How To Find The Force Of Friction Without Knowing The Coefficient Of Friction

Most people understand friction in an intuitive way. When you try to push an object along a surface, the contact between the object and the surface resists your push up to a certain pushing strength. Calculating the frictional force mathematically usually involves the "coefficient of friction," which describes how much the two specific materials "stick together" to resist motion, and something called the "normal force" that relates to the mass of the object. But if you don't know the coefficient of friction, how do you work out the force? You can achieve this either by looking up a standard result online or conducting a small experiment.

1. Set Up an Inclined Surface Using Similar Materials

Use the object in question and a small section of the surface you can move freely to set up an inclined ramp. If you can't use the whole surface or the whole object, just use a piece of something made from the same material. For example, if you have a tiled floor as a surface, you could use a single tile to create the ramp. If you have a wooden cupboard as an object, use a different, smaller object made from wood (ideally with a similar finish on the wood). The closer you can get to the real situation, the more accurate your calculation will be.

Ensure that you can adjust the incline of the ramp, by stacking up a series of books or something similar, so you can make small adjustments to its maximum height.

The more inclined the surface, the more the force due to gravity will work to pull it down the ramp. The force of friction works against this, but at some point, the force due to gravity overcomes it. This tells you the maximum force of friction for these materials, and physicists describe this through the coefficient of static friction (​μ​_static). The experiment allows you to find the value for this.

2. Conduct the Experiment

Place the object on top of the surface at a shallow angle that won't make it slide down the ramp. Gradually increase the incline of the ramp by adding books or other thin objects to your stack, and find the steepest incline you can hold it at without the object moving. You'll struggle to get a completely precise answer, but your best estimate will be close enough to the true value for the calculation. Measure the height of the ramp and the length of the base of the ramp when it's at this inclination. You're essentially treating the ramp as forming a right-angled triangle with the floor and measuring the length and height of the triangle.

3. Find the Friction Coefficient

The math for the situation works out neatly, and it turns out that the tangent of the angle of the incline tells you the value of the coefficient. So:

\(\mu_{static}=\tan{\theta}\)

Or, because tan = opposite / adjacent = length of base / height, you calculate:

\(\mu_{static}=(\text{length of base})/(\text{height of ramp})\)

Complete this calculation to find the value for the coefficient for your specific situation.

4. Calculating the Force of Friction

\(F=\mu_{static} N\)

Where the "​N​" stands for the normal force. For a flat surface, the value of this is equal to the weight of the object, so you can use:

\(F=\mu_{static} mg\)

Here, ​m​ is the mass of the object and ​g​ is the acceleration due to gravity (9.8 m / s²).

For example, wood on a stone surface has a friction coefficient of ​μ​_static = 0.3, so using this value for a 10 kilogram (kg) wooden cupboard on a stone surface:

\(F=\mu_{static} mg=0.3\times 10\times 9.8 = 29.4\text{ N}\)

1. Locate the Coefficient of Friction for Your Situation

Look online to find the coefficient of friction between your two substances. For example, a car tire on asphalt has a coefficient of ​μ​_static = 0.72, ice on wood has ​μ​_static = 0.05 and wood on brick has ​μ​_static = 0.6. Find the value for your situation (including using the sliding coefficient if you aren't calculating the friction from stationary) and make a note of it.

2. Multiply the Normal Force by the Coefficient

The following equation tells you the strength of the frictional force (with the static friction coefficient):

\(F=\mu_{static} N\)

If your surface is flat and parallel to the ground, you can use:

\(F=\mu_{static} mg\)

If it isn't, the normal force is weaker. In this case, find the angle of the incline ​θ​, and calculate:

\(F=\cos{\theta} \mu_{static} mg\)

For example, using a 1 kg block of ice on wood, inclined to 30°, and remembering that ​g​ = 9.8 m / s², this gives:

\(F=\cos{\theta} \mu_{static} mg=\cos{30}\times 0.05 \times 1 \times 9.8 = 0.424\text{ N}\)