CWatters said:
When I was at school (<1978) two of the subjects I studied were Physics and Applied Maths. There was some overlap which sometimes caused problems.
In physics class the syllabus taught us that friction was independant of contact area. That makes sense - you might think that increasing the contact area would increase friction but it also spreads out the load over a larger area so net effect could reasonably be zero.
However the Applied Maths syllabus seemed to use a different definition of the coefficient of friction. That required us to factor in the contact area. We did raise this contradiction with our teachers but were just told we had to remember which exam we were sitting in and answer accordingly!
We must have done because we passed with good grades.
I am interested in the background behind this since in my experience there were some differences of emphasis and rigour in Physics and Applied maths, but no outright conflicts as you describe.
Here is an extract from a University of London Applied maths text of the era concerning Friction.
" The mathematical discussion of the force of friction depends upon certain assumptions which are embodied in the so called laws of friction and are found to be in close agreement with experiment.
Law1
When two bodies are in contact the direction of the force of friction on one of them at its point of contact is opposite to the direction in which the point of contact tends to move relaticve to the other.
Law2
If the bodies are in equilibrium the force of friction is just suficient to prevent motion and may therefore be determined by applying the conditions of equilibrium of all forces acting on the body.
The amount of friction that can be exerted between two surfaces is limited and if the forces acting on the body are sufficiently great motion will ensue.
Hence we define limiting friction as the friction which is exerted when is onthe point of being broken by one body sliding over another.
The magnitude of limiting friction is assumed to be given by the following 3 laws:
Law3
The ratio of the limiting friction to the normal reaction between the two surfaces depends upon the substances from which the surfaces are composed, not on the magnitude of the normal reaction.
The ratio is usually given the Greek letter μ. Thus if the normal reaction is Rlb. wt., the limiting friction is μRlb. wt. and for given materials, polished to the same extent, μ is found to be constant and independent of R.
μ is called the coefficient of friction.
Law4
The amount of limiting friction is independent of the area of contact between the two surfaces and the shape of the surfaces, provided that the normal reaction is unaltered.
Law5
When the motion takes place the direction of friction is opposite to the direction of relative motion and independent of velocity. The magnitude of the force of friction is in a constant ratio to the normal reaction, but this ratio may be slightly less than when the body is just on the point of moving. "
All these laws are needed for the mathematical analysis of friction.
I have italicised Law 4 as it is quite clearly in line with Physics.