We have already seen several times that a physical rigid plane, when pushed by a body in contact with it, reacts with a normal force which is equal and opposite to the active force. In the example drawn in Fig.3.12the plane is horizontal and the active force, which is vertical, is simply the weight Fw of the block lying on the plane. The normal reaction N is vertical upwards.
We now apply to the block a force F parallel to the contact surface (horizontal in this particular case), by attaching a wire to the block and pulling. Suppose that we
Fw N F Ft F Ft Ft, max motion 0 0 (a) (b)
Fig. 3.12 aActive and constraint forces on a block, b friction force versus applied tangential force
gradually increase the tangential force starting from zero. We observe that initially, when F is not very strong, the block does not move, it is still in equilibrium. This implies that the resultant of the forces must still be zero, not only in the direction normal to the plane, where nothing is changed, but also in the tangent one, where now there is a force. The constraint must have developed also a force parallel to the contact surface, Ftequal and opposite to F, namely
Ft¼ F:
The force developed by the constraint parallel to the contact surface, when there is no motion, is called static friction.
If we continue to increase the tangential force on the block F, the tangential force by the constraint increases too, as long as the block does not move. This happens at a certain value of the active force, meaning that the friction force cannot be larger than a maximum value that we call Ft,max.
This behavior is followed in all cases in which two dry surfaces are in contact. In these conditions, it is experimentally found that the maximum value of the static friction is proportional to the normal force, namely that
Ft;max¼ lsN: ð3:21Þ
The proportionality constantµsis called the coefficient of static friction, which is clearly a dimensionless quantity.
We now study the motion of the block when the tangential applied force is larger than Ft,max. By measuring its acceleration, we infer that a tangential contact force Ft is present, which is in general somewhat smaller than Ft,maxas shown in Fig.3.12b. Also in the case of relative movements of the two contact surfaces, it is experi- mentally found that the tangential force by the constraint is proportional to the normal one. Its direction is always parallel and opposed to the velocity, namely
Ft¼ ldNut; ð3:22Þ
where utis the unit vector of the velocity. The dimensionless constantµdis called coefficient of kinetic friction.
Figure3.12b shows schematically the tangential force of the constraint versus the applied tangential force. We see that Ftgrows to be equal to the applied force up to Ft,max. Then, when the motion is started, it diminishes somewhat, as we have already noticed, and then remains approximately, but not exactly, constant. Notice that in the majority of the casesµd< µs but there are also opposite cases.
As a matter of fact, the static and dynamic friction forces are due to the inter- actions between the molecules on the surfaces of the two bodies. Consequently, Eqs. (3.21) and (3.22) are a macroscopic description of a complex microscopic situation. We observe that friction coefficients depend critically on the status of the surfaces in contact, on how they have been machined, on their cleanliness, etc. Notice carefully that the molecules on the surface of a body made of a certain
substance, for example copper or steel, are not only of that substance. Water is almost always present, oxidation too. One canfind mentioned values of the friction coefficients between, say, copper and copper, copper and steel, etc. But, there is no single copper on copper, etc. friction coefficient, for the just mentioned reasons.
As a matter of fact, for example in the case of a piece of copper, it is possible to obtain surfaces populated by copper molecules only. The piece must be processed with ad hoc procedures under a vacuum, because in the presence of air, copper will oxidize and water molecules will be deposited on the surface immediately. Now suppose we have produced two such blocks in a vacuum and put their surface in contact. They immediately stick one onto the other and you will not be able to separate them. They became a unique copper bock. How are molecules supposed to know to which block they belong?
Thefirst astronauts to land on the Moon observed this phenomenon. Putting two stones gathered from the soil in touch, they found them sticking together and difficult to separate, even if their surfaces were obviously irregular.
There is no universal mechanism at the origin of the friction between two contact surfaces. Consider the important case of two metal surfaces. Metallic surfaces can be worked to be extremely smooth. Even in these conditions, surfaces are not smooth if looked at nanometer scales. Figure3.13tries to show the surfaces as seen at a large magnification. The irregular patterns have a typical scale of 10 = 100 nm. When two surfaces are, we think, in contact, the contact is indeed only between the“crests” on the two sides. Consequently the surface really in contact, say Scis much smaller than the nominal surface S (typical values of Sc/S are between 10−4 and 10−5). However, the larger is the normal force N pushing the two surfaces one against the other, the larger is the number of crests touching each other. We can then understand why the friction force is proportional to N. We can also understand why it is independent of the area of contact. Suppose we keep N constant and double the contact macroscopic surface S. The action of the normal force will distribute on a doubled area and its effect on the crests per unit area will halve. The number of contacts per unit surface will halve too, but they will cover a twice as large area. The total number of contact has not varied. In conclusion, Sc is pro- portional to N and independent of S.
Fig. 3.13 Pictorial view of the contact surfaces between two metals, at nanometer scale
In the contact points the molecules of the two bodies interact strongly attracting each other and becoming, so to say, welded. To have one surface sliding on the other, these micro welding points must be broken. Again the necessary force is proportional to Scand consequently to N and independent of S.
What we have just described is relative to dry surfaces between solid bodies and has nothing to do with the friction between lubricated surfaces. In this case, afilm of liquid is present between solid surfaces, the molecules of which are far enough away from each other to have an interaction. In this case the friction is due to the viscosity of the lubricant (see Sect.3.6).
The rolling resistance or rolling friction is the force resisting the motion developed by the constraint, for example the support surface, when a cylindrical or spherical body, such as a reel or a ball, rolls on the surface. Figure3.14represents in cross section such a cylinder, say a reel, of radius r. We apply a force F to the axis of the reel parallel to the support plane and normal to the axis. We assume that the reel does not slide on the plane due to the static friction force. This type of motion is called pure rolling. When the reel rolls, it does that about an instanta- neous axis that is the contact generator in the considered instant. The moment of the applied force about the instantaneous rotation axis isτ = rF. The moment τ nec- essary to have the rolling at a constant angular velocity is experimentally found to be proportional to the magnitude of the normal force N, namely
s ¼ cN; ð3:23Þ
whereγ is the rolling resistance coefficient. Its physical dimension is a length, and is measured in meters. The applied moment is equal and opposite to the moment developed by the constraint.
The rolling resistance force is generally smaller than the dynamic friction. As a matter of fact it is due to quite complicated phenomena in the region of contact between the reel and the support plane. In Fig.3.14this region is shown as aflat area of longitudinal with δ. This is an idealization, because actually both the cylinder and the plane deform into shapes that are not forward-backwards sym- metrical. We are here simplifying a lot. We can say that on the contact area a number of the above considered “crests” of both bodies are in contact. The dif- ference is that now, to have movement, the microwelds are broken acting in a
δ
r N
F
Fig. 3.14 Schematics of the rolling resistance
direction normal, rather than parallel, to the surface. This requires, caeteris paribus, a smaller force.
Example E 4.3 Consider Fig. 3.15. A brick lies on an inclined surface, the incli- nation of which,α, can be varied. Given the coefficient of static friction µs, what is the maximum value ofα at which the brick remains still?
The forces on the brick are its weight mg and the force exerted by the constraint. The latter can be decomposed in a normal, N, and a tangential, Ft, component, which is the friction. For equilibrium the components of the resultant must be zero. Namely, Ft¼ mg sin a and N ¼ mg cos a. Hence Ft=N ¼ tan a. But, the static friction force cannot be larger thanµsN, and the no-slide condition isa arctan ls. The maximum angle, sayaf arctan lsis called the friction angle. For example, the slopes of the piles of sand or of the screes in the mountains naturally settle on the corresponding friction angle.
We have seen in Sect.2.11that friction forces are dissipative, and that their work is negative when their application point moves, because they are always in a direction opposite to the motion, see Eq. (2.41). Indeed, the friction forces are always such as to oppose the relative motion of the two bodies. This does not imply that the friction acting on a body would always act to slow it down, on the contrary it can also accelerate it.
As an example, let us consider our brick, of mass m, ling on the horizontal platform of a cart. The latter moves straight forward with constant acceleration a(see Fig.3.16) in the direction of its velocity v. If the acceleration of the cart is not too large, the block remains still relative to the platform; its motion is accelerated with the same acceleration a as the cart. It must be acted upon by a force equal to ma. But the only horizontal force acting on it is the friction Ft. Hence, Ft= ma. The friction accelerates the brick. We know that Ftcan be at most equal toµsN =µsmg.
mg mg Ft N N Ft α α Fig. 3.15 A brick on a slide
and the forces acting on it
N
Ft a
Fig. 3.16 A brick on an accelerating platform and the forces acting on it
Consequently the maximum acceleration of the cart at which the brick does not slide isµsg.
Notice that in this case the friction has the direction of the velocity, namely of the displacement. Consequently its work is positive. In the same way, when we start running we are accelerated by the friction force between our shoe soles and the ground, when a car accelerates the accelerating force is the friction between its reel and the road. Notice however, that in these cases the work of the friction force is zero, because the point of application does not move.