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The force of friction

Classical mechanics

The force of friction

Friction exists all around us. It brings objects to rest, keeps them at rest, decides if we slide or fall and is more integral to our daily life than we may think. Here’s an introductory look at friction.

A friction, as we know, is any conflict of interest concerning a particular event, idea or phenomenon. In physics, it retains this spirit in a slightly different, more “physical” form. Consider a ball rolling on the ground in your house or a crate skidding out of control in a shipyard. Both these objects, however different they may be, showcase a common experience: the objects under observation eventually come to a halt. In fact, almost every single object in the universe experiences a resistive force that opposes it’s state of motion, and when this is between two surfaces in contact, we call it friction. (Other forms of resistance, such as air resistance, tension, drag, terminal velocity etc. will be the subject of another article on resistive physical forces.)



Types of friction

First of all, it helps to understand the nature of frictional forces. This is simple enough: friction acts against a body, it is directed exactly opposite to the direction of motion of a body, and it acts parallel to the surface. The cause of frictional forces in everyday life are often attributed to the unevenness of surfaces, but this is a somewhat practical approach that is not universally true (look up cold welding).

Friction comes in various forms based on when it acts on a body. For example, consider a body at rest relative to a table. If you want to move the body by pushing it across the table, you begin by applying a force. The body may not move instantly, but as you keep increasing the force that you apply on the body, it eventually gives and begins to move. During the time when you attempted in vain to push the body, it was being acted upon by an external, opposing force specifically called static friction. This is the friction that, keeps a body at rest until a force that exceeds it has been applied to start the body’s motion.

Static friction is that friction which exists between two bodies when they are at rest relative to each other and which works towards keeping the bodies at rest.

Suppose you finally move the body, then you start to notice that the object is not moving as fast as you expect for the force that you apply, or, that some force seems to be opposing the motion of the body. This is called the kinetic friction of the body.

Kinetic friction is that friction which exists between two bodies when they are in motion relative to each other and which works against the state of motion of the body, attempting to bring it to rest.

One may be tempted to conclude that friction, either way, seems to prefer a state of rest. The wording should be made a little more carefully, however: friction opposes relative motion of two bodies. In this article we will maintain f_{s} and f_{k} to refer to static and kinetic friction respectively.

Static and kinetic friction

Static and kinetic friction

Coefficients of friction

The mathematical description of friction is afforded by what are called the coefficients of friction. In general terms, the coefficient, represented by \mu is the ratio of frictional force offered by the surface to the normal force existing between two surfaces. Indeed friction is proportional to this normal force, so called because it is perpendicular or “normal” to the two bodies in contact at any instant of time. Therefore,

    \begin{align*} \vec{f}_{friction}=\mu \vec{N} \end{align*}

Having established that a pair of objects at rest are kept at rest by static friction (or, alternatively, their state of no relative motion is retained), and that a pair of objects at rest are brought to rest by kinetic friction (or the state of relative motion is opposed), we realise that the coefficients associated with these two different forces must be different. Simple reasoning will quickly make this apparent: the same body must be acted by different magnitudes of friction depending on whether it is at rest or in motion, so, according to the above equation, with \vec{N} being constant and \vec{f}_{friction} varying by circumstance, there is no doubt that \mu should vary as well, proportionally.

In case of static friction, \mu is called the coefficient of static friction and, if we follow the same diagrammatic notation as we did above, we arrive at,

    \begin{align*} \vec{f_s}=\mu_{s} \vec{N} \end{align*}

where \mu_{s} is said coefficient. Note that this equation is dynamic, meaning it varies with the external force, \vec{F} being applied and the equation in the form stated is only valid for that instant when \vec{f_s}=\vec{f_s}_{max}, the maximum static friction.

Incidentally, this is not really a problem since we are rarely concerned with any circumstance other than that precise moment at which the object begins to move; that is to say, we are often interested in \vec{f_s}_{max}, which actually makes this equation useful — all other circumstances up to this moment of initial motion are considered to have proportionally less static friction given that the object remains at rest. This moment is called the threshold of motion: it is that point at which an applied force overcomes static friction and successfully makes an object move.

The case is same with the coefficient of kinetic friction:

    \begin{align*} \vec{f}_{k}=\mu_k \vec{N} \end{align*}

Although the equations for the static and kinetic coefficients seems analogous, there exist two key differences between them on a physical level. Firstly, kinetic friction is considered, for all practical purposes, constant over a wide range of applied forces. In fact this is what brings an object to rest as soon as no applied force acts on it to keep overcoming kinetic friction. Secondly, this constant kinetic friction is a little less than the maximum static friction. Why this happens is a complex, microscopic phenomenon having to do with surface roughness, but is rarely taken into consideration since, for all purposes in classical mechanics, the above set of equations adequately describe the frictional forces and their effects we see in our day-to-day life.

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V.H. Belvadi

V.H. Belvadi is a physics postgraduate specialising in solid-state physics and also interested in astroparticle physics. When he is free he makes photographs and short films, writes on his personal website, makes music, reads voraciously, or plays his violin.

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