The normal force is one component of the contact force between two objects, acting perpendicular to their interface. The frictional force is the other component, acting parallel to the plane of the interface between objects.

Friction tends to oppose any relative motion between objects, acting at the surfaces in contact.

If there is relative motion, the frictional force is the kinetic force of friction.

If there is no relative motion, the frictional force is the static force of friction.

Push a book across a table. The kinetic force of friction, f_k is the force that brings the book to rest, opposing the motion.

f_k is proportional to the normal force:

f_k = m_k N

m_k is known as the coefficient of kinetic friction. It is a dimensionless number that depends only on the two surfaces in contact. Typical values are 0.1 - 1.0, but could be higher or lower.

Static friction can be much less intuitive than kinetic friction. For a book at rest on a flat table, where the only other forces are the force of gravity and the normal, the static force of friction is zero. If you then push horizontally with a small force, static friction establishes an equal-and-opposite force that keeps the book at rest.

As you steadily increase your force, the force of friction increases to match you. Eventually your force exceeds the maximum force of friction, and the book moves.

The maximum force of static friction, f_s, is given by:

f_{s max} = m_s N

m_s is known as the coefficient of static friction.

m_s ³ m_k. In other words, it is harder to start something moving than it is to keep it moving once it's moving.

Static friction can be difficult to deal with because in general:

f_s £ m_s N

We will often deal with the limiting case, involving the maximum force of static friction, but always think before writing down

f_{s max} = m_s N. We will deal with cases where f_s < m_s N.

An easy way to measure the coefficient of static friction is to place two objects together and then tilt them until the top one slides. The angle at which one object starts to slip on the other is directly related to the coefficient.

When the two objects are horizontal there is no frictional force. As the objects are slowly tilted, the force of static friction must increase from zero to counteract the component of the force of gravity that acts along the interface.

Eventually, as the angle increases, that component of the force of gravity exceeds the maximum value of the force of static friction, and the top object slides off.

How does the angle at which this occurs relate to the coefficient of static friction?

Take the limiting case, the angle reached just before the block begins to slide, and draw the free-body diagram.

Here we can use:

f_s = f_{s max} = m_s N

because this is the angle at which the force of static friction equals its maximum value.

Use a coordinate system with +x down the slope and +y perpendicular to the slope.

Split the force of gravity into x and y components.

Apply Newton's second law twice.

SFx = m ax = 0	|	SFy = m ay = 0
mg sin(q) - fs = 0	|	N - mg cos(q) = 0
mg sin(q) = ms N	|	N = mg cos(q)

Substitute the second expression into the first:

mg sin(q) = m_s mg cos(q)

The factors of mg cancel. Re-arranging gives:

sin(q) cos(q)

=

tan(q)

=

ms

So, the coefficient of static friction is equal to the tangent of the angle at which the objects slide.

A similar method can be used to measure m_k. To do that you give the top object a push as you increase the angle. When the top object keeps sliding with constant velocity, the tangent of that angle is equal to m_k.

A simple model of what's happening at the microscopic level helps us understand friction. Viewed under a microscope, a surface generally looks rather rough, with hills and valleys. When you put two surfaces together, they actually make contact at very few places. When trying to move surfaces past each other, the high parts on each surface get stuck on one another.

Something about the interaction between the surfaces must change when the surface area decreases. The force is certainly the same. What changes?

With the same force, decreasing the area increases the pressure, basically squishing the surfaces closer together. Things balance so the force of friction is, to a first approximation, the same.