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I was recently wondering what would happen if the force sliding two surfaces against each other were somehow weaker than kinetic friction but stronger than static friction. Since the sliding force is greater than the maximum force of static friction, $F > f_s = \mu_s F_N$, it seems that the surfaces should slide. But on the other hand, if the force of kinetic friction is greater than the applied force, there'll be a net force $\mu_k F_N - F$ acting against the motion, suggesting that the surfaces should move opposite to the direction they're being pushed! That doesn't make sense.

The only logical resolution I can think of is that the coefficient of static friction can never be less than the coefficient of kinetic friction. Am I missing something?

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You can look at it from a more fundamental POV: the friction comes from the interaction between electron clouds--which can be rather bumpy in nature for a surface. So, for a body at rest, its bumps will have sunk into the dips on the surface it rests on. On the other hand, if the body is in motion, the bumps skate over the bumps of the surface. One can see that the body gets more force when it is stuck in the dip, as one has to remove it from the dip. –  Manishearth Mar 22 '12 at 11:23
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5 Answers

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The problem with this question is that static friction and kinetic friction are not fundamental forces in any way-- they're purely phenomenological names used to explain observed behavior. "Static friction" is a term we use to describe the observed fact that it usually takes more force to set an object into motion than it takes to keep it moving once you've got it started.

So, with that in mind, ask yourself how you could measure the relative sizes of static and kinetic friction. If the coefficient of static friction is greater than the coefficient of kinetic friction, this is an easy thing to do: once you overcome the static friction, the frictional force drops. So, you pull on an object with a force sensor, and measure the maximum force required before it gets moving, then once it's in motion, the frictional force decreases, and you measure how much force you need to apply to maintain a constant velocity.

What would it mean to have kinetic friction be greater than static friction? Well, it would mean that the force required to keep an object in motion would be greater than the force required to start it in motion. Which would require the force to go up at the instant the object started moving. But that doesn't make any sense, experimentally-- what you would see in that case is just that the force would increase up to the level required to keep the object in motion, as if the coefficients of static and kinetic friction were exactly equal.

So, common sense tells us that the coefficient of static friction can never be less than the coefficient of kinetic friction. Having greater kinetic than static friction just doesn't make any sense in terms of the phenomena being described.

(As an aside, the static/kinetic coefficient model is actually pretty lousy. It works as a way to set up problems forcing students to deal with the vector nature of forces, and allows some simple qualitative explanations of observed phenomena, but if you have ever tried to devise a lab doing quantitative measurements of friction, it's a mess.)

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"the force required to keep an object in motion would be greater than the force required to start it in motion." --I think we are missing the velocity with which it is to be kept in motion. Trying to keep an object in motion near its terminal velocity would require more force than to start it from zero to a value much less than the terminal velocity. Drag counts as a friction force right? –  user80551 Sep 10 '13 at 9:44
    
@user80551: This is all in the context of the Amontons model. The coefficients $\mu_k$ and $\mu_s$ only make sense within that model. In that model, there is no terminal velocity, and the frictional forces are independent of $v$ for $v\ne0$. –  Ben Crowell Sep 10 '13 at 15:18
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With static friction there is no movement between the surfaces and therefore no work is done, and so the two surfaces remain at the same temperature (ambient).

However, with dynamic friction there is work done and so one or both surfaces absorb the energy and heat up. With typical friction materials the maximum coefficient of friction is only achieved when the material is heated, and in many (not all) cases the dynamic coefficient of friction, at elevated temperature and with suitable contact pressure and sliding speed, is in fact HIGHER than the static coefficient of friction with lower (ambient) temperature and zero sliding speed with the same contact pressure.

Certainly for some types of materials it is demonstrably the case that the static friction coefficient can be lower than the quoted dynamic friction coefficient.

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With the simple model of friction, using coefficients of static and kinetic friction, you are right - the coefficient of kinetic friction cannot be higher than the coefficient of static friction.

In real world, the phenomenon of friction may be much more complex. Your reasoning points out that with the velocity very close to zero kinetic friction cannot be much higher than static friction. To be more accurate, if the velocity is infinitely small, the kinetic friction may be higher than static one, but then it may only be infinitely small! An example is air friction, which is zero when a body doesn't move and increases with velocity. In a simple model, for small velocities, air friction is just proportional to velocity.

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In a simple model for small velocities anything that starts at zero when v=0 is proportional to velocity. –  Mark Eichenlaub Nov 11 '10 at 13:05
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This answer is speculative - not based on my experience with friction.

Logically, there's no reason kinetic friction has to be velocity-independent. You could have kinetic friction that increases with velocity. That way, if you push on something with more force than static friction, the thing would accelerate up to some certain velocity at which kinetic friction equaled the applied force, and then accelerate no more.

If that speed were very slow, you could say that kinetic friction is greater than static friction for all normally-encountered speeds without a paradox.

However, as you pointed out, kinetic friction would have to be less than or equal to static friction for speeds right next to zero.

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The kinetic friction is usually not greater than the applied force.

You start moving an object which increases the static force to prevent any motion. However, you keep applying more force until you reach a maximum value for the static friction. Then, the object begins to move. However, the kinetic friction (which is produced when the object moves) is less than the applied force. Furthermore, the kinetic friction doesn't depend on the applied force, therefore, you don't need to apply more force to keep moving the object (actually, you need less force).

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I guess what I'm asking is whether the kinetic friction necessarily has to be less than the applied force at the point when static friction is exceeded. I've always assumed that it did have to be, until yesterday. –  David Z Nov 11 '10 at 5:49
    
Let me be more precise. The maximum value in which the static friction is 'defeated' is called 'starting friction'. When the applied force is enough to reach that point, the kinetic friction starts to play a role, however, the kinetic friction is usually less than the starting friction. Given that the applied force is greater than the starting friction, then usually the applied force exceeds the kinetic friction. It is said 'usually' because there are cases in which the kinetic friction is greater than the static friction. –  Robert Smith Nov 11 '10 at 6:33
    
By the way, I made a mistake in the answer. I said 'kinetic force' instead of 'kinetic friction'. I'm changing that. –  Robert Smith Nov 11 '10 at 6:36
    
I'm curious about these cases in which the kinetic friction is greater than the static friction. What sorts of physical systems are they, and what exactly happens when the applied force exceeds the starting friction but is less than the kinetic friction? –  David Z Nov 11 '10 at 12:33
    
@David: From what I've able to read, cases in which the kinetic friction is greater than the static friction, are caused by the chemical bonding between the surfaces. It appears to be an area of current research which involves more than the forces we are considering. Nevertheless, scholar.lib.vt.edu/theses/available/etd-07282008-135807/… (Page 41) provides an example of $\mu_{k}>\mu_{s}$. Trustworthy? I don't know. –  Robert Smith Nov 11 '10 at 20:25
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protected by Qmechanic Sep 10 '13 at 7:50

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