Coefficient of Friction

Date: 10/17/97 at 06:05:01
From: Stuart Cliff
Subject: Coefficient of Friction

From experiments carried out I have found that the formula "F=xR"
Where F = Frictional force
      x = coefficient of friction
      R = Reactional force

Does not hold true; that is, the angle of the slope "#" (and therefore 
x) needs to be decreased when the mass of the object is increased - 
yet the angle "#" should remain constant.

For the life of me, I am unable to figure out why (pure mathematics is 
my strength). Any ideas?

Thanks!

Date: 10/22/97 at 11:24:59
From: Doctor Mark
Subject: Re: Coefficient of Friction

Hi, Stuart

You don't say whether you are talking about static or kinetic 
friction, so I can't really figure out what your difficulty is.  
However, I can explain what the deal is with these two kinds of 
friction.

Static and kinetic friction both result from the "stickiness" between 
two objects. The origin of friction is not at all simple, and is the 
subject of the field of Rheology. That is, one does not generally 
derive the coefficient of friction, one measures it experimentally.

Kinetic friction is the easier of the two to understand. If one object 
is sliding along another, then the frictional force acting on one of 
the objects is given by F_k = mu_k times N, where the "_" denotes a 
subscript, mu is the coefficient of friction, and N is the "normal 
force" exerted by the other object on the object in question 
(actually, the component of this force perpendicular (hence the word 
"normal") to the interface between the two objects. 

Generally, one finds the normal force by using Newton's second law, 
F = ma (as a vector equation), and from that you can find the 
frictional force.  Note that the kinetic frictional force always has 
a direction opposite to that of the velocity of the object.

Static friction is more complicated. The static frictional force is 
the force of one object on another when the two objects are not 
sliding over one another. As a result, this is given not by an 
equality (an equation), but by an INequality, namely, that F_s is 
(less than or equal to) mu_s times the normal force. 

Generally, you find the static frictional force by (as always) 
applying F = ma, and demanding that the net force be ZERO (of course, 
because if it's static friction, the object is not sliding over the 
other one, and so the velocity is zero, and constant, so a is zero, 
and so the net force is zero). The direction of the static frictional 
force is always opposite to the direction in which the object would 
move, given the other forces, if there were no friction. 

Generally, what you do is find the normal force by demanding that the 
sum of the components of the forces on the object perpendicular to the 
boundary between the two objects is zero. You then find the static 
frictional force by demanding that the sum of the components of the 
forces on the object parallel to the  boundary between the two objects 
is zero. You then use the inequality to find some condition for the 
object to stay at rest with respect to the other object.

For instance, if it is a block sitting on a plane inclined at an angle 
B to the horizontal, then the Normal force is given by N = mg cos B.  
The static frictional force will be given by F_s = mg sin B. Both of 
these results are obtained by applying F = ma to the object; since it 
is at rest and staying at rest, a = 0. Using the inequality mentioned 
above, you find that the angle B must be such that tanB is less than 
or equal to mu_s in order for the  block to stay at rest on the 
incline. This means that if the angle is bigger than  arctan(mu_s), 
the block will start moving. At that time, kinetic friction will take 
over, and you have to redo the problem. Of course, this is one way of 
measuring the coefficient of static friction: put the  block on the 
plane, start increasing the angle, then find the angle A at which the 
block starts to move. The coefficient of static friction is given by 
tan A.

Generally, the coefficient of static friction is larger than that of
kinetic friction, as is obvious if you have ever tried to push 
something across a floor - it is harder to *get* the thing moving than 
it is to *keep* it moving (if you think about it, how could that *not* 
be true?!), and since the normal force has not changed, the 
coefficient of static friction must be greater than that of kinetic 
friction.

Depending on the constitution of the materials which are in contact, 
it is quite possible that the coefficient of friction could in fact 
depend on the mass of the object in question, since very dense 
materials would deform the surface they are in contact with, and that 
would increase the ratio of the frictional force to the mass (or 
normal force), i.e., the coefficient of friction. But for "hard" 
materials, that shouldn't be a very large effect. It probably does 
make a difference for materials which deform easily, such as the tires 
on a car.

Having said all of this, I don't know if I have really answered your
question, so please feel free to write back to me if you feel I have 
not.

-Doctor Mark,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/