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# Normal Force in Rolling Friction

by Ron Kurtus (15 November 2016)

The * normal force in rolling friction* is the perpendicular force pushing the wheel to the surface on which it is rolling. It is an essential part of the standard rolling friction equation.

Situations include a wheel rolling on a plane surface, a wheel with an axle on a plane surface, and a wheel acting as a roller between two objects.

Questions you may have include:

- What is the normal force for a wheel on an object?
- What is the normal force for a wheel with an axle?
- What is the normal force for a wheel between objects?

This lesson will answer those questions. Useful tool: Units Conversion

## Standard rolling friction equation

The standard friction equation for a wheel rolling on another surface is:

F_{r}= μ_{r}N

where:

**F**is the resistive force of rolling friction or rolling resistance_{r}**μ**is the coefficient of rolling friction for the two surfaces (Greek letter "mu" sub r)_{r}**N**is the normal or perpendicular force between the wheel and other surface

## Normal force for a wheel on an object

When a wheel is rolling on the level surface of another object, the normal force is typically the weight of the wheel.

Normal force is weight of the wheel

### On incline or curved surface

When the wheel is on an incline or curved surface, the normal force varies with the cosine of the angle between the weight and the direction perpendicular to the surface. Thus,

N = W*cos(α)

where

**N**is the normal force on the surface**W**is the weight is the wheel**α**(Greek letter small alpha) is the angle between**N**and**W****cos(α)**is the cosine of angle alpha

(See Friction on an Inclined Surface for more information.)

The normal force varies with the angle of the incline

## Normal force for a wheel with an axle

The normal force of a wheel that has an axle held in some other object—such as a cart or other vehicle—is the weight or the wheel plus the weight of the cart.

N = W1 + W2

where

**N**is the total normal force**W1**is the weight of the object on the axle of the wheel**W2**is the weight of the wheel, including its axle

Normal force is sum of wheel and object weights

## Normal force for rollers between objects

When wheels are placed between two objects, they act as rollers. There is rolling friction both on the top and bottom of the wheel. The normal force on the top is the weight of the top object, and the normal force on the bottom is the weight of the top object and the wheel.

Normal forces for roller configuration

Thus, the friction between the top object and the rollers is **F _{r1} = μ_{r}(W1)**, where

**W1**is the weight of the upper object and the normal force on each roller.

**F**is the friction force acting on the rollers.

_{r1}The friction of each roller on the bottom object is **F _{r2} = μ_{r}(W1 + W2)**, where

**W2**is the weight of the roller and

**W1 + W2**is normal force on the bottom object.

**F _{r2}** is the friction force that acts along with

**F**to cause the rollers to turn.

_{r1}## Summary

When a wheel is rolling on a plane surface, the normal force on the surface is the weight of the wheel, along with a cosine factor is the surface is on an incline. When a wheel with an axle is rolling on a plane surface, the normal force on that surface is the weight of the cart plus that of the wheel. In the case of rollers between two objects, the weight or the top object and the roller create the normal force.

Always try to do your best

## Resources and references

### Websites

**Friction Resources** - Extensive list

**Rolling friction and rolling resistance** - includes coefficients - Engineering Toolbox

**Rolling Friction** - simple explanation - Davidson College

### Books

**Top-rated books on Friction Science**

**Top-rated books on Friction Experiments**

## Questions and comments

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## Normal Force in Rolling Friction