Friction

Subject :- Engineering Mechanics

UNIT:-3
Friction

ELECTRO-THEROPY
Make a World, Difference.

By-
Rajkamal Ghorai (DEE-2013) (7278840303)
Tufan Pal (DEIE-2015) (9002554366)
Anup Kr. Dutta (DEE-2017) (8337850463)

ELECTRO-THEROPY 1
 Friction

 Definition: - When an external force is applied to a body to over on another body, an opposing force
exerted along the common contact surface of the two bodies. This opposing force is called friction or
frictional resistance or frictional force.
The force of friction depends upon the nature of surfaces in contact and always parallel to the surface
of contact. The direction of frictional force, acting at the point of contact of the surfaces, always acts in a
direction opposite to the direction of the applied force or in which the contact surface tends to move.
Let us consider a wooden block of weight 'W’ kept on a rough horizontal surface. Now, a force 'P' is
acting on the block as shown in fig.
Due to the applied, force 'P’ the block has the tendency to move in same direction of the force. Then
a force comes into play to oppose that force, this ' opposing force known as friction and will act in the
contact surfaces of block and that surface.
The forces acting on the block are:
P = applied force
W = Weight of the block
R = normal reaction
F = Frictional force.
Generally these four forces acting on a body when it moves on a rough surface. If the force P is so
small then it can just move the body then
P = F............ (i)
and R = W ..........(ii)
Hence, friction is a force.

 Unit: - Its unit will be KN, N etc.

 Graph between applied force and frictional force: - When

external force (P) is small the body does not move. As P increases, F
also increases till it attains its maximum value. This maximum value
of friction is called limiting static friction and is denoted by Fs. As P
is increased further, the friction force cannot balance it anymore and
the body starts sliding. As soon as the body sets in motion, the
magnitude of FS suddenly drops to a lower value Fk called kinetic
friction. This is due to the less chance of meshing between the
irregularities of the surfaces in contact when these surfaces move with
respect to each other.

The variation of frictional force is shown in Fig. The portion OA represents when the body is at rest
and F = P. The point A corresponds to the maximum static friction (Fs). The point B corresponds to the
kinetic friction (Fk). Beyond point B the graph is a straight line parallel to X-axis.

 Limiting frictional 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 frictional force.

ELECTRO-THEROPY 2
 Friction

 Angle of friction: - The angle between the normal reaction R and

the resultant of limiting friction (𝜇𝑅) and normal reaction R is known
as angle of friction.

It is denoted by symbol ‘𝝋’.

 Coefficient of friction: - It is de med as the ratio of limiting force of friction (F) to the normal reaction
(R) between the two bodies at contact.

It is denoted by symbol ‘𝜇’.

𝐹
Thus 𝜇 = ∴ F = 𝜇R
𝑅
In most of the cases value of 𝜇 is less than 1.
When there is limiting equilibrium coefficient of friction is called static and is denoted by 𝜇𝑠 . The
maximum value of the static friction force (Fs) is proportional to the normal component (N) of the reaction
of the surface.
So we have Fs = 𝜇𝑠 . N............. (i)
Where 𝜇𝑠 = constant called the co-efficient of static friction.

When there is a relative motion between the two bodies, the coefficient of friction is called kinetic and
` the denoted by 𝜇𝑘 . The magnitude of kinetic friction Fk may be written in the form
Fk = 𝜇𝑘 N...... (ii)
Where 𝜇𝑘 = constant known as the Co-efficient of kinetic friction.

Generally, for solid bodies, 𝝁𝒔 > 𝝁𝒌 .

 Angle of repose: - If a body is placed over an inclined plane whose angle of inclination is gradually
increased, then the angle at which the body just starts to move i.e. sliding down in the plane, is called
Angle of Repose.

 Relation between angle of friction & angle of repose: - Let, the inclination is increased to a
when the body just start to slide downwards.
If W = weight of body
R = Normal reaction
𝛼 = Angle of inclination
𝑅 𝜇𝑅
By Lami's Theorem, =
sin(90+ 𝛼) sin(180 − 𝛼)
Or, tan𝛼 = 𝜇 = tan𝜑 ∴𝛼=𝜑

∴ Angle of repose is equal to angle of friction.

 Cone of friction & its significance: - We have seen that when a body having
impending motion in the direction of force (P), then the resultant force (R) will make
𝐹
an angle of 𝑡𝑎𝑛−1 (𝑁) = 𝜑 with the normal reaction (where F = Limiting frictional
force, N = Normal reaction).

For limiting friction F = 𝜇N and the action at contact surfaces (A) makes an
angle of 𝑡𝑎𝑛−1 𝜇 with the normal reaction. This angle is denoted by 𝜑.

ELECTRO-THEROPY 3
 Friction

Thus µ = tan𝜑.
If the direction in which the body tends to move is gradually changed then the limiting frictional force
will always lie in the plane through A perpendicular to the normal reaction and the direction of the resultant
force (R = √𝑁 2 + 𝐹 2 ) at contact point will always lie on a cone with its vertex at A and axis along the line
of the normal reaction. Thus the resultant force (R) generates a right circular cone with semi vertical angle
equal to 𝜑. This cone is known as cone of friction of friction is not limiting, then the angle made by the
resultant force at contact point, will be less than 𝜑.

 Types of friction: - Generally friction are two types - (a) Static friction
(b) Dynamic friction.

Dynamic friction may be subdivided into another two types - (i) Sliding friction
(ii) Rolling friction.

(a) Static friction is a force that keeps an object at rest. It must be overcome to start moving the object.
(b) Dynamic friction is the force that must be overcome to maintain steady motion of one body relative to
another because they remain in contact.
(i) Sliding friction refers to the resistance created by two objects sliding against each other.
(ii) When a body rolls over another the frictional force developed is called the force of
rolling friction.

 Laws of Friction (Coulomb Laws): - The friction developed between two contact surfaces which
are non-lubricated is known as dry friction. The following are the laws of dry friction :
(a) The force of friction always act in a direction opposite to that in which the body tends move.
(b) The magnitude of frictional force is exactly equal to applied force, till the limiting value is reached.
(c) The limiting frictional force bears a constant ratio to the normal reaction between the two surfaces.
(d) The ratio between limiting friction and normal reaction is slightly less when there is motion.
(e) The force of friction is independent of the area of contact between the two surfaces.

 Advantages & disadvantages of friction: - Friction has both useful and harmful effects.
Some useful effects of friction are describe below:
1. It is impossible for everybody to walk or climed on any place if frictional force is absent there. If
anybody try to move in forward directions without absent of frictional force they will fall down.
2. If we pull out nail from wood the frictional force offer. The resistance to the force to pull out the nail
from the wood.
3. When a ladder is placed with its one end at the vertical wall and the other end at floor. If the frictional
force F1 and F2 of wall and floor respectively prevent the ladder from slipping.
4. When we do some work by our hand, things does not slip from our hand due to the effect of frictions.

Some harmful effect of frictions are described below:

1. Large amount of power is lost in friction in gear, engine parts, bearings etc.
2. Chance of worn out of machine tools due to frictional force.
3. Due to frictional effect of dry leaves on a forest, chance of great firing.
4. When a fluid run over a pipe there is a chance of produce of heat and this heat damage the pipe and
make an accident etc.

ELECTRO-THEROPY 4
 Friction

 Equilibrium of body on inclined plane subjected to forces parallel to inclined plane

and body tends to move upward: -
Resolving the forces parallel & perpendicular to the inclined plane,

P = 𝜇𝑅 + W sin 𝛼 …….. (i)

R = W cos 𝛼 …...... (ii)

Putting the value of R in the equation (i)

P = 𝜇W cos 𝛼 + W sin 𝛼
= tan 𝜑 W cos 𝛼 + W sin 𝛼
sin𝜑 cos𝛼 + cos𝜑 sin𝛼
=W( )
cos𝜑
sin (𝛼 + 𝜑)
=W( )
cos𝜑

𝐬𝐢𝐧(𝜶 + 𝝋)
∴ P=W( )
𝐜𝐨𝐬𝝋

 Equilibrium of body on inclined plane subjected to forces parallel to inclined plane

and body tends to move downward: -
Resolving the forces parallel & perpendicular to the inclined plane,

P + 𝜇𝑅 = W sin 𝛼 …….. (i)

R = W cos 𝛼 …...... (ii)

Putting the value of R in the equation (i)

P + 𝜇W cos 𝛼 = W sin 𝛼
Or, P = W sin 𝛼 - 𝜇W cos 𝛼
= W sin 𝛼 - tan 𝜑 W cos 𝛼
sin𝛼 cos𝜑 − cos𝛼 sin𝜑
=W( )
cos𝜑
sin (𝛼 − 𝜑)
=W( )
cos𝜑

𝐬𝐢𝐧(𝜶 − 𝝋)
∴ P=W( )
𝐜𝐨𝐬𝝋

 Ladder friction: - Consider a ladder AB resting on ground and

leaning against a wall as shown in fig. As the upper end of the ladder
tends to slip downwards, the frictional force between ladder and wall
(= 𝜇𝑆) will be upwards where S is the reaction between ladder and
wall. Similarly, as the lower end to slip away from the wall, the
frictional force between ladder and floor (= 𝜇R) will be towards the
wall, where R is the normal reaction between ladder and the floor.

For condition of equilibrium, ∑ 𝑉 = 0, ∑ 𝐻 = 0 & ∑ 𝑀 = 0.

If, W = Weight of ladder

Z = length of ladder

ELECTRO-THEROPY 5
 Friction

𝛼 = inclination with horizontal.

∑𝑉 = 0 ∴ R + 𝜇S = W …… (i)
∑𝐻 = 0 ∴ S = 𝜇R …… (ii)
𝑙
∑ 𝑀𝐴 = 0 ∴ W x 2 cos 𝛼 – S x l sin 𝛼 – 𝜇S x l cos 𝛼 = 0 ……. (iii)

From this three equations the problem can be solved.

ELECTRO-THEROPY 6