Key words: Friction, sliding, static, kinetic, coefficient, incline, Physics, Ron Kurtus, School for Champions. Copyright © Restrictions

# Sliding Friction on an Inclined Surface

by Ron Kurtus (revised 23 January 2015)

The sliding friction of an object on a flat inclined surface is different than the friction for the same object on a level surface. The reason is that the resistive force of friction is a function of the normal or perpendicular force of the object on the surface. When the surface is at an angle, that force is reduced according to the angle of inclination.

Questions you may have include:

• What is the general friction equation?
• What is the friction equation at an angle?
• How can static coefficient be determined?

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

## Friction equation

The resistive force of friction equals the coefficient of friction times the normal or perpendicular force. The general friction equation is:

Fr = μN

where:

• Fr is the resistive force of friction
• μ is the coefficient of friction for the two surfaces (Greek letter "mu")
• N is the normal or perpendicular force pushing the two objects together

If the object is not yet moving, the static coefficient of friction (μS) is used.

If the object is sliding, the kinetic coefficient of friction (μK) is used.

## Equation at an angle

When the object is on a level surface, the normal force is the sum of the weight (W) of the object plus any forces pushing the objects together. For the sake of simplicity, let's only consider the weight of the object. Thus, on a level surface, the friction is:

Fr = μW

However, when the object is situated on an incline, the normal force pushing the object against the surface is the weight times the cosine of the angle of inclination.

Fra = μWcos(a)

where

• Fra is the friction force on the incline at angle a
• a is the angle of inclination
• cos(a) is the cosine of angle a

The weight of the object is a vector quantity pointing downward. It can be divided into its vector components Wcos(a), perpendicular to the incline surface, and Wsin(a), parallel to the surface.

Forces on object on inclined surface

Note that Wsin(a) is a force that could cause the object to slide down the incline or ramp. When Wsin(a) is greater than μSWcos(a), the object will start to slide.

## Finding static coefficient of friction

One way to determine the static coefficient of friction is by changing the inclination of the ramp until the object just starts to slide. That is when the kinetic friction takes over from the static friction.

At that angle:

Wsin(a) = μSWcos(a)

μS = Wsin(a)/Wcos(a)

Thus:

μS = tan(a)

## Summary

The sliding friction of an object on a flat inclined surface is a function of the normal or perpendicular force of the object on the surface. When the surface is at an angle, that force is reduced according to the angle of inclination.

You can use this information to determine the static coefficient of friction between the surfaces.

Help others learn

## Resources and references

Ron Kurtus' Credentials

### Websites

Friction Concepts - HyperPhysics

Friction - Wolfram Research Science World

Friction Resources - Extensive list

### Books

The following books are available from Amazon.com.

#### Basic

Complete Idiot's Guide To Physics by Johnnie T. Dennis; Alpha (2003) \$18.95

What Is Friction? (Ages 4-8) by Lisa Trumbauer; Children's Press (CT) (2004) \$4.95

Friction Science and Technology (Mechanical Engineering Series) by Peter J. Blau; Marcel Dekker Pub. (1995) \$89.95

Physics of Sliding Friction (NATO Science Series E:) by B.N. Persson, E. Tosatti; Springer Pub. (1996) \$358.00

Do you have any questions, comments, or opinions on this subject? If so, send an email with your feedback. I will try to get back to you as soon as possible.

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www.school-for-champions.com/science/friction_on_inclined_surface.htm.

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