4. FRICTION
4.1 Laws of friction. We know from experience that when two bodies tend to
slide on each other a resisting force appears at their surface of contact which
opposes their relative motion. The laws of dry friction are best understood by the
following experiment. A block of weight W is placed on a horizontal surface
(Fig.4.1a). Suppose, now, that a horizontal force P is applied to the block
(Fig.4.1b). If P is small, the block will not move; some other horizontal force
must therefore exist, which balances P. This other force is the static-friction
force F, which is actually the resultant of a great number of forces acting over
the entire surface of contact between bodies. The nature of these forces is not
known exactly, but it is generally assumed that these forces are due to the
irregularities of the surface in contact and also, to a certain extent, to molecular
attraction. A detail examination of the nature of friction is complex physicomechanical problem lying beyond the scope of theoretical mechanics.
If the force P is increased, the friction force F also increases, continuing to
oppose P, until its magnitude reaches a certain maximum value Fm (Fig.4.1c). If
P is further increased, the friction force cannot balance it any more and the
block starts sliding. As soon as the block has been set in motion, the magnitude
of F drops from Fm to a lower value Fk . From then on, the block keeps sliding
with increasing velocity while the friction force, denoted by Fk and called
kinetic-friction force, remains approximately constant.
Experimental evidence shows that the maximum value of static-friction force
Fm is proportional to the normal component N of the reaction of the surface, i.e.
Fm = µsN
where µs is a constant called the coefficient of static friction.
(4.1)
Similarly, the magnitude Fk of the kinetic-friction force may be put in the form
Fk = µkN
where µk is a constant called the coefficient of kinetic friction.
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(4.2)
The coefficients of friction µs and µk do not depend upon the area of the surfaces
in contact. Both coefficients, however, depend strongly on the nature of the
surfaces in contact. Since they also depend upon the exact condition of the
surfaces, their value is seldom known with an accuracy greater than 5%.
Approximate values of coefficients of static friction are given in Table 4.1 for
various dry surfaces. The corresponding values of the coefficient of kinetic
friction would be about 25% smaller.
Table 4.1 Approximate values of coefficients of static friction for dry surfaces
Metal on metal
0,15 ÷ 0,35
Metal on wood
0,20 ÷ 0,60
Metal on leather
0,30 ÷ 0,60
Wood on wood
0,30 ÷ 0,70
Rubber on concrete
0,60 ÷ 0,90
Steel on ice
~0,03
From the description given above, it appears that four different situations may
occur when a rigid body is in contact with a horizontal surface:
a) The forces applied to the body do not tend to move it along the surface of
contact; there is no friction force (Fig.4.2a);
b) The applied forces tend to move the body along the surface of contact but are
not large enough to set it in motion. The friction force F which has developed
may be found by solving the equations of equilibrium for the body. Since there
is no evidence that the maximum value of static-friction force has been reached,
the equation (4.1) cannot be used to determine the friction force F (Fig.4.2b);
If the magnitude F of the friction force is smaller than its maximum value Fm ,
i.e. F < Fm = µsN, we say that the friction is not developed
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c) The applied forces are such that the body is just about to slide. We say that
motion is impending. The friction force F has reached its maximum value Fm,
and, together with the normal force N, balances the applied forces. Both the
equation of equilibrium and the equation (4.1) may be used. We also note that
the friction force has a sense opposite to the sense of impending motion
(Fig.4.2c);
d) The body is sliding under the action of the applied forces, and the equations
of equilibrium do not apply any more. However, F is now equal to the kineticfriction force Fk and the equation (4.2) may be used. The sense of Fk is opposite
to the sense of motion (Fig.4.2d).
Let us resume the above statements in a form of three Coulomb laws (1781!):
At the condition of impending motion and once motion has begun, it is possible
for given dry surfaces to relate the frictional and normal components of force so
that:
1. The total amount of friction which can be developed is independent of the
magnitude of the area of contact.
2. The total friction force which can be developed is proportional to the normal
force transmitted across the surface of contact.
3. For low velocities, the total amount of friction which can be developed is
practically independent of velocity. However, it is less than the frictional force
corresponding to impending motion.
Remarks:
1. Very often we limit our modelling task to either static or dynamic problem
only. In other words we don’t need to create a model for static and dynamic
purposes simultaneously. In those cases a friction force as well as a coefficient
of friction may be denoted simpler without lower indices, i.e. F and µ.
The latter simplified symbols will be used in what follows.
2. In a majority of Polish handbooks the letter T is used for denoting the friction
force (from the word tarcie).
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4.2 Angle of friction. It is sometimes found convenient to replace the normal
force N and the friction force F by their resultant R. Let us consider again a
block of weight W resting on a horizontal plane surface. If no horizontal force is
applied to the block, the resultant R reduces to the normal N (Fig.4.3a).
However, if the applied force P has a horizontal component Px which tends to
move the block, the force R will have a horizontal component F and, will form a
certain angle with the vertical (Fig.4.3b). If Px is increased until motion becomes
impending, the angle between R and the vertical grows and reaches a maximum
value (Fig.4.3c). This value is called angle of friction and denoted by φ.
From Fig.4.3c, we note that tan φ = Fm/N = µN/N. Thus finally
tan φ = µ
(4.3)
4.3 Problems involving dry friction. Problems involving dry friction are found
in many engineering applications. Some deal with simple situations particularly
when a single rigid body has only one point (or one plane) of contact with the
ground or another body (Fig.4.4a). There is also a simple situation when a single
body has more than one point of contact but an impending or actual motion in
one point of contact results in the same in all others contact points (Fig.4.4b,c).
This means that fully developed friction appears simultaneously in all contact
points. There are, however, more complicated situations when:
a) at the stage of modeling we have to select from an infinite number of possible
contact points only one or two actual points of contact (Fig.4.5a);
b) a single rigid body has two points of contact with other bodies but the
impending or actual motion may appear at one or two points and this is not
known in advance (Fig.4.5b)
c) a system of rigid bodies has more than one point of contact but the motion of
one member of a system does not necessarily affect the motion of others
members (Fig.4.5c)
Also, a number of common machines and mechanisms may be analyzed by
applying the laws of dry friction. These include wedges, screws, journal and
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thrust bearings, and belt transmissions. They can be studied in more advanced
texts and some of them will be considered in the final chapter of this lecture
entitled Selected Applications.
There is another problem caused by friction. It appears when a free-body
diagram is to be drawn. As long as friction is not involved for bodies in contact
an arbitrary sense may be given to all unknown reactions. This is not the case
when friction has to be taken into account (is involved). Now, both components
of a total reaction force, i.e. normal and tangential reaction– must be drawn with
an appropriate (correct) sense.
A relatively simple situation occurs when two bodies A and B are in contact
and tend to slide on each other (Fig.4.6a). The friction forces exerted,
respectively, by A on B and by B on A are equal and opposite (Newton’s third
law). It is important, in drawing the free-body diagram of one of the bodies, to
include the appropriate friction force with its correct sense. The following rule
should then be applied: the sense of the friction force acting on A is opposite to
that of the motion (or impending motion) of A as observed from B (Fig.4.6b).
The sense of the friction force acting on B is determined in a similar way
(Fig.4.6c). Note that the motion of A as observed from B is a relative motion.
A more difficult situation is presented in Fig.4.7a. Now a rotating disc is put
at a rough surface. In order to determine the correct sense of friction force in a
phase when the disc touches the ground, we need to establish the relative
velocity of the touch point of the disc with respect (relative) to the ground. The
actual friction force acts in a direction opposite to this velocity (see Fig.4.7b). In
fact, it is the friction force which initiates and then accelerates the translational
motion of the disc.
4.4 Rolling resistance – wheel friction. The wheel is one of the most important
inventions of our civilization. Its use makes it possible to move heavy loads with
relatively little effort. Because the point of the wheel in contact with the ground
at any given instant has no relative motion with respect to the ground, the wheel
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eliminates the large friction forces, which would arise if the load were in direct
contact with the ground. In practice, however, there exists some resistance to the
motion of the wheel. This resistance is mostly due to the fact that wheel and
ground deform, with the result that contact between wheel and ground takes
place, not at a single point, but over a certain area. This resistance is known as
the rolling friction or rolling resistance. The phenomenon of rolling resistance
and the law of rolling friction can be best understood by the following
experiment. Consider a wheel of weight W resting on a rough horizontal surface
(Fig.4.8a). Under the load W, both wheel and the ground deform slightly,
causing the contact between wheel and the ground to take place over a certain
area. Suppose, now, that a small horizontal force P is applied at center of wheel
(Fig.4.8b). Experimental evidence shows that the resultant R of the forces
exerted by the ground on the wheel over this area is applied at a point B, which
is not located directly under the center O of the wheel, but slightly in front of it.
Let the force P increases such that the wheel rolls at constant speed. To balance
the moment of W about B and to keep the wheel rolling at constant speed, the
force P must satisfy the following equation
Pr = Wb
(4.4)
where r is the radius of wheel and b is the horizontal distance between O and B.
The distance b is commonly called the coefficient of rolling resistance, which
has the dimension of length. For instance b = 0,5 mm for a wheel rolling on a
rail, both of which are made of mild steel. The value of b depends on the
material of the bodies and is determined experimentally.
The ratio b/r for most materials is much less than the coefficient of static
friction µs. That is why in mechanisms rolling parts (wheels, rollers, ball
bearings, etc.) are preferred to sliding parts.
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