rolling friction is a force, the unit of ​F​_k,r is newtons. When you're solving problems involving a rolling body, you'll need to look up the specific coefficient of rolling friction for your specific materials. Engineering Toolbox is generally a fantastic resource for this type of thing (see Resources).

As always, the normal force (​F​_n) has the same magnitude of the weight (i.e., ​mg​, where ​m​ is the mass and ​g​ = 9.81 m/s²) of the object on a horizontal surface (assuming no other forces are acting in that direction), and it is perpendicular to the surface at the point of contact. ​If the surface is inclined​ at an angle ​θ​, the magnitude of the normal force is given by ​mg​ cos (​θ​).

Calculating rolling friction is a fairly straightforward process in most cases. Imagine a car with a mass of ​m​ = 1,500 kg, driving on asphalt and with ​μ​_k,r = 0.02. What is the rolling resistance in this case?

Using the formula, alongside ​F​_n = ​mg​ (on a horizontal surface):

\(\begin{aligned}
F_{k,r} &= μ_{k,r}F_n\)
\(&= μ_{k,r} mg\)
\(&= 0.02 × 1500 \;\text{kg} × 9.81 \;\text{m/s}^2\)
\(&= 294 \;\text{N}
\end{aligned}\)

You can see that the force due to rolling friction seems substantial in this case, however given the mass of the car, and using Newton's second law, this only amounts to a deceleration of 0.196 m/s². I

f that same car was driving up a road with an upwards incline of 10 degrees, you'd have to use ​F​_n = ​mg​ cos (​θ​), and the result would change:

\(\begin{aligned}
F_{k,r} &= μ_{k,r}F_n\)
\(&= μ_{k,r} mg \cos(\theta)\)
\(&= 0.02 × 1500 \;\text{kg} × 9.81 \;\text{m/s}^2 × \cos (10 °)\)
\(&= 289.5 \;\text{N}
\end{aligned}\)

Because the normal force is reduced due to the incline, the force of friction reduces by the same factor.

You can also calculate the coefficient of rolling friction if you know the force of rolling friction and the size of the normal force, using the following re-arranged formula:

\(μ_{k,r} = \frac{F_{k,r}}{F_n}\)

Imagining a bicycle tire rolling on a horizontal concrete surface with ​F​_n = 762 N and ​F​_k,r = 1.52 N, the coefficient of rolling friction is:

\(\begin{aligned}
μ_{k,r} &= \frac{F_{k,r}}{F_n} \
&=\frac{1.52 \;\text{N}}{762 \;\text{N}} \
&= 0.002
\end{aligned}\)