Amontons' first law of friction

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Name

This law is sometimes termed Amontons' first law of friction after Guillaume Amontons, but is often attributed to Leonardo da Vinci, and is hence sometimes termed Leonardo da Vinci's law of friction. Often, the law is studied without an explicit name, but rather, by its statement.

Statement

The friction force experienced by two dry bodies whose planar surfaces are in contact is independent of the area of contact between the bodies.

Amonton's first law of friction is implicit in the Coulomb model of friction, and is closely related to Amonton's second law of friction, which says that the friction force is (limiting value for static friction, actual value for kinetic friction) proportional to the normal force.

Apparent counterexamples

Formulation of apparent counterexample	Resolution	Possible empirical tests
Rolling a wheel is easier than dragging a box of the same weight and material	Rolling is a fundamentally different operation from dragging because there is no slipping at the surface of contact (so static friction operates). Dragging a wheel is just as hard as dragging a box	?
With the same material, the same type of surface, and the same shape, objects with larger area encounter more friction	Area is related to volume (assuming a similar shape), which is related to mass (assuming the same density), which is related to weight (since $g$ is the same), which in turn affects the normal force experienced to balance the weight, which then affects the friction force that can operate.	Same object, same kind of surface, different sides of different areas; for more, expand below.
In case of multiple supports, the support with larger contact area contributes more in friction	The larger contact area, as well as further geometrical details, affect the relative distribution of the normal force. It is not the contact area per se that is playing a role.

Below are more details of these apparent counterexamples.

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Surface area and rolling

One might initially suppose that rolling wheels is easier than dragging boxes of the same material because the surface area of contact of a wheel with the ground is less. This is, however, not the real reason. In fact, dragging a wheel is as difficult as dragging a box of the same weight. The reason why rolling a wheel is easier is because during rolling, there is no actual slipping of the contact surfaces against each other, so friction does not oppose the rolling. In fact, friction helps with rolling.

(There is another force, called rolling friction, which is of much smaller magnitude and does oppose rolling. This is a component of the normal force that acts to oppose rolling because of minor deformations of the surface).

Area appears to be a proxy for weight?

It is true (Amonton's second law of friction) that the magnitude of the friction force (limiting value for static friction, actual value for kinetic friction) is dependent on the magnitude of the normal force between the surfaces. In situations where the surface of contact is between a movable object and a fixed object, the surface is horizontal, and the earth's gravitational force is the only external acting force on the movable object, the normal force equals and cancels out the gravitational force. In many related situations, the magnitude of gravitational force, which is the weight of the movable object, plays a role in determining the magnitude of the normal force, and hence, the magnitude of the friction force.

For two objects of the same shape, the one with the larger base area also has larger volume. If they have the same density of material, the object with larger volume has larger mass, and hence larger weight. Thus, bigger objects may appear to encounter larger frictional forces -- however, the bigness of concern here is weight (determined by mass) rather than the area of the surface in contact.

Two tests of this would be:

• Take an object with surfaces of different sizes (such as a cuboid with different side lengths) and consider the magnitude of friction force with different choices of face as base. (Note that we have to make sure the cuboid is not so lopsided as to create other effects such as toppling).
• Take two objects of the same weight but with different base areas.

Multiple supports

Suppose a block is resting horizontally on two tables, with some part of the base of the block on one table and some part on the other table. In this case, the weight of the block is counterbalanced by the normal forces exerted by the two tables. In other words, the sum of the upward normal forces exerted by each of the tables equals the weight of the block.

The way the weight is distributed between the two normal forces depends on the geometry as well as the distribution of mass of the block, among other things. Thus, assuming other things the same, there may be a clear numerical relation between the fraction of the area made to rest on one table and the normal force exerted by that table. This dependence, however, is not due to area but due to the extent to which that table feels the weight of the block.

References

Instructional video links

Video link (on course page)	Video link to correct start time (not necessarily on course page)	Segment	Contextual information	Transcript link	Transcript segment
MIT OCW lecture on friction by Walter Lewin	link	09:59 -- 11:10 (watch earlier and later for full context)	Does not explicitly name the law but discusses it and uses an experimental demonstration by using the angle of repose as a proxy for estimating the limiting coefficient of static friction and showing that this is independent of area.	same as video link	From "Now comes something..." to "...touch the critical surfaces."

Other online resources

• The law of Leonardo (da Vinci) on Nano World, has accompanying demonstration.