Applying Newton's Three Laws: Frictional Forces

Another quite common force is frictional force. Like the normal force, it is caused by direct contact between surfaces. However, while the normal force is always perpendicular to the surface, the frictional force is always parallel to the surface. To fully describe the cause of friction requires knowledge beyond the scope of classical mechanics. For our purposes, it is enough to know that friction is caused by electrical interactions between the two surfaces on a microscopic level. These interactions always serve to resist motion, and differ in nature according to whether or not the surfaces are moving relative to each other. We shall examine each of these cases separately.

Static Frictional Forces

Consider the example of two blocks, one resting on top of the other. If friction is present, a certain minimum horizontal force is required to move the top block. If a horizontal force less than this minimum force is applied to the top block, a force must act to counter the applied force and keep the block at rest. This force is called the static frictional force, and it varies according to the amount of force applied to the block. If no force is applied, clearly there is no static frictional force. As more force is applied, the static frictional force increases until it reaches a certain maximum value; once the horizontal force exceeds the maximum frictional force the block begins to move. The frictional force, defined as F_s^max, is conveniently proportional to the normal force between the two surfaces:

F_s^max = μ_sF_N

The constant of proportionality, μ_s is called the coefficient of static friction, and is a property of the materials that are interacting (i.e. two interacting rough materials will have a higher value of μ_s than two smooth materials).

This equation for maximum static frictional force contains a lot of information, and a few remarks must be made for clarification.

The equation seems to be relating two vectors, F_s^max and F_N. This relation is valid only for the magnitudes of the vectors, not the direction. In fact, the two vectors will always be perpendicular.
The equation introduces the concept of the coefficient of static friction. This constant varies from material to material, but does not depend on the orientation of the material on the surface. For example, if a block of wood is set on a concrete platform, μ_s is the same whether the block is on its side, its front, or its top. In other words, the coefficient does not change according to the surface area of contact.
Since the equation does not specify a direction for the frictional force, it must be stated and understood that the frictional force always acts in the opposite direction as the force applied to the object.
It is vitally important to remember that this equation only gives the maximum static frictional force, which corresponds to the maximum force that can be applied to a body before it moves. If a lesser force is applied to the body, a frictional force less than the maximum force counteracts the original force.

Though it is rather surprising that frictional force and normal force are related in such a simple manner, physical intuition tells us that they should be directly related. Consider again a block of wood on a concrete platform. The normal force is given by the weight of the wood. If an additional downward force is applied to the wood (producing a greater normal force) the surfaces are actually in closer contact than they were before, and the resulting electrical interactions are stronger. Thus, intuitively, a greater normal force yields a greater frictional force. Our intuition agrees with the equation.

Kinetic Frictional Forces

Once a force is applied to an object that exceeds F_s^max, the object begins to move, and static frictional forces no longer apply. The moving object does still experience a frictional force, but of a different nature. We call this force the kinetic frictional force. The kinetic frictional force always counteracts the motion of the object, and is independent of speed. No matter the speed of the object (as long as v≤ 0) it experiences the same frictional force. Also, for the same reasons as explained with static friction, the kinetic frictional force is proportional to the normal force:

F_k = μ_kF_N

This equation is of the same form as that for maximum static frictional force, and defines the coefficient of kinetic friction, μ_k, which has the same properties as μ_s, but a different value. μ_k is a property of the interacting materials, and, like μ_s, is independent of orientation of the objects. The only significant difference between the two friction equations is that the first measures the friction between two stationary objects and its value is dependent on the force applied to one, while second measures a frictional force that only exists when one of the objects is moving and which is not depend on the force applied to the block. Finally, when comparing static with kinetic friction, it must be noted that μ_s is always greater in value than μ_k. Simply stated, this means that it takes less force to keep a block moving than to start its motion.

These two types of friction, like the normal force, arise whenever two objects are in direct contact. Often both kinetic and static friction apply to a given situation, as an object might start at rest (when static friction applies) then begin to move (when kinetic friction applies). Though friction applies in so many situations, it is often ignored in order to simplify the situation. Unless friction is explicitly stated to be present in a given problem, in can be ignored. That said, friction remains one of the most widely used applications of Newton's Laws.

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