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Laws of Dry Friction and Coefficients of Friction

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Equations

(Eq1)     Fm = μsN	Magnitude of the maximum static friction force
(Eq2)     Fk = μkN	Magnitude of the kinetic friction force

Nomenclature

symbol	description
Fm	maximum force
Fk	kinetic friction force
μs	coefficient of static friction
μk	coefficient of kinetic friction
N	normal component of the reaction surface

Explanation

Friction is both good and bad: it allows objects to move, such as in walking or driving, yet can result in a lot of wasted energy through the generation of heat, such as in engines, and in other areas.

Friction can be understood simply by observing the following picture of a cinder block sitting on asphalt:



Zooming in, it can be seen that there are irregularities in the surfaces of both materials, and when they are moved against each other, these small bits are pushing against each other. It is the summation of all of these horizontal forces causing resistance that make it hard to push an object with a lot of friction. When an object is pushed the bits may break off, or they may bump over each other. The weight of the object, or any vertical forces that are pushing down on the object, will influence the friction because as vertical downward force is increased, it becomes harder to overcome those horizontal forces between all the little small bits interconnected with each other between the contacting surfaces.

The laws of dry friction may be explained with a wooden block of weight W that is placed on a horizontal concrete plane surface as shown below.



The forces acting on the block are its weight W and the reaction of the surface. Since the weight has no horizontal component, the reaction of the surface also has no horizontal component. The reaction is therefore normal to the surface and is represented by N as shown above.

Now a horizontal force P is applied to the block as shown in the following figure:



If P is small enough, the block will not move. This only means that some other horizontal force must therefore exist, which balances P. This other force is the static-friction force F, which is actually the resultant of numerous forces acting over the entire surface of contact between the block and the plane, as indicated in the first picture in this lesson. It is generally assumed that these forces are due to the irregularities of the surfaces in contact and, to a certain extent, to molecular attraction.

If the force P is increased, the friction force F also increases, continuing to oppose P until its magnitude reaches a certain maximum value F_m, as shown in the following plot:



If P is further increased, the friction force cannot balance it any more and the block starts sliding. A more accurate representation of the previous plot may be the following:



F and P are relatively proportional until F_m is reached, and motion begins. F_k may be shown as a wavy and bumpy line because the frictional force varies at the micro-level interactions, as seen in the first picture in this lesson, which may not be consistent.

Notice in the following figure that, as the magnitude F of the friction force increases from 0 to F_m, the point of application a of the resultant N of the normal forces of contact moves to the right, so that the couples formed, respectively, by P and F and by W and N remain balanced.



If N reaches point b, which is fixed at the bottom corner of the block, before F reaches its maximum value F_m, the couples will no longer be balanced and the block will tip about point b before the block can start sliding.

As soon as the block has been set in motion, the magnitude of F drops from F_m to a lower value F_k as shown below and in the above plots:



This is because there is less interpenetration between the irregularities of the surfaces in contact when these surfaces move with respect to each other. From then on, the block keeps sliding with increasing velocity while the friction force is now denoted by F_k, which is called the kinetic friction force. This can be observed in the plot above. The kinetic friction force remains approximately constant throughout the motion of the block.

Experimental evidence shows that the maximum value F_m of the static friction force is proportional to the normal component N of the reaction of the surface. The equation is:

(Eq1)

Fm = μsN

Where μ_s is a constant called the coefficient of static friction. Similarly, the magnitude F_k of the kinetic friction force may be put in the form:

(Eq2)

Fk = μkN

Where μ_k is a constant called the coefficient of kinetic friction. The coefficients of friction μ_s and μ_k do not depend upon the area of the surfaces in contact. Both coefficients, however, depend strongly on the nature of the surfaces in contact. Since they also depend upon the exact condition of the surfaces, their value is seldom known with an accuracy greater than 5 percent. The corresponding values of the coefficient of kinetic friction would be about 25 percent smaller.

It is presumed that four different situations can occur when a rigid body is in contact with a horizontal surface:

1.	No friction (Px = 0)	F = 0 N = P + W		The forces applied to the body do not tend to move it along the surface of contact; there is not friction force.
2.	No motion (Px < Fm)	F = Px F < μsN N = Py + W		The applied forces tend to move the body along the surface of contact but are not large enough to set it in motion. The friction force F which has developed can be found by solving the equations of equilibrium for the body. Since there is no evidence that F has reached its maximum value, the equation Fm = μsN cannot be used to determine the friction force.
3.	Motion impending (Px = Fm)	Fm = Px Fm = μsN N = Py + W		The applied forces are such that the body is just about to slide, that is, motion is impending. The friction force F has reached its maximum value Fm and, together with the normal force N, balances the applied forces. Both the equations of equilibrium and the equation Fm = μsN can be used. Also note that the friction force has a sense opposite to the sense of impending motion.
4.	Motion (Px > Fm)	Fk < Px Fk = μkN N = Py + W		The body is sliding under the action of the applied forces, and the equations of equilibrium do not apply any more. However, F is now equal to Fk and the equation Fk = μkN may be used. The sense of Fk is opposite to the sense of motion.

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Next Lesson: Angles of Friction⇒