Analysis: Basic model of dry friction

How can we represent friction? Often, we can use the simple model devised by two French researchers– Coulomb and Amontons. In fact, this model of friction is often called Coulomb friction. Here’s how it goes. First, we need an FBD. We’ll use the experiment with the book on the table. The graph plots the friction force F against pushing force P. As we increase force P, it is at first balanced by an increasing force F, the friction force. At first, the friction force balances the push, and the book stays put. It’s in equilibrium. But then there comes a point where F can no longer increase. It has reached its limiting value, and the book moves.

Once it moves, the friction force often drops to a lower level. Our friction model specifies that this limiting value before it moves is given by the equation P equals mu s N where mu s is the coefficient of the limiting static friction, and N is the normal force. Normal means at right angles to the surface. The coefficient mu s is assumed to be independent of the load and independent of the surface area in contact. These assumptions are often rough and ready, but the results are useful all the same. When the force P exceeds the limiting friction, the book will accelerate, and the friction force will drop to the value known as the kinetic friction force.

Our model specifies that this value is given by F equals mu k N. mu k is the coefficient of kinetic friction. Again, this is rough and ready, but it is useful. Now let’s look at the book sliding down an incline. Here is the FBD. Notice that the axes are aligned along the incline and perpendicular to the incline. That’s often a help. We can split the weight force into components– one down incline, and one perpendicular to the incline or normal to it. If we now apply equilibrium along the incline and then perpendicular to the incline, we can get expressions for F, N, and, hence, F divided by N.

Pause the video and try to find F, N, and F divided by N in terms of mg, the weight, and the angle alpha.

Here are the results. Now we will consider the case of impending motion. That is when an object is just on the point of moving. And when the angle is for the special case of impending motion, we substitute a new symbol, phi, for alpha and rearrange to get F divided by N equals 10 phi. Now F divided by N is also mu, the coefficient of friction. So if we know phi, by a test for example, we can find mu. Similarly, if we find the angle when the book will just keep moving when it’s given a push, then we can find the coefficient of kinetic friction. Here’s another useful way of describing friction behaviour.

phi is called the angle of friction either static or kinetic as appropriate. Now let’s look at the tipping box.