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Co-efficient of Friction


What is the definition of the dynamic coefficient of friction?

If a body of mass W rests on a plane horizontal surface and needs a force F to move it horizontally across the surface in a sliding mode then the coefficient of friction is numerically equal to F/W. It is usually given the symbol µ.


Coefficients of Friction

However, there are two coefficients:


i) The limiting coefficient of static friction. This is based on the force required to start the body moving.

ii) The dynamic coefficient of friction. This is based on the force required to keep the body moving at constant velocity across the surface.


With many combinations of materials, µ static > µ dynamic, but for rubber and plastics in particular, µ dynamic may increase with velocity and become greater than µ static.

How is friction developed?

There are essentially two components of friction that create the horizontal/lateral forces that occur in sliding.


The first is an interaction between the peaks and troughs inherent in the micro roughness and indeed macro roughness of the two sliding surfaces. These peaks are sometimes referred to as asperities. It is a form of interlocking of the surfaces which requires a particular level of force to overcome it.


The second component of friction is a molecular adhesion between the two surfaces. Whilst not a major component in very rough surfaces, it becomes the major component when the surfaces are smooth, and in particular when one surface is rubber or plastic. With a smooth surface such as glass and a very slightly damp, soft rubber (the slight dampness appears to enhance the adhesion) one can achieve far higher values of µ than can be obtained with even the roughest surfaces.

Do materials have a unique value of coefficient of friction?

The coefficient of friction is dependent on both the material of the surface and that of the body which is sliding over it. One might argue that if both these were the same material then one would get a unique value for that material. Unfortunately, this is not generally true since it can depend on the degree of roughness of each of the surfaces.

Does µ depend on contact pressure?

With many hard, non-resilient materials, such as metals, ceramics and timber, µ is independent of the contact pressure over a relatively wide range. This unfortunately does not extend to any system where one or both surfaces are rubber or plastic. Depending on the hardness of the rubber/plastic one can get some degree of independence of contact pressure, but it is over a much restricted range of pressures.

How does the presence of a contaminant such as water affect µ?

Not only does the presence of a contaminant such as water or oil affect the value of µ, but also the manner in which it affects µ is very dependent upon factors such as the contact pressure, the size and shape of the contact area, the velocity of movement and the viscosity of the liquid. This makes the measurement of µ in wet conditions much more complex. The failure to understand this complexity has resulted in different models of slip test machine giving totally different test results for the same surface (see next section).

How do dry contaminants affect µ?

Dry contaminants such as dust, powders, etc., can well affect µ by various different mechanisms. These contaminants can act like tiny spheres and have a ball-bearing effect. They can effectively destroy the adhesion factor in the creation of friction forces (see B4). They can fill in the troughs of the roughness, thus changing the nature of the surface. It all depends on the dry contaminant itself and the nature of the two sliding surfaces.

Is slip resistance the same as the coefficient of friction?

Slip resistance is the dynamic coefficient of friction of a walking surface as determined in relation to a given contaminant and a given shoe heel material, as would be experienced by a slipping pedestrian. It should always be quoted in relation to the contaminant, e.g. water wet or dry/clean. If no heel material is quoted, it is generally assumed in the UK to be in relation to a standardised rubber known as Four S.

Which is more critical, the shoe sole or the heel?

Slips generally take place either when the heel of the ‘front’ foot lands or as the sole of the ‘rear’ foot pushes off. The latter rarely leads to a fall, it is merely an annoyance to the pedestrian. It frequently happens when pushing a heavily loaded trolley in a supermarket.


The most dangerous slip occurs as the heel lands on the floor. At this precise moment, the foot/leg needs restraint to stop it moving forwards as it lands and if it does not find it, the foot/leg slips forwards uncontrollably. Indeed, unless the pedestrian is able to somehow stop the slip by stumbling, or by some other means, once the foot has moved about 100mm or even less, the geometry of the body (the angle of the legs) becomes such that more and more restraint is required. The slipping system thus becomes very rapidly out of control.


The ‘classic’ slip is for the leading leg to shoot forward, the rear leg buckles at the knee and that foot also shoots forward due to the increase in lateral loading on the foot caused by the forward bending of the knee. The body then falls vertically to the ground with the person usually landing on their bottom or the base of their spine. Usually they land almost exactly over the spot where their foot originally slipped.

The coefficient of friction is a dimensionless quantity symbolized by the Greek letter μ and is used to calculate the force of friction (static or kinetic).

The coefficient of static friction is defined as the ratio of the maximum static friction force (F) between the surfaces in contact to the normal (N) force. The coefficient of kinetic friction is defined as the ratio of the kinetic friction force (F) between the surfaces in contact to the normal force: Ff/N.

The two types of friction are static and kinetic friction.

Coefficient of Friction from Wikipedia

Both static and kinetic coefficients of friction depend on the pair of surfaces in contact; their values are usually determined experimentally. For a given pair of surfaces, the coefficient of static friction is usually larger than that of kinetic friction; in some sets the two coefficients are equal, such as teflon-on-teflon.

The friction force is directed in the opposite direction of the resultant force acting on a body. In the case of kinetic friction, the direction of the friction force may or may not match the direction of motion: a block sliding atop a table with rectilinear motion is subject to friction directed along the line of motion; an automobile making a turn is subject to friction acting perpendicular to the line of motion (in which case it is said to be 'normal' to it). A motionless body is subject to static friction. The direction of the static friction force can be visualized as directly opposed to the force that would otherwise cause motion, were it not for the static friction preventing motion. In this case, the friction force exactly cancels the applied force, so the net force given by the Vector sum, equals zero. It is important to note that in all cases, Newton's first law of motion holds.

While it is often stated that the coefficient of friction (COF) is a "material property," it is better categorized as a "system property." Unlike true material properties (such as conductivity, dielectric constant, yield strength), the COF for any two materials depends on system variables like temperature, speed, atmosphere, as well as on geometric properties of the interface between the materials. For example, a copper pin sliding against a thick copper plate can have a COF that varies from 0.6 at low speeds (metal sliding against metal) to below 0.2 at high speeds when the copper surface begins to melt due to frictional heating. The latter speed, of course, does not determine the COF uniquely; if the pin diameter is increased so that the frictional heating is removed rapidly, the temperature will drop, the pin remains solid and the COF rises to that of a 'low speed' test.


Friction is the force resisting the relative motion of two surfaces in contact or a surface in contact with a fluid (e.g. air on an aircraft or water in a pipe). It is not a fundamental force, as it is derived from electromagnetic forces between atoms and electrons, and so cannot be calculated from first principles, but instead must be found empirically. When contacting surfaces move relative to each other, the friction between the two objects converts kinetic energy into thermal energy, or heat. Friction between solid objects is often referred to as dry friction or sliding friction and between a solid and a gas or liquid as fluid friction. Both of these types of friction are called kinetic friction. Contrary to popular credibility, sliding friction is not caused by surface roughness, but by chemical bonding between the surfaces.[1] Surface roughness and contact area, however, do affect sliding friction for micro- and nano-scale objects where surface area forces dominate inertial forces.[2] Internal friction is the motion-resisting force between the surfaces of the particles making up the substance.


From Wikipedia




Factors affecting the friction between surfaces


Dry surfaces
1. For low surface pressures the friction is directly proportional to the pressure between the surfaces. As the pressure rises the friction factor rises slightly. At very high pressure the friction factor then quickly increases to seizing
2. For low surface pressures the coefficient of friction is independent of surface area.
3. At low velocities the friction is independent of the relative surface velocity. At higher velocities the coefficent of friction decreases.

Well lubricated surfaces
1. The friction resistance is almost independent of the specific pressure between the surfaces.
2. At low pressures the friction varies directly as the relative surface speed
3. At high pressures the friction is high at low velocities falling as the velocity increases to a minimum at about 0,6m/s. The friction then rises in proportion the velocity 2.
4. The friction is not so dependent of the surface materials
5. The friction is related to the temperature which affects the viscosity of the lubricant

Please refer to... Surface Friction Notes

Static Coefficient of Friction

The static friction coefficient (μ) between two solid surfaces is defined as the ratio of the tangential force (F) required to produce sliding divided by the normal force between the surfaces (N)

μ = F /N

For a horizontal surface the horizontal force (F) to move a solid resting on a flat surface

F= μ x mass of solid x g.

If a body rests on an incline plane the body is prevented from sliding down because of the frictional resistance. If the angle of the plane is increased there will be an angle at which the body begins to slide down the plane. This is the angle of repose and the tangent of this angle is the same as the coefficient of friction.


Sliding Coefficient of Friction

When the tangential force F overcomes the frictional force between two surfaces then the surfaces begins to slide relative to each other. In the case of a body resting on a flat surface the body starts to move. The sliding frictional resistance is normally different to the static frictional resistance. The coefficient of sliding friction is expressed using the same formula as the static coefficient and is generally lower than the static coefficient of friction..

The Coefficient of Friction

Friction is "the resistance an object encounters in moving over another" (OED).

It is easier to drag an object over glass than sandpaper. The reason for this is that the sandpaper exerts more frictional resistance. In many problems, it is assumed that a surface is "smooth", which means that it does not exert any frictional force. In real life, however, this wouldn't be the case. A "rough" surface is one which will offer some frictional resistance.

Limiting Equilibrium

Imagine that you are trying to push a book along a table with your finger. If you apply a very small force, the book will not move. This must mean that the frictional force is equal to the force with which you are pushing the book. If the frictional force were less that the force produced by your finger, the book would slide forward. If it were greater, the book would slide backwards.

If you push the book a bit harder, it would still remain stationary. The frictional force must therefore have increased, or the book would have moved. If you continue to push harder, eventually a point is reached when the frictional force increases no more. When the frictional force is at its maximum possible value, friction is said to be limiting. If friction is limiting, yet the book is still stationary, it is said to be in limiting equilibrium. If you push ever so slightly harder, the book will start to move. If a body is moving, friction will be taking its limiting value.

In summary:

The frictional force between two objects is not constant, but increases until it reaches a maximum value. When the frictional force is at its maximum, the body in question will either be moving or will be on the verge of moving.
The Coefficient of Friction

The coefficient of friction is a number which represents the friction between two surfaces. Between two equal surfaces, the coefficient of friction will be the same. The symbol usually used for the coefficient of friction is m, where 0 ≤ m ≤ 1 .

The maximum frictional force (when a body is sliding or is in limiting equilibrium) is equal to the coefficient of friction × the normal reaction force.


F = mR

Where m is the coefficient of friction and R is the normal reaction force.

This frictional force, F, will act parallel to the surfaces in contact and in a direction to oppose the motion that is taking/ trying to take place.

A particle of mass 5 kg is at limiting equilibrium on a rough plane which is inclined at an angle of 30 degrees to the horizontal. Find the coefficient of friction between the particle and the plane.

Resolving up the plane:
F - 5gsin30 = 0

Resolving perpendicular to the plane:
R = 5gcos30

In limiting equilibrium, so F = mR
5gsin30 = m5gcos30

m = sin30/cos30 = 0.577 (3sf)



Do all machines give the same answer?

The simple answer is no. In dry conditions the differences tend to be relatively small, but significant differences can be found in wet readings.


The reason for this was discovered about 15 years ago by the UK’s Health & Safety Laboratory and the designer of SlipAlert, who were both working independently on the problem.

Why do machines give different readings in the wet?

The simple answer is that the film of water that becomes trapped between the slider and the surface being measured produces an uplift on the slider and reduces the effective friction between the slider and the surface.


Understanding precisely how a film of water produces uplift and how it is going to affect the machine is much more difficult. This is because the uplift depends on several factors, all of which need to be taken into account when designing the machine. What the machine designer has to do is to ensure that the combination of those factors on his machine produces exactly the same proportional uplift as a pedestrian experiences when their heel slips across the same surface.


Are there any machines that have been specifically designed to react correctly to the hydrodynamic film?

As far as we are aware, the only machine that has been designed to react to the hydrodynamic film in the same way as a pedestrian’s heel reacts on the same surface is SlipAlert.


The TRL Pendulum was not designed to do so, but by pure chance does react correctly. It is for this reason that measurements using the Pendulum have been found from many years experience to correlate with actual slipping accidents.


As one would logically expect, SlipAlert therefore correlates with the Pendulum (see graph).


Could I convert the readings from another type of machine to give the same readings as a Pendulum or SlipAlert?

Unfortunately, because the matter is so complex, the answer is no. The problem is that two surfaces tested by machine A may seem identical, but the Pendulum and SlipAlert will indicate that they are quite different. Conversely, two surfaces that give very different readings on machine A may in fact have very similar slipping properties. There is simply no easy means of unravelling the complexity of the way that the uplift is produced and how it affects the final reading to make one machine correlate with another by use of a simple factor.


How important is the slider?

The slider on the machine is designed to replicate the heel of the pedestrian. If you are investigating an accident, it is helpful to use a material that is similar to the heel on the shoe worn by the pedestrian who slipped. In simply monitoring a floor or testing it to see how it is likely to behave in use there are two schools of thought.


One school suggests using a standardised rubber such as Four S. This means that your results can be compared directly with someone else’s. Whilst this can be very useful in terms of standardisation, there are three problems with Four S. The first is that it does not wear well and needs to be reprepared and replaced quite frequently. The second is that in the wet it does need to be very carefully prepared to ensure it is quite smooth, otherwise one gets an occasional significantly different reading. The third is that it represents a relatively good heel in terms of its performance. This means that one cannot tell whether the floor will be satisfactory against the whole range of heel materials that are considered to be acceptable in respect of their frictional qualities. Ideally, the standard heel should be at the lower end of the range rather than towards the top end; as yet no one has put forward such a ‘standard’ slider material.


The second school of thought is to test the floor using a range of heel materials. This can be quite illuminating, but because there are only two rubbers that have been standardised it can lead to disagreements.

What about roughness measurements?

The roughness of the surface certainly has an effect on friction in both the wet and dry states. If one really understands the way that it does so, particularly in the partially lubricated wet situation, it will become apparent that measurements of roughness are not likely to lead directly to the slip resistance of the surface or be such that one could reliably make specific judgements about the surface using that data alone.


When the concept of the hydrodynamic film was first put forward, the measurement of roughness seemed a logical step forward that might assist in the measurement of slip resistance. However, if one goes back to first principles and starts manipulating the mathematical equation that relates to lubrication and at the same time considers how friction is itself developed, one begins to realise that although a roughness measurement of the surface might give a not unreasonable indication of the wet slip resistance, equally it might give a totally incorrect indication. Dr Malcolm Bailey has tested several hundred surfaces and found this to be true. The latest version of the UK Slip Resistance Group Guidelines gives the clear warning:


‘It should be noted that the micro roughness of one surface may have the same numerical value as measured as that of another surface but be quite different in profile, as illustrated by the difference in profiles shown in Figure 1.


For this reason, roughness readings should not be used in isolation but linked with other salient information, such as Pendulum readings from the particular material being tested or specified.’


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Man slipping - should have used a SlipAlert!

Why people slip...

Slips generally take place either when the heel of the ‘front’ foot lands or as the sole of the ‘rear’ foot pushes off. Usually people land almost exactly over the spot where their foot originally slipped.



1 in a million...

We all walk differently, so some of us are more prone to slipping than others. Less than one person in 1 million is likely to slip if the floor surface has a dynamic co-efficient of friction greater than 0.39. For this reason, 0.4 is usually regarded as the criretia for safety of a floor used regularly by many people.



Many a slip...

People can slip or trip or fall on any surface, even on surfaces that are not slippery. But if you are responsible for the safety of people - your staff, customers or other visitors - then you should test your floors regularly.





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