Derivation of static & kinetic friction formula

We know formula of static friction & kinetic friction = μsN and μkN.

I have searched online a lot about them.

I do know what is the formula & how to use it but I couldn’t find any article onto how are these formulas actually derived.

Like what was the 1st step, then the 2nd & then we got a coefficient. So, how was the formula contemplated & derived?

• Thank you everyone for your answers. I am very grateful. I am reading your answer & will comment the difficulties I am getting in some time. Dec 24, 2021 at 15:56

Coulomb friction is an empirical law. It was observed that the force required to move an object was proportional to its weight.

Regarding static and kinetic friction it is possible to use graphs like to following to see the difference between the two coefficients.

The rugged part of the kinetic friction is from overcoming the surface roughness between of the contact surfaces (and its more a stochastical quantity)

I think the formulas are empirical. But saying 'empirical' raises two questions:

• why is $$\mu_k$$ smaller than $$\mu_s$$
• why is $$\mu_k$$ independent of velocity

You can make a loose unscientific analogy to address the first one by grouping the various phenomena that generate friction forces into "reversible" and "irreversibe", in terms of thermodynamics. The reversibles ones, by conservation of energy, net out to 0 cumulative effect when moving fast enough to avoid "sticking". Somewhat like the way if you drive into a smooth dip in the road, like a small valley, the vehicle gets its energy and momentum back on the way out. But if you come to a full stop at the bottom, you now have to do work to climb out, i.e. the object experiencing friction must overcome both groups of forces to get going again.

The second one is equivalent of there being a constant amount of energy expended per unit distance to produce $$\mu_K$$ -- the forces from the irreversible effects. This from Force = Energy/Distance. But again, why should it be a constant amount of energy per distance?

Here is a much more in-depth answer from the physics stack:

https://physics.stackexchange.com/questions/16633/why-are-there-both-static-and-kinetic-friction

Spoiler - the explanation is still being debated.

We can do it even by our crude luggage weight scale.

• Put our test cube weighing say 10 kg on the surface we want to test and hook it to the scale.

• Pull horizontally gradually, but evenly, it won't move but the scale will show force in the spring increasing gradually, until the sample jerks and starts to be pulled

• Record the force just before the jerk and the lower force scale shows after the sample is moving at a constant speed.

• Divide these forces by the weight of 10kg, you get $$\mu_s \ and \ \mu_k.$$

Lets talk about a material first. Assume you have Copper sheet/bar, and you conduct an experiment on it. You apply a force and track its behavior during the application of force, like deformations. You can also calculate the stresses and strains when applying a quasi-static force so that you would have numerous points, using which you can plot a graph, like force against displacement, and/or then stress against strain. What is the purpose of plotting a graph? It was noted that no matter what kind of geometry you have or how huge the force is which you are applying, you will still get the same stress-strain curve for Copper (although the force-displacement curve can change). This coined the term Elastic Modulus (linear coefficient which relates the stress with strain), which was then declared as a material-dependent property only, because the experiments proved that.

Now coming back to your question, you have a certain material (like a box) which moves against another material (like ground). The boxes are usually manufactured through the same process which means that the roughness on the boxes faces throughout the world can be assumed to be the same. Same goes for the ground. When you displace a box against ground, the experiments unravelled that this much force is required to actually begin displacing a box of a certain weight. It was also discovered that the force required changes linearly with linear change in weight by experiments. And as a result, coefficient of friction for static and kinematic cases were discovered which relates the frictional force with the weight of the object by a linear relation. This was confirmed by several experiments, that for the same two materials in contact (like the box and ground), same linear relation holds and the coefficient of friction (i.e. the slope of frictional force against weight) remains constant.

When you change either of the objects, or its manufacturing process, or do any other thing which can change either of the objects' surface roughness, then the frictional force against weight relation will change (i.e. their slope will change) but still the relation will be linear (as confirmed by extensive experimentations).