How To Determine The Minimum Coefficient Of Static Friction

Friction is a force that opposes motion. Physicists distinguish between static friction, which acts to keep a body at rest, and kinetic friction, which acts to slow down its motion once it begins moving. The force exerted by static friction (​_F_s_​) is proportional to the perpendicular force exerted by a body against the surface along which it's moving, which is called the normal force (​_F_N_​). The proportionality factor is called the coefficient of static fraction, which is usually denoted by the Greek letter mu with a subscript ​s​ (​_µ_s_​). The mathematical relation is:

\(F_s=\mu_s F_N\)

This coefficient depends on the characteristics of the two surfaces that are in contact with each other. It has been tabulated for a number of different materials. If you can't find ​µ​_​s​ for the materials you're using, you can determine it with a simple experiment.

Use an Inclined Plane

A simple way to determine ​_µ_s​ is to place the object in question on an inclined plane made of the same material as the surface you're studying. Gradually increase the angle of the incline until the object starts to slide. Record that angle. You can immediately find ​µ_s_​ because it is equal to the tangent of the angle. Here's why:

As you raise the incline, the force of gravity acting on a body of mass ​m​ has a horizontal and a vertical component. Applying Newton's Law to each of these just before the body starts to move, you find the horizontal component (which acts in the ​x​-direction) to be ​_F_x_​ = ​_ma_x_​. The same is true in the ​y​ direction: ​_F_y_​ = ​_ma_y_​.

The acceleration in the ​x​-direction, ​_ma_x_​, is equal to the force of gravity, which is mass times acceleration due to gravity (​g​) times the sine of the angle (​ø​) formed at the fulcrum of the incline. Since the body is not moving, this is equal to the opposing force of static friction, and you can write:

\(1. mg\sin{\theta}=F_s\)

The ​y​-direction component of force, ​_ma_y_​, is equal to the cosine of the angle times the mass times the acceleration due to gravity, and this must equal the normal force, since the body isn't moving,

\(2. F_N=mg\cos{\theta}\)

Remember that ​_F_s​ = ​µ_sF_N_​. Substitute for ​_F_s_​ in equation (1):

\(mg\sin{\theta}=\mu_s F_N\)

and use the equality to equation (2) to substitute for ​_F_N_​:

\(mg\sin{\theta}=\mu_s mg \cos{\theta}\)

The term "​mg​" cancels from both sides:

\(\mu_s=\frac{\sin{\theta}}{\cos{\theta}}=\tan{\theta}\)