LESSON 7:
FRICTION
ENGINEERING MECHANICS
In this chapter, the concepts related to friction are explained and
the laws of friction are presented. Application of these laws to
many engineering problems including wedge and rope/ belt
friction is illustrated.
Friction involves contact between two bodies. Tangential forces
generated between contacting surfaces are called frictional
forces. When a body moves or tends to move over another
body, a force opposing the motion develops at the contact
surfaces. This force, which opposes the movement or the
tendency of movement, is called frictional force or simply
friction. Friction is due to the resistance offered to motion by
minutely projecting particles at the contact surfaces. In some
types of machines & processes retarding effects of frictional
forces are to be minimized. Examples are bearings of all types,
gears, the flow of fluid in pipes etc. In some situation the
effects of friction needs to be maximized. Examples are brakes,
clutches, wedges etc.
1. Sliding Friction
It is the friction experienced by a body when it slides over the
other body.
2. Rolling Friction
It is the friction experienced by a body when it rolls over a
surface.
It is experimentally found that the magnitude of limiting
friction bears a constant ratio to the normal reaction between
the two surfaces and this ratio is called Coefficient of Friction.
Thus in Fig. 7.1,
F
Coefficient Friction =N, Where
F = limiting friction, and
N = normal reaction between the contact surfaces.
Coefficient of friction is denoted by 
Types of Friction
 
N
F
1. Dry Friction
Dry friction occurs when the unlubricated surfaces of two
solids are in contact under a condition of sliding or a tendency
to slide. This type of friction is also called Coulomb’s friction,
which will be studied in this lesson.
2. Fluid Friction
Fluid friction occurs when adjacent layers in a fluid (liquid or
gas) are moving at different velocities. This motion causes
frictional forces between fluid elements, and these forces depend
upon the relative velocity between layers.
3. Internal Friction
Internal friction occurs in all solid materials which are subjected
to cyclical loading.
Fig. 7.1
Frictional Force
Frictional force has the remarkable property of adjusting itself
in magnitude to the force producing or tending to produce the
motion so that the motion is prevented. However, there is a
limit beyond which the magnitude of this force cannot increase.
If the applied force is more than this maximum frictional force,
there will be movement of one body over the other. This
maximum value of frictional force, which comes into play,
when the motion is impending, is known as limiting friction.
It may be noted that when the applied force is less than the
limiting friction, the body remains at rest and such frictional
force is called Static Friction, which may have any value
between zero and the limiting friction. If the value of the
applied force exceeds the resistance experienced by the body
while moving is known as dynamic or kinetic friction.
Dynamic friction is usually found to be some what less than the
limiting friction. Dynamic Friction may be grouped into the
following two:
Laws of Friction
The principles discussed above are mainly due to the
experimental studies by Coulomb (1781) and Morin (1831).
These principles constitute the laws of dry friction and are listed
as follows:
1. The force of friction always acts in a direction opposite to
that in which the body tends to move.
2. Till the limiting value is reached, the magnitude of friction is
exactly equal to the force which tends to move the body;
3. The magnitude of the limiting friction bears a constant ratio
to the normal reaction between the two surfaces;
4. The force of friction depends upon the roughness /
smoothness of the surfaces;
5. The force of friction is independent of the area of contact
between the two surfaces;
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ENGINEERING MECHANICS
6. After the body starts moving, the dynamic friction comes
into play, the magnitude of which is less than that of
limiting friction and it bears a constant ratio with normal
force. This ratio is called coefficient of dynamic friction.
Angle of Friction
Consider a block as shown in Fig. 7.2 subject to a pull P. Let F
be the frictional force developed and N the normal reaction.
Thus, at the contact surface, the reactions are F and N. They can
be combined graphically to get the reaction R which acts at angle
q to normal reaction. This angle q , called the angle of friction, is
given by:
If f is the value of angle , when motion is impending,
frictional force will be limiting friction and hence
F
tan  =N=  = tan 
F
tan 
or
N
As frictional force increases the angle q increases and it can reach
maximum value a when limiting value of friction is reached. At
this stage
F

tan
7.1
N
and this value of a is called angle of limiting friction. Hence, the
angle of limiting friction can be defined as the angle between the
resultant reaction and the normal to the plane on which the motion of
the body is impending.
 = 
Thus, the value of angle of repose is the same as the value of
limiting angle of friction.
Cone of Friction
When a body is to impend to move in the direction of P, the
frictional force will be limiting friction and the resultant reaction
R will make limiting friction angle a with the normal as shown
in Fig 7.4. If the body is having impending motion in some
other direction, again the resultant reaction makes limiting
frictional angle a with the normal in that direction. Thus, when
the direction of force P is gradually changed through 360°, the
resultant R generates a right circular cone with semi central angle
equal to .
If the resultant reaction is on the surface of this inverted right
circular cone, whose semi-central angle is limiting frictional angle
(a), the motion of the body is impending, if the resultant is
within this cone the body is stationary. This inverted cone with
semi-central angle equal to limiting frictional angle a is called
Cone of friction.
Angle of Repose
We have seen that when grains (food grain, soil, sand, etc.) are
heaped, a cone like shape is formed. There exists a limit for the
inclination of the surface. Beyond this inclination the grains
starts rolling down. This limiting angle up to which the grains repose
(sleep) is called the angle of repose.
If we consider a block of weight W resting on an inclined plane,
which makes an angle q with the horizontal as shown in Fig.
7.3. When q is small the block will rest on the plane. If q is
increased gradually a stage is reached at which the block starts
sliding. This angle between those two contact surfaces is called
the angle of repose.
Thus, the maximum inclination of the plane on which a body, free from
external forces, can repose (sleep) is called Angle of Repose.
Now, consider the equilibrium of the block shown in Fig 7.3.
Since the surface of contact is not smooth, not only normal
reaction, but frictional force also develops, since the body tends
to slide downward, the frictional resistance will be up the plane.
 forces normal to the plane = 0, gives
N = W cos 
7.2
 forces parallel to the plane = 0, gives
F = W sin 
7.3
Fig. 7.4
Having studied the basic terms of friction, let us apply them to
a simplified case. We try to find out the force that is required to
cause the motion to impend in case of a system of to blocks
connected together by a cord placed over a pulley as shown in
Fig. 7.5, where m = 0.2 .
F
Fig. 7.5
Dividing Eqn. 7.3 Eqn. 7.2 we get,tan  =N
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ENGINEERING MECHANICS
Free body diagrams of the blocks are as shown in Fig. 7.5(b).
Consider the block of 750 N,
 forces normal to the plane = 0,
i.e.
N – 750 cos 60° = 0,
1
N = 375 N
Since the motion is impending, from law of friction,
3. Two identical blocks A & B are connected by a rod and they
rest against vertical and horizontal planes respectively as
shown in Fig. 7.8. If sliding impends when m = 45,
determine the coefficient of friction, assuming it to be same
for both floor and wall.
[Ans. 0.0414]
1
F =  N = 0.2x375 = 75 N
 forces parallel to the plane = 0,
1
1
T – F – 750 sin 60° = 0,
1
T = 75+750 sin 60° = 724.52 N
Considering 500 N body:
v = 0,
N – 500 + P sin 30° = 0
2
Fig. 7.8
N + 0.5 P = 500
From law of friction,
2
Notes
F = 0.2 N = 0.2(500-0.5P) = 100 – 0.1 P
2
2
H = 0, P cos 30° - T – F = 0,
2
P cos 30° - 724.52 – 100 + 0.1 P = 0
P = 853.52 N
Problems
1. The block A shown in Fig. 7.6 weighs 2000 N. The cord
attached to A passes over a frictionless pulley and supports a
weight equal to 800 N. The value of coefficient of friction
between A and the horizontal plane is 0.35. Solve for
horizontal force P,
a. if the motion is impending towards the left, and
b. if the motion is impending towards the right.
Fig. 7.6
2. A 3000 N block is placed on an inclined plane as shown in
Fig. 7.7. Find the maximum value of W for equilibrium if
tipping does not occur. Assume coefficient of friction as 0.2.
[Ans. 1014.96 N]
Fig. 7.7
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