Just something I have been wondering about while ploughing through some very dull mechanics work:

Think of it this way, whenever an object say a box is sliding down an inclined plane, there is a very small resistance to the motion down the plane. If this force was any greater than 1 then the box would halt.The only meaningful force that pulls the box down the plane is its own weight.

If the coefficient of friction was greater than 1, wouldn't the box slide up the hill?

The coefficient of friction m is only an approximation and thus has its limitation. For most everyday applications, the approximation works well. One example where the approximation breaks down is given by the Freyman Lectures - a pure copper/copper boundary.

Suppose you have 2 copper rods and you do the physical experiment of determining m, then you will find that m is less than 1. That is due to the fact that there are various impurities on the surface, such as grease, dirt, oxide layers, microbes, etc. Now, suppose you can remove all these impurities both on the surface and inside the copper rod (e.g. stick the rod into a vacuum, scraping the oxide layer, performing infinite number of electrolysis), so that you have 100 "pure" copper - as pure as we can get in reality). Now rest one rod onto another. What you will find is that the rods stick together even when the position is vertical. This means that the coefficient of friction is very large, as virtually no normal reaction force exists.

Furthermore, the value of m is not a constant. It takes some constant static value but when the surface is moving, the dynamic value of m is actually dependent on the speed of which the surfaces move with respect to another. Typically for very small speed, m_dynamic < m_static.

Kerwin

If an object is on a slope of angle A from the horizontal and is not sliding, then R = m g cosA and F = m g sinA. Looking at angles between 0 and 90, m g sinA will be positive, so m ³ F/R m ³ tanA. Since tan A tends to infinity as A ®90, the coefficient of friction can be very high for an object on a steep slope. The reason that an object will not slide up a slope if it has a high coefficient of friction is that the frictional force is less than or equal to this coefficient of friction multiplied by the natural reaction.
Coefficients of friction seem to be less than 1 in mechanics questions to make the questions more realistic; you don't need to drag sandpaper up a slope very often.

Tom.

Thanks everyone