B. Tech. -4th Semester (MECH. ENGG.

) THEORY OF MACHINES (4ME4-07)

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UNIT -II
FRICTION DRIVES: TYPES AND LAWS OF FRICTION
LECTURE NO.- 1
Introduction
It has been established since long, that the surfaces of the bodies are never perfectly
smooth. When, even a very smooth surface is viewed under a microscope, it is found to
have roughness and irregularities, which may not be detected by an ordinary touch. If a
block of one substance is placed over the level surface of the same or of different
material, a certain degree of interlocking of the minutely projecting particles takes place.
This does not involve any force, so long as the block does not move or tends to move. But
whenever one block moves or tends to move tangentially with respect to the surface, on
which it rests, the interlocking property of the projecting particles opposes the motion.
This opposing force, which acts in the opposite direction of the movement of the upper
block, is called the force of friction or simply friction. It thus follows, that at every joint in
a machine, force of friction arises due to the relative motion between two parts and
hence some energy is wasted in overcoming the friction
Types of Friction
In general, the friction is of the following two types:
1. Static friction. It is the friction, experienced by a body, when at rest.
2. Dynamic friction. It is the friction, experienced by a body, when in motion. The dynamic
friction is also called kinetic friction and is less than the static friction.
It is of the following three types :
(a) Sliding friction. It is the friction, experienced by a body, when it slides over another
body.
(b) Rolling friction. It is the friction, experienced between the surfaces which has balls or
rollers interposed between them.

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B. Tech. -4th Semester (MECH. ENGG.) THEORY OF MACHINES (4ME4-07)
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(c) Pivot friction. It is the friction, experienced by a body, due to the motion of rotation
as in case of foot step bearings.
The friction may further be classified as :
1. Friction between unlubricated surfaces,
The friction experienced between two dry and unlubricated surfaces in contact is known
as dry or solid friction. It is due to the surface roughness. The dry or solid friction includes
the sliding friction and rolling friction as discussed above.
2. Friction between lubricated surfaces
When lubricant (i.e. oil or grease) is applied between two surfaces in contact, then the
friction may be classified into the following two types depending upon the thickness of
layer of a lubricant.
1. Boundary friction (or greasy friction or non-viscous friction). It is the friction,
experienced between the rubbing surfaces, when the surfaces have a very thin layer of
lubricant. The thickness of this very thin layer is of the molecular dimension. In this type
of friction, a thin layer of lubricant forms a bond between the two rubbing surfaces. The
lubricant is absorbed on the surfaces and forms a thin film. This thin film of the lubricant
results in less friction between them. The boundary friction follows the laws of solid
friction.
2. Fluid friction (or film friction or viscous friction). It is the friction, experienced between
the rubbing surfaces, when the surfaces have a thick layer of the lubrhicant. In this case,
the actual surfaces do not come in contact and thus do not rub against each other. It is
thus obvious that fluid friction is not due to the surfaces in contact but it is due to the
viscosity and oiliness of the lubricant.
Limiting Friction
Consider that a body A of weight W is lying on a rough horizontal body B as shown in Fig.
1.1 (a). In this position, the body A is in equilibrium under the action of its own weight W,
and the normal reaction RN (equal to W) of B on A. Now if a small horizontal force P1 is
applied to the body A acting through its centre of gravity as shown in Fig. 1.1 (b), it does

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B. Tech. -4th Semester (MECH. ENGG.) THEORY OF MACHINES (4ME4-07)
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not move because of the frictional force which prevents the motion. This shows that the
applied force P1 is exactly balanced by the force of friction F 1 acting in the opposite
direction.

Figure 1.1 Limiting friction

If we now increase the applied force to P=as shown in Fig. 1.1 (c), it is still found to be in
equilibrium. This means that the force of friction has also increased to a value F 2 = P2.
Thus every time the effort is increased the force of friction also increases, so as to become
exactly equal to the applied force. There is, however, a limit beyond which the force of
friction cannot increase as shown in Fig. 1.1 (d). After this, any increase in the applied
effort will not lead to any further increase in the force of friction, as shown in Fig. 1.1 (e),
thus the body A begins to move in the direction of the applied force. This maximum value
of frictional force, which comes into play, when a body just begins to slide over the
surface of the other body, is known as limiting force of friction or simply limiting friction.
It may be noted that when the applied force is less than the limiting friction, the body
remains at rest, and the friction into play is called static friction which may have any value
between zero and limiting friction.
Laws of Static Friction
1. The force of friction always acts in a direction, opposite to that in which the body tends
to move.
2. The magnitude of the force of friction is exactly equal to the force, which tends the
body to move.

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B. Tech. -4th Semester (MECH. ENGG.) THEORY OF MACHINES (4ME4-07)
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3. The magnitude of the limiting friction (F ) bears a constant ratio to the normal reaction
(RN ) between the two surfaces. Mathematically F/RN = constant
4. The force of friction is independent of the area of contact, between the two surfaces.
5. The force of friction depends upon the roughness of the surfaces.
Laws of Kinetic or Dynamic Friction
1. The force of friction always acts in a direction, opposite to that in which the body is
moving.
2. The magnitude of the kinetic friction bears a constant ratio to the normal reaction
between the two surfaces. But this ratio is slightly less than that in case of limiting friction.
3. For moderate speeds, the force of friction remains constant. But it decreases slightly
with the increase of speed.
Laws of Solid Friction
1. The force of friction is directly proportional to the normal load between the surfaces.
2. The force of friction is independent of the area of the contact surface for a given normal
load.
3. The force of friction depends upon the material of which the contact surfaces are
made.
4. The force of friction is independent of the velocity of sliding of one body relative to the
other body.
Laws of Fluid Friction
1. The force of friction is almost independent of the load.
2. The force of friction reduces with the increase of the temperature of the lubricant.
3. The force of friction is independent of the substances of the bearing surfaces.
4. The force of friction is different for different lubricants.
Coefficient of Friction
It is defined as the ratio of the limiting friction (F) to the normal reaction(RN ) between
the two bodies. It is generally denoted by µ. Mathematically, coefficient of friction, µ

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B. Tech. -4th Semester (MECH. ENGG.) THEORY OF MACHINES (4ME4-07)
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Limiting Angle of Friction

Figure 1.2 Limiting angle of friction

Consider that a body A of weight (W) is resting on a horizontal plane B, as shown in Fig.
1.2. If a horizontal force P is applied to the body, no relative motion will take place until
the applied force P is equal to the force of friction F, acting opposite to the direction of
motion. The magnitude of this force of friction is F = µ.W = µ.RN , where R= is the normal
reaction. In the limiting case, when the motion just begins, the body will be in equilibrium
under the action of the following three forces :
1. Weight of the body (W),
2. Applied horizontal force (P),
3. Reaction (R) between the body A and the plane B
The reaction R must, therefore, be equal and opposite to the resultant of W and P and
will be inclined at an angle φ to the normal reaction R N. This angle φ is known as the
limiting angle of friction. It may be defined as the angle which the resultant reaction R
makes with the normal reaction RN. tan φ = F/RN = µ RN / R= µ
Angle of Repose
Consider that a body A of weight (W) is resting on an inclined plane B, as shown in Fig.
1.3. If the angle of inclination α of the plane to the horizontal is such that the body begins
to move down the plane, then the angle α is called the angle of repose. A little
consideration will show that the body will begin to move down the plane when the angle
of inclination of the plane is equal to the angle of friction (i.e. α = φ). This may be proved
as follows:

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B. Tech. -4th Semester (MECH. ENGG.) THEORY OF MACHINES (4ME4-07)
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Figure 1.3 Angle of response

The weight of the body (W) can be resolved into the following two components:
1. W sin α, parallel to the plane B. This component tends to slide the body down the
plane.
2. W cos α, perpendicular to the plane B. This component is balanced by the normal
reaction (RN ) of the body A and the plane B.
The body will only begin to move down the plane, when
W sin α = F = µ.RN = µ.W cos α …………………………………………………………...(∵ RN = W cos α)
∴ tan α = µ = tan φ or α = φ ........................................................................(∵ µ = tan φ)
Minimum Force Required to Slide a Body on a Rough Horizontal Plane

Figure 1.4 Minimum force required to slide a body

Consider that a body A of weight (W) is resting on a horizontal plane B as shown in Fig.
1.4. Let an effort P is applied at an angle θ to the horizontal such that the body A just
moves. The various forces acting on the body are shown in Fig. 1.4. Resolving the force P
into two components, i.e. P sin θ acting upwards and P cos θ acting horizontally.

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B. Tech. -4th Semester (MECH. ENGG.) THEORY OF MACHINES (4ME4-07)
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Now for the equilibrium of the body A,

RN + P sin θ = W
or RN = W – P sin θ ………………………………………………………………………………………………….....(i)
P cos θ = F = µ.RN …………………………………………………………………………….....(ii) ...(∵ F = µ.RN )
Substituting the value of RN from equation (i),we have
P cos θ = µ (W – P sin θ) = tan φ (W – P sin θ) ................................................(∵ µ = tan φ)
sin 
 (W  P sin  )
cos 

P cos θ .cos φ = W sin φ – P sin θ.sin φ

P cos θ.cos φ + P sin θ.sin φ = W sin φ
P cos (θ – φ) = W sin φ .........................................[cos θ. cos φ + sin θ.sin φ = cos (θ – φ)]
W sin 
P .................................................................................................................(iii)
cos(   )

For P to be minimum, cos (θ – φ) should be maximum, i.e.

cos (θ – φ) = 1 or θ – φ = 0° or θ = φ
In other words, the effort P will be minimum, if its inclination with the horizontal is equal
to the angle of friction.
∴ Pmin = W sin θ ....…………………………………………………………………………[From equation (iii)]
Friction of a Body Lying on a Rough Inclined Plane
Consider that a body of weight (W) is lying on a plane inclined at an angle α with the
horizontal, as shown in Fig. 1.5 (a) and (b).

Figure 1.5 . Body lying on a rough inclined plane.

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B. Tech. -4th Semester (MECH. ENGG.) THEORY OF MACHINES (4ME4-07)
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1. Considering the motion of the body up the plane

Let W = Weight of the body,
α = Angle of inclination of the plane to the horizontal,
φ = Limiting angle of friction for the contact surfaces,
P = Effort applied in a given direction in order to cause the body to slide with uniform
velocity parallel to the plane, considering friction,
P0 = Effort required to move the body up the plane neglecting friction,
θ = Angle which the line of action of P makes with the weight of the body W,
µ = Coefficient of friction between the surfaces of the plane and the body,
RN = Normal reaction, and
R = Resultant reaction.
When the friction is neglected, the body is in equilibrium under the action of the three
forces, i.e. P0, W and RN , as shown in Fig. 1.6 (a). The triangle of forces is shown in Fig.
1.6 (b). Now applying sine rule for these three concurrent forces,
P0 W W sin 
 or P0  ...........(i)
sin  sin(   ) sin(   )

Figure 1.6 . Motion of the body up the plane, neglecting friction.

When friction is taken into account, a frictional force F = µ.R N acts in the direction
opposite to the motion of the body, as shown in Fig. 1.7 (a). The resultant reaction R

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B. Tech. -4th Semester (MECH. ENGG.) THEORY OF MACHINES (4ME4-07)
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between the plane and the body is inclined at an angle φ with the normal reaction R N.
The triangle of forces is shown in Fig. 1.7 (b). Now applying sine rule,
P W W sin(   )
 or P  ............(ii)
sin(   ) sin(  (   )) sin(  (   ))

Figure 1.7. Motion of the body up the plane, considering friction.

2. Considering the motion of the body down the plane
Neglecting friction, the effort required for the motion down the plane will be same as for
the motion up the plane, i.e
W sin 
P0  ...........(iii )
sin(   )

Figure 1.8. Motion of the body down the plane, considering friction.
When the friction is taken into account, the force of friction F = µ.RN will act up the plane
and the resultant reaction R will make an angle φ with RN towards its right as shown in
Fig. 10.9 (a). The triangle of forces is shown in Fig. 1.7 (b). Now from sine rule,

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B. Tech. -4th Semester (MECH. ENGG.) THEORY OF MACHINES (4ME4-07)
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P W W sin(   )
 or P  ............(iv)
sin(   ) sin(  (   )) sin(  (   ))

Efficiency of Inclined Plane

The ratio of the effort required neglecting friction (i.e. P0 ) to the effort required
considering friction (i.e. P) is known as efficiency of the inclined plane. Mathematically,
efficiency of the inclined plane,
P0

P
Let us consider the following two cases:
1. For the motion of the body up the plane
Efficiency,
P0 W sin  sin(  (   ))
  
P sin(   ) W sin(   )
sin  sin  cos(   )  cos  sin(   )
 
(sin  cos   cos  sin  ) sin(   )
cot(   )  cot 

cot   cot 
2. For the motion of the body down the plane
Since the value of P will be less than P0 , for the motion of the body down the plane,
therefore in this case,
P0 W sin(   ) sin(   )
  
P sin(  (   )) W sin 
sin(   ) sin  cos   cos  sin 
 
sin  cos(   )  cos  sin(   ) sin 
cot   cot 

cot(   )  cot 

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B. Tech. -4th Semester (MECH. ENGG.) THEORY OF MACHINES (4ME4-07)
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B. Tech. -4th Semester (MECH. ENGG.) THEORY OF MACHINES (4ME4-07)
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