FRICTION

Friction of RIGID bodies can be approximated by 3 rules:
1. The force of friction is directly proportional to the applied load. (Amontons 1st Law)
2. The force of friction is independent of the apparent area of contact. (Amontons 2nd Law)
3. Kinetic friction is independent of the sliding velocity. (Coulomb's Law)

Lecture Notes: Friction.pdf Friction.one

Coulomb Friction

Coulomb friction, (named after Charles-Augustin de Coulomb), is a model used to calculate the force of dry friction. It is an approximation to what happens in real life, according to the 3 rules in the summary box above. It is meant to be used on rigid bodies, since soft and flexible materials (like rubber tyres for example) are more sensitive to the area of contact (See point 2 above).
The definition is below;

where

is the friction force.

is the coefficient of friction. (depends on materials)

is the normal force exerted between the surfaces, (which is equal to W in this diagram).

This formula is a rule of thumb giving an approximation of an extremely complicated physical interaction. In many cases, the relationship between normal force and frictional force is not exactly linear (the frictional force is not entirely independent of the contact area of the surfaces - especially so for soft materials like rubber). The Coulomb approximation works best for relatively hard, rigid materials.

Static Friction

When the two surfaces are not moving, the friction is slightly higher.

= coefficient of static friction. Motion cannot begin until the applied force is higher than the maximum friction force

Kinetic Friction

When the two surfaces are moving, the friction usually goes down a little.

= coefficient of kinetic friction. Friction force always opposes the direction of motion and is considered to be constant.

The force of friction is always exerted in a direction that opposes movement (for kinetic friction) or potential movement (for static friction) between the two surfaces.

Coefficients of Friction

The coefficient of friction

, is a dimensionless (scalar) value so it has no units. (This is because it is a ratio between two forces). The coefficient of friction depends on the materials used; for example, ice on steel has a low coefficient of friction, while rubber on pavement has a high coefficient of friction. Coefficients of friction range from near zero to greater than 1 – such as a soft rubber tyre on rough concrete.
The coefficient of friction is measured experimentally, and cannot be found through calculations. Rougher surfaces tend to have higher effective values. Most dry materials in combination have friction coefficient values between 0.3 and 0.6. Values outside this range are rarer, but teflon, for example, can have a coefficient as low as 0.04. A value of zero would mean no friction at all, which is impossible in any practical sense – even magnetic levitation vehicles have drag. Rubber in contact with other surfaces can yield friction coefficients from 1 to 2.

The normal force

The normal force is the net force compressing two parallel surfaces together, and is always perpendicular to the surfaces.
In the simple case of a mass resting on a horizontal surface, the only component of the normal force is the force due to gravity, so the normal force = weight.

= W = mg.

If the object is on an inclined plane, the normal force is less, because less of the force of gravity is perpendicular to the face of the plane. Therefore, the normal force, and ultimately the frictional force, is determined using force components.
Note: There may be forces other than gravity - like springs.

In a typical friction problem, we must first determine

and then multiply by

to give us the friction force

Block on a ramp (top) and corresponding free body diagram of just the block (bottom).

Approximate coefficients of friction

The most slippery solid known, discovered in 1999, dubbed BAM (for the elements boron, aluminum, and magnesium), has an approximate coefficient of friction of 0.02, about half that of Teflon.

Angle of friction

Another way to define the friction is by the angle of the opposing force.
The friction angle

is defined as;

tan

So tan

Angle of repose

When a body is on an incline, there is a maximum angle that can be reached before it will begin to slide. That maximum angle is called the angle of repose, (Sometimes the angle of repose refers to the maximum slope of granular material instead. Never mind).

It is defined as:

where

is the angle from horizontal and

is the static coefficient of friction between the objects.

The angle of repose occurs when

The block will not slide as long as

(friction force) is greater than

(parallel force).
At the point where motion begins, we can use that angle to calculate the coefficient of friction.

Friction on inclined plane (interactive)

Notes:
Fw = Weight Force = m * g
Fn = Normal Force = Fw * cos

Fp = Parallel Force = Fw * sin

Ff = Friction Force =

* Fn
Note that in reality the friction force acts at the interface between the two surfaces, but that in most cases it can be simplified as if it acts through the Centre of Gravity of the block. (Move the Simple/Realistic slider)

Q: When is this simplification not good enough?
A: If the block tips over.
If friction is high and the block is tall, as angle is increased the Parallel force and the Friction force produce an increasing couple. Once the moment from this couple is larger than the stabilizing moment of the Normal force, the block will tip over.

Additional Information

Coefficient of Friction (More complete)

(i) Friction force is not effected by the area of contact. This is very weird. In fact, when Coulomb first presented this conclusion to the Academy Francaise, he was thrown out of the room.
(ii) Friction force is proportional to the normal force.
(iii) Surface roughness has hardly any effect on friction.
(iv) Dust. dirt or liquids have a huge effect on friction. Even a little moisture on the surfaces can reduce friction by 20-30%. If there’s a thin layer of grease on the surfaces it can cut friction by a factor of 10.
(v) Friction forces are strongly related to the materials. Some materials like to bond with each other (metals generally bond well to other metals, for example) and so have high friction forces. Some materials (e.g. Teflon) don’t bond well to other materials.
(v) As the surfaces start to slide, the friction force usually drops slightly.
(vi) The kinetic friction force usually decreases very slightly with speed

Note that there are some exceptions to these rules. For example, extremely microscopic can be attracted by Van der Waals forces. Also, rubbers don’t obey all the rules listed above.

Material	Approx friction coefficient
Clean metals in air	0.8-2
Clean metals in wet air	0.5-1.5
Steel on soft metal (lead, bronze, etc)	0.1-0.5
Steel on ceramics (sapphire, diamond, ice)	0.1-0.5
Ceramics on ceramics (eg carbides on carbides)	0.05-0.5
Polymers on polymers	0.05-1.0
Metals and ceramics on polymers (PE, PTFE, PVC)	0.04-0.5
Boundary lubricated metals (thin layer of grease)	0.05-0.2
High temperature lubricants (eg graphite)	0.05-0.2
Hydrodynamically lubricated surfaces (full oil film)	0.0001-0.0005

From Engineering Materials Ashby and Jones, Pergammon, 1980

Static vs Dynamic Friction

The coefficient of friction for a pair of materials is often quoted as STATIC and DYNAMIC values. From the typical diagram below, dynamic friction isusually lower than static (although this is nothing compared to the effect of a little moisture).

Examples of static friction being higher than dynamic friction

1. Stick-slip.
This difference creates the "stick-slip" action, which can be very annoying when you are trying to make something move smoothly from a stopped position. E.g. A lightly loaded air cylinder.

Animation from http://www.engin.brown.edu/courses/en3/Notes/Statics/friction/friction.htm

2. Anti-lock brakes.
ABS was first developed for aircraft landing wheels and was showed a consistent reduction in stopping distance. A vehicle with locked wheels takes longer to stop than one at the threshold of locking (just before lock-up). The difference can be about 10% in ideal conditions, and up to 30% on a slippery surface.
On some very loose surfaces, ABS increases the stopping distance because locked wheels actually "dig in". (The main advantage of ABS, of course, is that the driver can steer.)

3. Stopping a sliding object (angle of repose)
When an inclined plane reaches the angle of repose= atan(

), the object will start sliding. Decreasing the angle slightly below repose angle will not stop it! This is because kinetic friction is lower than static friction. It will only stop once the angle is below the Kinetic angle of repose = atan(

), which could be a few degrees lower.

Questions:

Questions (Ivanoff old edition)

Homework Assignment:
Do all questions 7.1 to 7.20