Friction · Minimum pulling Force P to move the body up on an inclined plane By Resolving P in the...

Friction

FRICTION AND FRICTIONAL FORCE

If we slide or try to slide a body over a surface the

motion is resisted by a bonding between the body

and the surface. This resistance is represented by a

single force and is called friction.

The force of friction is parallel to the surface and

opposite to the direction of intended motion.

Force of friction is partially independent of

microscopic area of surface in contact and relative

velocity between them. (if it is not high)

www.gangwarinstitue.com Contact : 8400-582-582, 8604-582-582

However, it depends on the nature of material of

the surface in contact (force of adhesion) and

their roughness and smoothness.

Normally with increase in smoothness friction

decreases. But if the surface area are made too

smooth by polishing and cleaning the bonding

force of adhesion will increase and so the

friction will increase resulting in 'Cold welding'

Friction is a non conservative force, i.e. work

done against friction is path dependent.


In its presence mechanical energy is not conserved. Thus

friction reduces efficiency of a machine.

It is a general misconception that friction always opposes

the motion. No doubt friction opposes the motion of a

moving body but in many cases it is also the cause of

motion.

In moving a person or Vehicle pushes the ground

backwards (action) and the rough surface of ground

reacts and exerts a forward force due to friction which

causes the motion if there had been no friction there will

be slipping and no motion


There are three types of frictional force depending

upon the nature of relative motion between the

bodies.

Types of Friction

1. Static Friction

The opposing force that comes into play when one

body tends to move over the surface of another, but

the actual motion has yet not started is called static

friction.


P

R

F

mg

Fig. 5.1

If applied force is P and the body remains at rest then static

friction F = P.

If a body is at rest and no pulling force is acting on it, force

of friction on it is zero.

Static friction is a self-adjusting force because it changes

itself in accordance with the applied force and is always

equal to net external force.

2. Limiting Friction

If the applied force is increased, the force of static friction also

increases. If the applied force exceeds a certain (maximum)

value, the body starts moving. This maximum value of static

friction upto which body does not move is called limiting

friction.www.gangwarinstitue.com Contact : 8400-582-582, 8604-582-582

The magnitude of limiting friction between any

two bodies in contact is directly proportional to the

normal reaction between them.

Direction of the force of limiting friction is always

opposite to the direction in which one body is at the

verge of moving over the other.

Coefficient of static friction : (a) ms is called coefficient of

static friction and is defined as the ratio of force of

limiting friction and normal reaction


Dimension :

Unit : It has no unit.

Value of m depends on material and nature of

surfaces in contact that means whether dry or

wet ; rough or smooth polished or non-

polished.

Value of m does not depend upon apparent area

of contact.


0 0 0[M L T ]

3. Kinetic or Dynamic Friction

If the applied force is increased further and sets the

body in motion, the friction opposing the motion is

called kinetic friction.

Kinetic friction depends upon the normal reaction.

where is called the coefficient of

kinetic friction.

Value of depends upon the nature of surface in

contact.

Kinetic friction is always lesser than limiting friction


k k kF RorF R m km

km

k l k sF F m m

i.e. coefficient of kinetic friction is always less than

coefficient of static friction. Thus we require more

force to start a motion than to maintain it against

friction. This is because once the motion starts

actually ; inertia of rest has been overcome. Also

when motion has actually started, irregularities of

one surface have little time to get locked again into

the irregularities of the other surface.

Kinetic friction does not depend upon the velocity

of the body.


Types of kinetic friction

Sliding friction : The opposing force that comes

into play when one body is actually sliding over

the surface of the other body is called sliding

friction. e.g. A flat block is moving over a

horizontal table.

Rolling friction : When objects such as a wheel

(disc or ring), sphere or a cylinder rolls over a

surface, the force of friction that comes into play

is called rolling friction.www.gangwarinstitue.com Contact : 8400-582-582, 8604-582-582

• Rolling friction is directly proportional to the normal

reaction (R) and inversely proportional to the radius (r)

of the rolling cylinder or wheel.

mr is called coefficient of rolling friction. It would

have the dimensions of length and would be measured

in metre.

• Rolling friction is often quite small as compared to the

sliding friction. That is why heavy loads are transported

by placing them on carts with wheels.


rolling r

RF

r m

• In rolling the surfaces at contact do not rub

each other.

• The velocity of point of contact with respect to the

surface remains zero all the times although the

centre of the wheel moves forward.

Graph Between Applied Force and Force of Friction

• Part OA of the curve represents static friction (Fs).

Its value increases linearly with the applied force

• At point A the static friction is maximum. This

represent limiting friction (F1).



A C B

Fl Fk

Forc

e o

f fr

icti

on

Applied force O

Fs

Fig. 5.2

5. Friction is a Cause of Motion

It is a general misconception that friction always

opposes the motion. No doubt friction opposes the

motion of a moving body but in many cases it is also

the cause of motion. For example :

• While moving, a person or vehicle pushes the

ground backwards (action) and the rough surface of

ground reacts and exerts a forward force due to

friction which causes the motion. If there had been no

friction there will be slipping and no motion.


During cycling, the rear wheel moves by the force communicated

to it by pedalling while front wheel moves by itself. So, when

pedalling a bicycle, the force exerted by rear wheel on ground

makes force of friction act on it in the forward direction (like

walking). Front wheel moving by itself experience force of

friction in backward direction (like rolling of a ball). [However, if

pedalling is stopped both wheels move by themselves and so

experience force of friction in backward direction].


Action

Friction

Fig. 5.3

If a body is placed in a vehicle which is accelerating,

the force of friction is the cause of motion of the

body along with the vehicle (i.e., the body will

remain at rest in the accelerating vehicle until

If there had been no friction between

body and vehicle, the body will not move along with

the vehicle.


While pedalling Pedalling is stoped

Fig. 5.4

sma mg).m

From these examples it is clear that without friction motion

cannot be started, stopped or transferred from one body to

the other.

6. Advantages & Disadvantages of Friction

Advantages of friction

Walking is possible due to friction.

Two body sticks together due to friction.

Brake works on the basis of friction.www.gangwarinstitue.com Contact : 8400-582-582, 8604-582-582

ma msmg

a

Fig. 5.5

Writing is not possible without friction.

The transfer of motion from one part of a machine to other

part through belts is possible by friction.

Disadvantages of friction

Friction always opposes the relative motion between any

two bodies in contact. Therefore extra energy has to be

spent in over coming friction. This reduces the efficiency of

machine.

Friction causes wear and tear of the parts of machinery in

contact. Thus their lifetime reduces.

Frictional force result in the production of heat, which

causes damage to the machinery.www.gangwarinstitue.com Contact : 8400-582-582, 8604-582-582

7. Methods of Changing Friction

We can reduce friction

By polishing.

By lubrication.

By proper selection of material.

By streamlining the shape of the body.

By using ball bearing.

Also we can increase friction by throwing some sand on

slippery ground. In the manufacturing of tyres, synthetic

rubber is preferred because its coefficient of friction with

the road is larger.


Angle of Friction

Angle of friction may be defined as the angle which the

resultant of limiting friction and normal reaction makes with

the normal reaction.

By definition angle is called the angle of friction


P

R

F

mg

S

Fig. 5.8

Hence coefficient of static friction is equal to tangent

of the angle of friction.

Resultant Force exerted by surface on Block


Hence the range of S can be given by,

Angle of Repose


2mg S mg 1 m

Thus the coefficient of limiting friction is equal to the

tangent of angle of repose.

As well as i.e. angle of repose = angle of friction.


R

mg cos

F

mg sin

mg

Fig. 5.9

CALCULATION OF REQUIRED FORCE IN

DIFFERENT SITUATION

Minimum pulling Force P at an angle from the

horizontal


By resolving P in horizontal and vertical direction (as

shown in figure)


P

Fig. 5.10

R

P sin

P cos F

W

Fig. 5.11

Minimum pushing Force P at an angle from the

horizontal

By Resolving P in horizontal and vertical direction (as

shown in the figure)


P

Fig. 5.12


R

P sin

P cos F

W

Fig. 5.13

Minimum pulling Force P to move the body up on an

inclined plane

By Resolving P in the direction of the plane and

perpendicular to the plane (as shown in the figure)


WsinP

cos( )

P

Fig. 5.14


R + P sin

W cos

F + W sin

P cos

Fig. 5.15

W

Minimum Force to move a body in downward direction

along the surface of inclined plane




Wsin( )P

cos( )

P

Fig. 5.16


F

R + P sin

P cos

+

W sin W cos

W

Fig. 5.17

Minimum Force to avoid sliding of a body down on an

inclined plane




Wsin( )P

cos( )

P

Fig. 5.18


F + P cos R + P sin

W sin W cos W

Fig. 5.19

Minimum Force to move a body in downward direction

along the surface of inclined plane

Let the force P be applied at an angle with the

horizontal.


perpendicular to the plane (as shown in the figure)www.gangwarinstitue.com Contact : 8400-582-582, 8604-582-582

P

Fig. 5.20


R + P sin

P cos F

mg

Fig. 5.21


m

1

21 m

Fig. 5.22

i.e. For minimum value of P its angle from the

horizontal should be equal to angle of friction


ACCELERATION OF A BLOCK AGAINST

FRICTION

1. Acceleration of a Block on Horizontal Surface

When body is moving under application of force

P, then kinetic friction opposes its motion. Let a

is the net acceleration of the body


R

P Fk

mg

ma

Fig. 5.23

2. Acceleration of a Block Sliding down over a

rough inclined planeWhen angle of inclined plane is more than angle of

repose, the body placed on the inclined plane slides

down with an acceleration a.


R

P Fk

mg

ma

Fig. 5.23

3. Retardation of a block sliding up over a rough

inclined plane

When angle of inclined plane is less than angle of

repose, then for the upward motionwww.gangwarinstitue.com Contact : 8400-582-582, 8604-582-582

F

mg sin mg cos mg

R

ma

Fig. 5.24

The

WORK DONE AGAINST FRICTION

1. Work done over a rough inclined surface

If a body of mass m is moved up slowly on a

rough inclined plane through distances, then


mg sin + F mg cos

mg

R ma

Fig. 5.25

The

2. Work done over a horizontal surface

In the above expression if we put = 0 then

Work done = force × distance = F × s = µ mg s

It is clear that work done depends upon


mg sin + F mg cos

mg

R ma

s

Fig. 5.26

P F

mg

R

s

Fig. 5.27

Weight of the body.

Material and nature of surface in contact.

Distance moved.

MOTION OF TWO BODIES ONE RESTING ON

THE OTHER

When a body A of mass m is resting on a body B

of mass M then two conditions are possible.

A force F is applied to the upper body,

A force F is applied to the lower body


We will discuss above two cases one by one in the following

manner :

1. A Force F is applied to the upper body, then following four

situations are possible

When is no friction


F m A

M B L

Fig. 5.28

2. If Friction is present between A and B only and

applied force is less than limiting friction (F<Fl)



applied force is greater than limiting friction

(F>Fl)

In this condition the two bodies will move in the

same direction (i.e. of applied force) but with

different acceleration. Here force of kinetic

friction will oppose the motion of A while

cause the motion of B.


A B

Fa a

M m

kmgm


Ak amFF

i.e. m

FFa k

A

m

mgFa k

A

)( m

Free body diagram of A

Bk aMF

i.e. M

Fa k

B

M

mga k

B

m

Free body diagram of B

F A

Fk

maA

FK

MaB

B

4. If there is friction between B and Floor


FK

MaB

B

Fl

A Force F is applied to the lower body, then following

four situations are possible

1. When there is no Friction

B will move with acceleration (F/M) while A will

remain at rest (relative to ground) as there is no pulling

force on A.www.gangwarinstitue.com Contact : 8400-582-582, 8604-582-582


F

m A

M B L

Fig. 5.30


(F>Fl)



mgma kA m

i.e. ga kA m

Free body diagram of A

Bk MaFF

i.e. M

mgFa k

B

][ m

Free body diagram of B

A

Fk

maA

FK

MaB

B F

Note : As both the bodies are moving in the same

direction.

Acceleration of body A relative to B will be


MOTION OF AN INSECT IN THE ROUGH

BOWL

The insect crawl up the bowl, up to a certain height h

only till the component of its weight along the bowl is

balanced by limiting frictional force.

Let m = mass of the insect, r = radius

of the bowl, µ = coefficient of friction

for limiting condition at point A


Fl

mg sin

mg

mg cos

A

R

r O

y

h

Fig. 5.31


m2

m1

T

R

m1g

Fl

T

Fig. 5.32 m2g

MINIMUM MASS HUNG FROM THE STRING TO

JUST START THE MOTION

When a mass m1 placed on a rough horizontal

plane. Another mass m2 hung from the string

connected by frictionless pulley, the tension (T)

produced in string will try to start the motion of

mass m1.


Note : In the above condition Coefficient of friction

When a mass m1 placed on a rough inclined plane

Another mass m2 hung from the string connected by

frictionless pulley, the tension (T) produced in string will

try to start the motion of mass m1.


The


m2

R T

m1g sin + F m1g cos

T

m2g

m1

m1g

Fig. 5.33

2

1

tancos

m

mm

MAXIMUM LENGTH OF HUNG CHAIN

A uniform chain of length l is placed on the table

in such a manner that its l' part is hanging over

the edge of table without sliding. Since the chain

have uniform linear density therefore the ratio of

mass and ratio of length for any part of the chain

will be equal.


COEFFICIENT OF FRICTION BETWEEN A

BODY AND WEDGE

A body slides on a smooth wedge of angle and its

time of descent is t.


l

( l – l )

Fig. 5.34

S

Smooth wedge

S

Rough wedge

Fig. 5.35 Fig. 5.36

If the same wedge made rough then time taken by it to

come down becomes n times more (i.e. nt). The length of

path in both the cases are same.


STOPPING OF BLOCK DUE TO FRICTION

1. On horizontal road

Distance travelled before coming to rest:

1.1 Distance travelled before coming to rest

A block of mass m is moving initially with

velocity u on a rough surface and due to friction,

it comes to rest after covering a distance S.


v = 0 S

u

Fig. 5.37

1.2 Time taken to come to rest

2. On inclined road

When block starts with velocity u its kinetic energy will

be converted into potential energy and some part of it

goes against friction and after travelling distance S it

comes to rest i.e. v = 0.www.gangwarinstitue.com Contact : 8400-582-582, 8604-582-582


u

v = 0

S

Fig. 5.38

STOPPING OF TWO BLOCKS DUE TO FRICTION

When two masses compressed towards each other and

suddenly released then energy acquired by each block

will be dissipated against friction and finally block

comes to rest

i.e., F × S = E

[Where, F = Friction, S = Distance covered by block, E

= Initial kinetic energy of the block]


m1

A

m1

B

m2 m2

S1 S2

Fig. 5.39

VELOCITY AT THE BOTTOM OF ROUGH

WEDGE

A body of mass m which is placed at the top of the

wedge (of height h) starts moving downward on a

rough inclined plane.

Loss of energy due to friction = FL (Work against

friction)

PE at point A = mgh


u = 0

L

m

m

B

A

v

h

Fig. 5.40

By the law of conservation of energy

STICKING OF A BLOCK WITH ACCELOERATED

CART

When a cart moves with some acceleration toward right

then a pseudo force (ma) acts on block toward left.

This force (ma) is action force by a block on cart.www.gangwarinstitue.com Contact : 8400-582-582, 8604-582-582


F

mg

m R ma

M

a

F

CART

m

This is the minimum acceleration of the cart so that block does

not fall. and the minimum force to hold the block together

STICKING OF A PERSON WITH THE WALL OF ROTOR

A person with a mass m stands in contact against the wall of a

cylindrical drum (rotor). The coefficient of friction between the

wall and the clothing is µ. If Rotor starts rotating about its axis,

then person thrown away from the centre due to centrifugal

force at a particular speed , the person stuck to the wall even

the floor is removed, because friction force balances its weight in

this condition.



FC R

F

mg

Fig. 5.42

A block is lying static on the floor. The maximum

value of static frictional force on the block is 10N.

If a horizontal force of 8N is applied to the block,

what will be the frictional force on the block;

(A) 2N (B) 18N

(C) 8N (D) 10N.


01

A box of mass 2 kg is placed on the roof of a car. The

box would remain stationary until the car attains a

maximum acceleration. Coefficient of static friction

between the box and the roof of the car is 0.2 and g =

10 ms–2. This maximum acceleration of the car, for the

box to remain stationary, is;

(A) 8 ms–2 (B) 6 ms–2

(C) 4 ms–2 (D) 2 ms–2


02

The coefficient of static friction, between block A of mass 2

kg and the table as shown in the figure is 0.2. What would

be the maximum mass value of block B so that the two

blocks do not move? The string and the pulley are assumed

to be smooth and massless. (g = 10m/s2);

(A) 2.0 kg

(B) 4.0 kg

(C) 0.2 kg

(D) 0.4 kg


03

A force of 19.6 N when applied parallel to the

surface just moves a body of mass 10 kg kept on a

horizontal surface. If a 5 kg mass is kept on the first

mass, the force applied parallel to the surface to just

move the combined body is;

(A) 29.4 N (B) 39.2 N

(C) 18.6 N (D) 42.6 N


04

A body of weight 50N placed on a horizontal surface is

just moved by a force of 28.2N. The frictional force

and the normal reaction are;

(A) 10N, 15N

(B) 20N, 30N

(C) 2N, 3N

(D) 5N, 6N


05

A rough vertical board has an acceleration ‘a’ so

that a 2 kg block pressing against it does not fall.

The coefficient of friction between the block and the

board should be;

(A) > g/a

(B) < g/a

(C) = g/a

(D) > a/g.


06

a

2 kg

A stone weighing 1 kg and sliding on ice with a

velocity of 2 m/s is stopped by friction in 10 sec.

the force of friction (assuming it to be constant)

will be;

(A) –20N (B) –0.2N

(C) 0.2 N (D) 20N.


07

A block of mass 2 kg is kept on the floor. The coefficient

of static friction is 0.4. If a force F of 2.5 Newtons is

applied on the block as shown in the figure, the frictional

force between the block and the floor will be

(A) 2.5 N

(B) 5 N

(C) 7.84 N

(D) 10 N.


08

F

A block of mass m lying on a rough horizontal plane is

acted upon by a horizontal force P and another force Q

inclined at an angle to the vertical. The block will remain

in equilibrium, if the coefficient of friction between it and

the surface is


09

Pm

Q

What is the maximum value of the force F such

that the block shown in the arrangement, does not

move;

(A) 20 N

(B) 10 N

(C) 12 N

(D) 15 N.


10

F

m = 3kg60

o

m= 1

2 3

A block of mass 0.1 kg is held against a wall by

applying a horizontal force of 5 N on the block.

If the coefficient of friction force acting on the

block is;

(A) 2.5 N (B) 0.98 N

(C) 4.9 N (D) 0.49 N.


11

A per or mass ‘m’ is lying on a piece of paper of mass M

placed on a rough table. If the coefficient of friction

between the pen and paper, and the paper and table are

m1 and m2, respectively, then the minimum horizontal

force with which the paper has to be pulled for the pen

to start slipping is given by;

(A) (B)

(C) (D)


12

1 2(m M)( )g m m 1 2(m M )gm m

1 2{m (m M) }gm m 1 2m( )gm m

Two masses A and B of 10 kg and 5 kg respectively are

connected with a string passing over a frictionless pulley

fixed at the corner of a table as shown. The coefficient of

static friction of A with table is 0.2. The minimum mass of

C that may be placed on A to prevent it from moving is;

(A) 15 kg

(B) 10 kg

(C) 5 kg

(D) 12 kg.


13

A 20 kg block is initially at rest on a rough horizontal

surface. A horizontal force of 75 N is required to set

the block in motion. After it is in motion, a horizontal

force of 60 N is required to keep the block moving

with constant speed. The coefficient of static friction is

(A) 0.38 (B) 0.44

(C) 0.52 (D) 0.60


14

A system consists of three masses m1, m2 and m3 connected by

a string passing over a pulley P. The mass m1 hangs freely

and m2 and m3 are on a rough horizontal table (the coefficient

of friction = m). The pulley is frictionless and of negligible

mass. The downward acceleration of mass m1 is (Assume m1

= m2 = m3 = m);


15

A mass m rests on another mass M. The co-efficient of

static friction between the surfaces of m and M is m. M

rests on smooth frictionless horizontal plane. The

maximum force applied horizontally on M for which m

will move along with M without slipping is;


16

A conveyor belt is moving at a constant speed of 2m/s. A

box is gently dropped on it. The coefficient of friction

between them is m = 0.5 The distance that the box will

move relative to belt before coming to rest on it, taking g

= 10ms–2, is;

(A) Zero (B) 0.4 m

(C) 1.2 m (D) 0.6 m


17

A gramophone record is revolving with an angular

velocity . A coin is placed at a distance r from the

centre of the record. The static coefficient of friction is

. The coin will revolve with the record if;


18

A block of mass m is in contact with the cart C as shown

in the figure The coefficient of static friction between the

block and the cart is m. The acceleration of the cart

that will prevent the block from falling satisfied;


19

a

mC

In the figure given, the system is in equilibrium. What

is the maximum value that W can have if the friction

force on the 40 N block cannot exceed 12.0 N;

(A) 3.45 N

(B) 6.92 N

(C) 10.35 N

(D) 12.32 N


20

30o

W

40 N

The backside of a truck is open and a box of 40 kg is

placed 5m away from the rear end. The coefficient of

friction of the box with the surface of the truck is 0.15.

The truck starts from rest with 2m/s2 acceleration.

Calculate the distance covered by the truck when the

box falls off;

(A) 20 m (B) 30 m

(C) 40 m (D) 50 m.


21

A body of weight 64 N is pushed with just enough force

to start it moving across a horizontal floor and the same

force continues to act afterwards. If the coefficients of

static and dynamic friction are 0.6 and 0.4 respectively,

the acceleration of the body will be (Acceleration due to

gravity = g)

(A) g/6.4 (B) 0.64 g

(C) g/32 (D) 0.2 g


22

An ice cube is kept on an inclined plane of

angle 30°. Coefficient of kinetic friction

between block and incline plane is . What

is acceleration on block

(A) Zero (B) 2 m/s2

(C) 1.5 m/s2 (D) 5 m/s2


23

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