How do you calculate coefficient of friction?

First, it's important to remember that there are two kinds of coefficient of friction. The first is the coefficient of static friction (##\mu_s##), which determines the maximum force that friction can provide to prevent an object from beginning to move. The second is the coefficient of kinetic friction (##\mu_k##), which determines on objects that are already moving. In general the coefficient of static friction will be greater than the coefficient of kinetic friction (##\mu_s > \mu_k##).

The third important thing to remember is that coefficients of friction depend on both of the two materials that are in contact. For example, the coefficient of static friction for steel on brass, steel on rubber, and rubber on brass will, in general, all be different.

To calculate the coefficient of static friction between two materials, you can place an object made of one material on a surface made of the other material. Then begin to slowly incline the surface. Measure the angle of elevation, ##\theta_c##, when the object first begins to break free and slide down the surface. The inverse tangent of that "critical angle" will be the coefficient of static friction.

##\tan^{ -1}(\theta_c) =\mu_s##

See if you can derive why this is the case by considering a free body diagram of the object on the inclined plane. Balance the component of gravity acting along the plane with the frictional force, which will be equal to the normal force times ##\mu_s##.

To determine the coefficient of kinetic friction, lower the angle slightly, and adjust it until giving the object a very slight push causes it to slide down the plane at constant velocity. The inverse tangent of that slightly smaller angle is the coefficient of kinetic friction.

Try this experiment out with a small object like a coin on the cover of a hardcover book, or the lid of a laptop computer.