Friction  (Page 5/5)

The external force along the contact surface exactly equals to kinetic friction during the motion. In this case, the body is subjected to a balanced external force system (including friction) having zero net force. As such, the body will continue moving with whatever initial velocity is given to the body. What it means that we apply external force, $F>F_s$ , momentarily to impart initial velocity to the body and then subsequently maintain uniform velocity by external force, F, such that $F=F_k$ .

A typical friction force - time plot for the situation is shown in the figure. Here, we have considered that external force along the contact surface is increased slowly till it equals maximum static friction, then external force is adjusted simply to maintain velocity of the body.

The maximum static friction coefficient, ${\mu }_s$ , and kinetic friction coefficient, ${\mu }_k$ , are usually close values for a pair of given surfaces and may be taken to be equal as an approximation.

2: Accelerated motion :

If we maintain external force along the contact surface in excess of kinetic friction, then body undergoes accelerated motion.

The free body diagram is shown here in the figure. Let “a” be the acceleration towards right.

$$\begin{array}{l}\sum F_x=F-F_k=ma\\ \Rightarrow a=\frac{F-F_k}{m}\end{array}$$

Laws of friction

The properties of friction as explained above are summarized in three laws of friction as :

1: If a force fails to initiate the motion of a body in contact with another surface, then static friction is equal to the component of net external force parallel to the surface of the contact surface.

$$\begin{array}{l}{f}_s=F_{||}\end{array}$$

2: The static friction has a maximum value (Fs), which is given by :

$$\begin{array}{l}{F}_s={\mu }_{s}N\end{array}$$

where ${\mu }_s$ is the coefficient of static friction and "N" is the magnitude of normal force. The maximum static friction ( $F_s$ ) and normal force (N) are mutually perpendicular to each other . The coefficient of static friction depends on the nature of the surfaces in contact.

3: If the body begins to move along the surface, then friction force, called kinetic friction, is reduced slightly, which is given by :

$$\begin{array}{l}{F}_k={\mu }_{k}N\end{array}$$

where " ${\mu }_k$ " is coefficient of kinetic friction and depends on the nature of surfaces in contact. Experimentally, it is found that ${\mu }_k<{\mu }_s$ .

Area of contact and friction

We observe that area of contact does not appear anywhere in our consideration of friction. Though, we might generally believe that a greater contact area should offer greater friction and would be difficult to move.

In order to understand the absence of area in our consideration, let us consider the different shapes of bodies of equal mass in contact with a given surface.

Irrespective of the difference in contact areas, the friction in all three cases is same.

If we recall, friction results from the temporary joints formed between the contact surfaces. Thus, friction depends on (i) the numbers of contact points and (ii) the force inducing joints at these contact points.

The actual contact area may be much less - only up to 40 % of the total surface area in the case of ordinary plane surface. This means that friction does not depend on the total area, but only the part of the area which is actually in contact with the other surface. This fact is incorporated in our consideration through "coefficient of friction" – which represents the mutual characteristic of two surfaces – not of one surface. It largely accounts for the numbers of contact points between two surfaces and hence is characteristic of a pair of surfaces in contact.

On the other hand, formation of joints at the contact points depend on the distribution of normal force at contact points. Normal force, in turn, is distributed across the surface. If the density of contact points throughout is uniform, we can say that formation of joints depends on the normal force per unit area. Normal force is equal to the product of normal force per unit area and area as given here :

$$\begin{array}{l}N=\left(\frac{N}{A}\right)A\end{array}$$

It is clear that by considering normal force we have implicitly accounted for the area of contact. In the nutshell, we can say in a very general way that (i) coefficient of friction largely accounts for the contact points between a given pair of surfaces and (ii) normal force accounts for the distribution of force at contact points across the whole area.