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# The Fracas over Friction

Professor Flipper, is explaining to the class the physics of friction. Despite the fact that he is a penguin, the audience is captivated by his eloquent oratory style.

Prof. Flipper
Now we come to an important force. The force of friction. Now Newton discovered his three laws of motion, and together with Newton's law of universal gravitation, which we will learn about later in the course, they explained the motion of the heavenly bodies: the moon, mercury, Venus, the earth, mars, Jupiter, Saturn, Uranus, Neptune, and Pluto, as they travel relentlessly around the celestial sphere. In addition to this, Newton's laws could explain the motion of many comets including Halley's comet that was believed to presage the invasion of a tribe of penguins that would take over the world. The fact that Newton achieved a nearly complete quantitative understanding of their motion, is considered to be one of human-kind's greatest achievements. The small precession of the orbit of mercury, 43 arc-seconds per century, was explained by Einstein several centuries later, using his theory of general relativity. Penguins such as myself have taken a keen interest in humans' relentless search for the truth, but we tend to be more practical minded. We have to be, living in Antarctica. Life is hard there even if you're equipped with a sharp beak, excellent eyesight, and a my beautiful pair of flippers. No, it's cold down there, let me tell you, and slippery too! So we are more concerned with slipping into the water off the relative warmth of our icebergs than with the orbital period of Uranus. That is why the force of friction is so important. It keeps you from slipping and sliding around the place. It is a wonderful force, and I would have surely slid of a cliff by now if it wasn't for this force.

To demonstrate how it works, consider an inclined plane made out of steel. Lets take this metal block and place it on the plane. The plane makes an angle with the horizontal. When the plane is flat, that is , the block sits on the plane, motionless. Now lets increase a little. Interesting, the block stays there. This is the force of friction at work. Without this force, any block would accelerate down the inclined plane, but the force of friction, as pictured above, applies just the right force to keep this block motionless. Along the direction of the inclined plane, the force of gravity has a component . If the acceleration in the x direction is zero, then the net force must be zero by Newton's second law. Therefore the force of friction must be exerting a force in the x direction. Now what happens if we increase the angle some more? will increase, but look! The block continues to stay still. Therefore the frictional seems to be equal to . Let's draw a graph of this:

Here we have the frictional force f as a function of the applied force. Along the red line the block is completely motionless or static. Because the block isn't moving, this is called static friction, and we'll denote the force by the symbol . It's direction is along the plane opposite to that of the applied force.

Now let's increase the angle just a smidgen more. Look, the block is sliding down the inclined plane! Notice it is not sliding freely, clearly the force of friction is having a considerable effect on its motion still. So in this case we'll call this friction kinetic friction, , because it involves an object in motion.

Let's draw in this part of the friction curve:

Here I've drawn the kinetic part in blue. It's been found experimentally to be pretty flat at low speeds and have close to what is shown.

Clearly we have a transition between two different kinds of behavior: sticking, and slipping. Let's draw the transition region in green:

What is the threshold for unsticking? That is, what is the maximum static friction force that can be sustained without the block giving way? Well countless brilliant humans and penguins have pondered this problem and have come up the formula relating it to the normal force N:

That is, the maximum static frictional force that can be sustained is proportional to the strength of the normal force. Now the coefficient of proportionality is this symbol , which is often called the coefficient of static friction. Normally it is a number of around one or less, and depends on the nature of the two surfaces. A shoe on a banana peel will be expected to have a small coefficient of static friction, where as Good running shoes are designed to have good traction and hence a large coefficient of static friction.

How about the value of the kinetic friction, ? Again this has been the subject of much cogitation and the answer is

Again it's proportional to the normal force and the coefficient of proportionality is called , the coefficient of kinetic friction. Experimentally it appears that .

Another miraculous thing about this is that these formulas only depend on the kind of material, not the surface area in contact. It kind of makes sense. If the surface area was small, like if you were wearing high heeled shoes instead of loafers, the pressure would be greater, so it would dig in to the ground more, thus compensating for less contact area.

So now I have elucidated the complete curve which is presented below:

The professor squawks, and the class sighs in awe at the miracle of what they've witnessed. Surely he must be the best penguin teacher on campus. But quite unexpectedly, a strange person walks in, wearing what appears to be an English policeman uniform, an ``English bobby''. He sits quietly down in the front row, and this seems to perturb the penguin, his voice becomes more bird-like but he continues with his lecture anyway. He squawks now more frequently.

Prof. Flipper
Now we come to an elegant means of determining the coefficients of static and kinetic friction. What we'll do is determine them from watching a block on this inclined plane.

Now let's figure out how to measure the coefficient of static We have the free body of this situation as shown:

The setup is identical to a previous example see 1.5.1. There instead of a frictional force, we had a string bearing a tension T, but the effect is the same, to keep the block in place. Eq. 1.9 says and eq. 1.11 becomes . At the angle where the block is about to slide, the frictional force is given by eq. 1.16 and we therefore have

which says that .

Now let's demonstrate this. Ah, the block starts slipping right here at 31 degrees, if I'm not mistaken.

Policeman
Umm, Professor-Penguin, would you be so kind as to repeat that demonstration, I want to check that you are obtaining a reproducible experimental result

Prof. Flipper
Class time is a rather precious commodity. I regret that I don't have (squawk) time to do a detailed scientific test of this, but I can assure you that it is quite reproducible.

Policeman
Let me introduce myself to the class, I'm one of the Physics Police. Our lot is not a happy one. We have to go around the world policing questionable scientists to make sure they don't mislead the rest of us. Now the professor here has been warned about giving this friction lecture before. He knows that the consequences are rather serious for demonstrating unreproducible experiments. He seems to have this urge to make unfounded experimental claims for reasons not very clear to a simple policeman such as myself.

Prof. Flipper
I assure you, that I'm now a completely reformed penguin. I did take what you said very seriously and my course now is (squawk squawk) completely (squawk) honest.

Policeman
Well then I must insist that you retry this friction experiment. Otherwise I'll have to call in Albert the polar bear. He's waiting outside, and hasn't tasted a penguin in weeks.

Prof. Flipper
Well, I'd be more than glad to redo this test again.

The penguin tries it again but this time it slips down at a different angle, 39 degrees

Prof. Flipper
Well that's very odd, it seems to have started to slip at a different angle after all, it must be that the surface is contaminated.

Policeman
Oh I have enough of your excuses penguin! You said the same thing last time, and every time it fails to be reproducible. Not only that but you said the answer was independent of the surface area. You know that's a load of nonsense! I have just about enough of you Professor(???) Flipper. Say where did you get your doctorate from anyway?

Prof. Flipper
University of Antarctica.

Policeman
Never heard of the place, but I'm afraid I'm going to have to arrest you for lying about physics.

Prof. Flipper
Wait, but the students love me, and they love my book, it's almost as popular as Tipler, Haliday and Resnick and Serway! We all have to say these things, don't you understand? No one wants to hear that friction is very complicated and difficult to understand quantitatively! All introductory books make the same claims. It's easier this way on the students!

Policeman
It's that kind of attitude that have kept penguins penguins! The first thing you should make clear to students is that in order to get anywhere in science, you have to be completely honest, and admit when things don't work quite the way you'd like them to. We don't want you instilling any more of your dishonest attitudes in these impressionable minds. Off to Antarctica with you before I sick Albert on you, you good for nothing bird! So let this be a lesson to all you students! You behave like the penguin and the physics police will be watching you too! We don't care if it doesn't make a good story, what's important is that it be honest. So when you do problems involving friction, remember that most of what you'll be doing is extremely approximate and you should regard it as only a rough guide to what you'd expect.

The curve the penguin drew figure 1.6 doesn't really exist. The green region in particular is bogus. It is not at all reproducible. It depends on whether you start with a static object and increase the applied force, or you start with a moving object and decreases the applied force. The physics of friction is a very interesting but complex subject and is currently under investigation by a lot of researchers, that are hopefully a good deal more honest than Professor Flipper.

Next: Example Up: Forces Previous: pulleys

Joshua Deutsch
Wed Jan 7 17:12:17 PST 1998