# Friction Experiment: Measure Static Sliding Coefficient of Friction with a Ramp

by Ron Kurtus (revised 21 November 2016)

The goal of this experiment is to measure the static sliding coefficient of friction between two surfaces by using a ramp and measuring its inclination.

The idea is that if you put a solid object on a ramp and start to tilt the ramp upward, there is a point where the object will start to slide. That is the angle where the force of gravity is strong enough to overcome the static sliding friction.

By simply knowing the angle or the inclination, you can then calculate the static sliding coefficient of friction between the two materials. You can cover the ramp with various materials to determine different coefficients.

(Also see: Sliding Friction on an Inclined Surface)

Questions you may have include:

• What materials are needed?
• What are the steps to take?
• How are the calculations made?

This lesson will answer those questions.

## Materials

1. A flat board to be used as a ramp
2. Optional covering material for the ramp
3. Objects to slide down the ramp

You need to be able to change the angle of the ramp.

## Steps

1. Measure the weight of your object
2. Place the ramp on the ground and put the object on the ramp
3. Slowly raise one end of the ramp until the object just starts to slide
4. Measure the height (A) and length (B) of the inclination, as in the drawing below Object on a ramp

The resulting static sliding coefficient of friction is:

μss = A/B

### Different combinations

You can use different combinations of materials to measure their coefficients of friction. For example, you can use a:

• Wooden board and a brick to calculate the kinetic coefficient of friction between wood and brick material
• Sheet of iron on the board and an iron block to slide down the ramp
• Sheet of iron with film of oil on it and an iron block to slide down the ramp
• Covering of wet linoleum and a shoe to see how slippery a floor can be

There are many combinations that you can measure.

## Explanation

Although the equation to find the static coefficient of friction is very simple, the principles behind it require some knowledge of Mathematics.

### Physics background

The coefficient of friction between two surfaces is a number that determines how much force is required to move an object that is held back by friction when the two surfaces are pressed together.

The standard sliding friction equation is:

Fss = μssN

where

• Fs is the resistive force of sliding friction
• μs is the coefficient of sliding friction for the two surfaces (Greek letter "mu")
• N is the normal force

When the object is on an incline, the normal force is:

N = W*cos(β)

where

• W is the weight of the object
• β is the angle of inclination (Greek letter "beta")
• cos(β) is the cosine of angle β

Thus, the equation for sliding friction is:

Fs = μsW*cos(β)

Since W = mg, the equation becomes:

Fs = μsmg*cos(β)

where

• m is the mass of the object
• g is the acceleration due to gravity

### Gravity force for object on incline

The force from gravity acting down the incline is the weight times the sine of the angle of inclination:

Fg = W*sin(β)

where

• Fg is the force from gravity pulling the object down the incline
• sin(β) is the sine of angle β

Since W = mg, the equation becomes:

Fg = mg*sin(β)

The point at which the object starts to move:

Fss = Fg

μsmg*cos(β) = mg*sin(β)

μs*cos(β) = sin(β)

μs = sin(β)/cos(β)

Since

sin(β) = A/C

cos(β) = B/C

tan(β) = A/B

## Derivation

The friction equation is Fr = fr x N, where Fr is the resistive force of friction or the amount of force required to overcome friction, fr is the coefficient of friction between the two surfaces, and N is the normal or perpendicular force pushing the two surfaces together. If the force pushing to surfaces together is gravity, then N equals the weight of the upper object.

#### Static and kinetic friction

For a sliding object, the static coefficient of friction results in the force required to start the object moving. Once the object is sliding at a steady rate, the kinetic coefficient of friction results in the force required to keep the object moving at that velocity.

#### Using ramp

A clever way to determine the static coefficient of friction is to start an object sliding down a ramp. The component of gravitational force that causes the object to just start moving is equal to the resistive force to keep the object stationary. That is the static force of friction.

Note that you must record what the two surfaces are. The coefficient of friction is always for two surfaces. For example, you could find the friction between wood and steel, wood on wood, rubber on wet pavement, and so on.

Knowing the force required to overcome the friction and the force pushing the object onto the ramp, will allow you to determine the static coefficient of friction.

### Mathematics

The coefficient of friction is calculated using trigonometry. Consider the triangle in the drawing below. Angles involved

C is the length of your ramp, which is inclined at an angle a and is at a height of A. The length of the sides of the triangle are A, B, and C. The relationship between the sides are the trigonometric functions sine of angle a,  which is abbreviated sin(a), cosine of a or cos(a) and tangent of a or tan(a).

Since sin(a) = A / C and cos(a) = B / C, then sin(a) / cos(a) = tan(a).

### Components of gravity

When an object that weighs W is on a ramp, the force of gravity can be divided into components in perpendicular directions.

#### Normal force component

The force pushing the object against the surface of the ramp is reduced because of the incline. The normal force N = W x cos(a), as show in the picture below. In the case where there is no incline, a = 0 degrees and N = W. Components on ramp

#### Component down the ramp

The component of gravity is pulling the object along the ramp is F = W x sin(a).

#### Object starts to move

Now when the angle a become steep enough, the object starts to move and F = Fr, which is the force of static friction required to start the object moving.

But you know that Fr = fr x N.

And for the object on the ramp, N = W x cos(a).

Thus W x sin(a) = fr x W x cos(a).

Using a little Algebra, we get fr = sin(a) / cos(a) or fr = tan(a).

Finally, since tan(a) = A / B, we have fr = A / B.

So, all you need to know is the angle the object starts to slide or the lengths of its sides, and you can easily determine the coefficient of friction between the two surfaces.

## Summary

You can measure the amount of static friction and the coefficient of friction of an object by seeing when it starts to slide on a ramp.

Be clever

## Resources and references

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Inclined Planes - The Physics Classroom

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