Friction
The force which opposes the movement or the tendency of movement is called
Frictional force or simply friction. It is due to the resistance to motion
offered by minutely projecting particles at the contact surfaces. However, there
is a limit beyond which the magnitude of this force cannot increase.
If the applied force is more than this limit, there will be movement of one body
over the other. This limiting value of frictional force when the motion is
impending, it is known as Limiting Friction.
When the applied force is less than the limiting friction, the body remains at
rest and such frictional force is called Static Friction, which will be having
any value between zero and the limiting friction.
If the value of applied force exceeds the limiting friction, the body starts
moving over the other body and the frictional resistance experienced by the
body while moving is known as Dynamic Friction. Dynamic friction is less
than limiting friction.
Dynamic friction is classified into following two types:
a) Sliding friction
b) Rolling friction
Sliding friction is the friction experienced by a body when it slides over the
other body.
Rolling friction 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 two surfaces and this ratio is
called Coefficient of Friction.
F
Coefficient of friction =
N
where F is limiting friction and N is normal reaction between the contact surfaces.
Coefficient of friction is denoted by µ.
F
Thus,
N
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Laws of friction
1. The force of friction always acts in a direction opposite to that in which 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 of contact and this ratio is called coefficient
of friction.
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.
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 the block shown in figure resting on a horizontal surface and subjected to
horizontal pull P. Let F be the frictional force developed and N the normal reaction.
Thus, at contact surface the reactions are F and N. They can be graphically combined
to get the reaction R which acts at angle θ to normal reaction. This angle θ called the
angle of friction is given by
F
tan
N
As P increases, F increases and hence θ also increases. θ can reach the maximum value
α when F reaches limiting value. At this stage,
F
tan
N
This value of α is called Angle of Limiting Friction. Hence, the angle of limiting
friction may be defined as the angle between the resultant reaction and the normal to
the plane on which the motion of the body is impending.
Angle of repose
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Consider the block of weight W resting on an inclined plane which makes an angle θ
with the horizontal. When θ is small, the block will rest on the plane. If θ is gradually
increased, a stage is reached at which the block start sliding down the plane. The angle
θ for which the motion is impending, is called the angle of repose. Thus, the maximum
inclination of the plane on which a body, free from external forces, can repose is called
Angle of Repose.
Resolving vertically,
N = W. cos θ
Resolving horizontally,
F = W. sin θ
F
Thus, tan
N
If ɸ is the value of θ when the motion is impending, the frictional force will be limiting
friction and hence,
F
tan
N
tan
Thus, the value of angle of repose is same as the value of limiting angle of repose.
Cone of friction
When a body is having impending motion in the direction of force P, the
frictional force will be limiting friction and the resultant reaction R will make
limiting angle α with the normal.
If the body is having impending motion in some other direction, the resultant
reaction makes limiting frictional angle α with the normal to 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 α.
27
Problem 1: Block A weighing 1000N rests over block B which weighs 2000N as
shown in figure. Block A is tied to wall with a horizontal string. If the coefficient of
friction between blocks A and B is 0.25 and between B and floor is 1/3, what should be
the value of P to move the block (B), if
(a) P is horizontal.
(b) P acts at 30˚ upwards to horizontal.
Solution: (a)
Considering block A,
V 0
N1 1000 N
Since F1 is limiting friction,
F1
0.25
N1
F1 0.25 N1 0.25 1000 250 N
H 0
F1 T 0
T F1 250 N
Considering equilibrium of block B,
V 0
N 2 2000 N1 0
N 2 2000 N1 2000 1000 3000 N
F2 1
N2 3
F2 0.3N 2 0.3 1000 1000 N
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H 0
P F1 F2 250 1000 1250 N
(b) When P is inclined:
V 0
N 2 2000 N1 P.sin 30 0
N 2 0.5 P 2000 1000
N 2 3000 0.5 P
From law of friction,
1 1 0.5
F2 N 2 3000 0.5 P 1000 P
3 3 3
H 0
P cos 30 F1 F2
0.5
P cos 30 250 1000 P
3
0.5
P cos 30 P 1250
3
P 1210.43N
Problem 2: A block weighing 500N just starts moving down a rough inclined plane
when supported by a force of 200N acting parallel to the plane in upward direction.
The same block is on the verge of moving up the plane when pulled by a force of 300N
acting parallel to the plane. Find the inclination of the plane and coefficient of friction
between the inclined plane and the block.
V 0
N 500.cos
F1 N .500 cos
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H 0
200 F1 500.sin (1)
200 .500 cos 500.sin
V 0
N 500.cos
F2 N .500.cos
H 0
500sin F2 300 (2)
500sin .500 cos 300
Adding Eqs. (1) and (2), we get
500 = 1000. sinθ
sin θ = 0.5
θ = 30˚
Substituting the value of θ in Eq. 2,
500sin 30 .500 cos 30 300
50
0.11547
500 cos 30
30
