Class content > Kinds of Forces > Resistive forces

Prerequisites

• Formulation of Newton's Laws as foothold principles
• Kinds of forces
• Springs
• Tension
• Resistive forces  

When the surfaces of two solid objects rub together, they tend to stick, opposing the rubbing.  In our Newtonian framework, we interpret such an influence on the motion of the two objects as a force.  In order to be able to use this force, we need to understand how this force behaves and what affects how big it is.

The phenomenology of friction

To get an idea of how friction behaves, let's consider a simple experiment.  Let's hook a force probe up to a block sitting on a table and slowly increase how hard we pull.  The data from the force probe can then be displayed in LoggerPro and we can get a first idea of how the force of friction behaves.  At first, even though we are pulling, the block doesn't move.  As we increase our pull, it eventually breaks away and starts to slide.  We then pull to keep it going at about a constant speed.  The graph of the force is shown below.

This is rather strange behavior, but we can begin to make sense of it if we consider a free body diagram of the block.  The block is being pulled to the right by the tension force of the string (which we assume is the same as the force being measured by the force probe on the other end of the string -- see the discussion of massless strings in Tension Forces). When it is not moving, the forces on it must be balanced, so the force we are measuring should be equal and opposite to the friction force exerted on the box by the table (by Newton 2).  When it is moving at a constant velocity, the same thing must be true, since the box is not accelerating.  The free-body diagram will look like this, with the friction and tension forces equal in magnitude:

The block only accelerates for a very short period of time -- say from about 32 s to 33 s on the graph.  At this time, the force shown on the force probe must be greater than the force of friction -- to speed the object up.

This is a tricky and surprising result!  The force of friction doesn't seem to have a fixed value, but seems instead to adjust to a maximum value in response to a force trying to slide the one object over the other, and then, once the sliding starts, to drop down to a lower value!  The fact that the friction force is not just determined by the position of the object but by what is being done to it, is a dangerous bend.  We identify these two different forces with different names.

• The maximum value the friction force can have when the objects are NOT sliding is called the force of static friction. 
• The value the friction force has when the two surfaces are sliding over each other is called the force of kinetic friction.
  

The mechanism of friction

Finding out the phenomenology of friction from observing how it behaves is one thing.  But we would like to make sense of what's happening.  Why should it behave like that? In order to understand what's going on, let's try to figure out some mechanism that might be responsible for this behavior.

In fact, the underlying mechanism of friction is quite complex and research on the topic is still going on today.  If we think about two surfaces trying to rub on each other, and imagine looking at the surfaces through a microscope, we might guess that there are two basic phenomena that are happening: molecular attractions and interlocking of rough edges.  

Molecular attractions

We know from chemistry that atoms attract each other when they are close enough (but not too close) by Van der Waals forces.  Some of our friction could come from the atoms in the surfaces of the two objects attracting each other.  That this does happen is shown by the phenomenon of Johansson (gauge) Blocks.  These are metal blocks that are polished to be so smooth that when their surfaces are slid together they stick quite strongly. But the surfaces have to be polished smooth to a fare-thee-well to make this happen and they have to be made of the same material.  This suggests that something else is happening in ordinary friction.

Interlocking of rough edges

Since we know that typically smoothing surfaces produces LESS friction rather than more, we might imagine that each surface has little bumps that get tangled up with the bumps on the other surface.  The key idea here is a sideways (shear) spring.  Our simple picture of an extension-compression spring can be extended easily to a spring that bends sideways as shown in the picture below.  Imagine a thin metal fiber that has one end imbedded in a block of wood.  If I press sideways on the top of the fiber, it will bend as shown.  The more I press, the more it will bend; like a spring, but sideways.

Now suppose our two surfaces that are experiencing friction have lots of little protuberances like this that can bend. We've assumed that these can be two different substances, so we've colored one red, the other blue.  On the left, we show the objects just sitting on each other.  There is no sideways force.  On the right we show some force applied to slide the top object to the right and the bottom object to the left.  The little rough edges from the two objects now encounter each other and try to bend each other.  As the little protuberances bend in response to the applied force, they begin to exert a force back, just like a spring. The result will be that there will be a tiny shift in the relative position of the two objects (these little fingers are microscopic), but the two objects will exert a force back on each other that opposes the force that is trying to slide the surfaces over each other.  This will provide the static friction force that opposes the object's motion.

If the applied force gets large enough, the little irregularities will slide over each other and the objects will begin to move.  The force of the occasional interfacing of the fingers will provide the kinetic friction force that resists the object's motion when it is actually moving (macroscopically).

This model gives a reasonable mechanism for the funny behavior of the friction force -- the fact that it starts at 0, then builds up to match an applied force, and finally drops down to a constant value as the objects slide over each other.

The equations of friction

We can set up an experiment to measure how the friction force depends on various situations as shown in the figure at the right.  Of course we would use a force probe to pull the blue block in order to measure the force between the red surface and the blue surface.  (What does the FBD for the blue block look like?) Here's what is found: The maximum static friction force (force that the surfaces can exert on each other before the block starts to move) is directly proportional to the Normal force squeezing the two surfaces together. The proportionality constant depends on the two surfaces. The maximum static friction force does NOT depend on the area of the surfaces in contact. The maximum static friction force is greater than the kinetic friction force (friction while sliding).

These results are summarized in the equations for friction below.  The symbols "μ" are called the coefficients of friction.

All the symbols in these equations tell us important conceptual ideas about friction.

1. The fact that we have two equations tells us that friction forces when the surfaces are not sliding over each other and sliding over each other are different.
2. The "≤" in the static friction equation tells us that the static friction equation only gives us a MAXIMUM.  In general the static force of friction may be less -- whatever it needs to be to prevent motion, possibly even 0.
3. The N tells us that it's how hard the two surfaces are being squeezed together that determines the (maximum) friction force.
4. The sub and superscripts on the coefficients of friction tells us that they are different for static and kinetic friction and that they depend on the two surfaces involved.
5. The fact that there is no area in the equation tells us that the surface area of contact does not affect the friction force!

 Workout: Friction

Joe Redish 9/27/11