Range of the Coefficient of Friction

Date: 10/18/2000 at 17:34:17
From: Steve Kapa
Subject: Can Coefficient of Friction be > 1 

My college physics professor stated that the coefficient of friction 
can be greater than one. I asked him how that could be. He *assigned* 
the coefficient of friction (mu) a value of 5 and plugged it into the 
formula f = mu * N.

I told him he couldn't just assign a value to mu, and asked him how he 
got that. After further discussion he got frustrated, closed (I should 
say slammed) his book, and then dismissed class. My question is, can 
you *prove* that the coefficient of friction (a ratio, from my 
understanding) can't be greater than 1?

Thank you.

Date: 10/23/2000 at 16:39:56
From: Doctor Wolfson
Subject: Re: Can Coefficient of Friction be > 1

Hi Steve,

Interesting question. Let's look at an example to see how "very 
sticky" surfaces behave:

Let's say, for convenience, that we're on a planet where g = 1 m/s^2, 
and we have a high-friction inclined slope of 60 degrees, with a 
high-friction mass of m = 1 kg that we are pulling downward (assisting 
gravity) with just the right force to compensate for friction and 
cause it not to accelerate at all. And let's say that the force we 
have to apply is

     (5 - sqrt(3))/2 N ~= 1.64 Newtons.

Does this seem plausible so far? We're just picking the numbers to use 
as parameters.

The force due to gravity is mg sin(theta), or 1*1*sqrt(3)/2. So this 
force, combined with the one we are providing, yields a total downward 
force of 5/2.

Since the object doesn't accelerate, friction must be equal and 
opposite to the downward force:

     (mu)*m*g*cos(theta) = 5/2
     (mu)*1*1*1/2 = 5/2
      mu = 5

As it turns out, even though most materials have values of mu 
considerably less than 1, this actually isn't a requirement, and mu 
isn't limited to the [0, 1) range. Physics doesn't "malfunction," nor 
does friction start speeding the object up backwards, just because mu 
is greater than 1. Incidentally, the example I gave can be thought of 
as a rough definition of mu_(kinetic) - it is the value that makes the 
friction equation balance with the amount of force required to prevent 
frictional deceleration.

I hope this helps. Feel free to write back if you'd like further 
clarification.

- Doctor Wolfson, The Math Forum
  http://mathforum.org/dr.math/   

Date: 10/23/2000 at 16:50:02
From: Doctor Ian
Subject: Re: Can Coefficient of Friction be > 1

Hi Steve, 

The thing is, it's easy to think of examples where the "coefficient of 
friction" would be greater than 1 - a bulldozer on dirt, for example, 
or Scotch tape on glass, or velcro - but in most of these cases, 
there is some question whether it's proper to describe what's going 
on as "friction," rather than something like "adhesion."

If I pound a piton into a mountain, it won't slide, but is it really 
friction that prevents it from sliding? Friction is sort of a catch- 
all category - if we can't explain a resistive force in any other 
way, then we call it friction.

However, in looking around the Web, I came across the following URL:

  The Coefficient of Friction, Coulomb (Static) and Dynamic (Kinetic)
  Friction (MathEngine Fast Dynamics Toolkit)
  http://www.mathengine.com/sdk1/Developers/SDK/FastDynamics/Docs/PhysicsNotes/FrictionCoefficient.html   

which contains the following coefficients:

   Aluminum on Aluminum      1.3
   Copper on Copper          1.3
   Iron on Iron              1.0
   Rubber on Steel           1.6

The first three can perhaps be explained in terms of something other 
than "friction" (e.g., "galling," which is the phenomenon that 
requires the frame and slide of a pistol to be made from different 
materials), but that's not the case for the fourth.

I guess you could call this an "existence proof" - we know that the 
coefficient can be greater than one, because there exists a pair of 
materials for which that is the case.  

I hope this helps. I know that Dr. Wolfson provided a different proof. 
Write back if you're not happy with either of our answers. And thank 
you for a very interesting question. We've all had a lot of fun 
thinking about it.  

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/