chapter 1 friction

8/13/2019 Chapter 1 Friction

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Static 3rdsecondary Chapter OneFr iction-2-

Movement

Friction is "the resistance an object encounters in moving over another" . It is easier to drag

an object over glass than sandpaper. The reason for this is that the sandpaper exerts more

frictional resistance. In many problems, it is assumed that a surface is "smooth", which meansthat it does not exert any frictional force. In real life, however, this wouldn't be the case. A

"rough" surface is one which will offer some frictional resistance

The fr iction force is denoted byF

F

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The opposite figure represents a body

rests on a rough horizontal plane, thisbody is acted by a force .

Notes: as weight of the body increases then the friction increases and the normal reaction

increases.

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Imagine that you are trying to push a book along a table with your finger. If you apply a very

small force, the book will not move. This must mean that the frictional force is equal to the force

with which you are pushing the book. If the frictional force were less that the force produced byyour finger, the book would slide forward.

If you push the book a bit harder, it would still remain stationary. The frictional force must

therefore have increased, or the book would have moved. If you continue to push harder,

eventually a point is reached when the frictional force increases no more. When the frictional

force is at its maximum possible value, here, fr iction is said to be limi ting. If friction is

limiting, yet the book is still stationary, it is said to be in limiting equilibrium. If you push everso slightly harder, the book will start to move. If a body is moving, friction will be taking its

limiting value.

Friction FF

L imiting fr iction


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Properties of fr iction

(1) The force of friction always acts in the direction opposite to the direction the body is tendingto move.

(2) The force of friction increases with the tangential force that tends to move the body so that

the two force are equal provided that the body is in a state of equilibrium.

(3) The magnitude of the force of friction increases up to a certain limit which it does not

exceed. At this value, the motion is just about to begin and the friction is then called the

limiting friction. When motion takes place the magnitude of the force of friction is nearly

equal to its maximum value in other words during motion the friction is a limiting friction.(4) The magnitude of the limiting friction bears a constant ratio to the normal reaction. This

ratio depends on the nature of the two surfaces in contact and is independent of their shapesand masses.

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The coefficient of friction is a number which represents the friction between two surfaces, it is

the ratio between the magnitude of the limiting frictionF

F and the magnitude of the normal

reaction R and it is denoted by FFR

Note: for most applications, 1 except for rare materials

So lets understand what does this mean :

A very simple example: which of the following is easier ? To lift a disk or to push itI think pushing it is much easier than lifting it

So lifting here tends to R and pushing it tends toF

F , so usually FF R

And from that, we can say that 1 for most applications and problems we will discuss

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Study

Coeff icient of fr iction


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FF

R'

R

1F

2F

F

1

i F R where Coefficient of friction and R the normal reaction

Note use this rule when the body is , this is called a limiting friction

ii

Equi libr ium rule:

About to move rule :

about to move

Not about

F F R friction is not limiting

2

you can use instead of and to use lami's rule

3 Study

It is the angle between th

2 2

F

F

to move rule :

Resultant reaction R' = F + R

Imp. R' F R

Angle of fr iction

F

e resultant reaction R'

the resultant between F and R

and the normal to the plane on which

body is about to move

Then:

don't use the resultant reaction R'

FF

Tan = = R

Imp.

except when is given

4

If a body is placed on a rough inclined plane with the horizontal

by angle and the body is about to move by its own weight

Special case on an inclined rough plane :

F

only

Then Tan Tan

If the direction of movement is not mentioned:

least force to move the body means the movement is down F and F are together

largest force to mo

Imp

F

1 2

ve the body means the movement is up F and F are opposite

5

If two horizontal forces F and F are acted on the body rests on a rough horizontal plane,

Special case on a hor izontal rough plane :

2 2 2

F 1 2 1 2

F

F

and the angle between them is , then:

F F F 2F F Cos

Where F R

Or F R

I f the body is not about to move

I f the body is about to move

Rules of fr iction


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F

F

T 1.5 kg.wt F T 1.5 kg.wt 1

1and R 6 kg.wt So R 6 2 2

3

Then from 1 and 2 : F R

the body is in Equilibruim

Then the friction is not limiting and not about to move

o o

F F

o

F

F 2Sin30 2.5 F 2.5 2Sin30 1.5 kg.wt 1

and R 2Cos30 3 kg.wt

R 0.9 3 1.559 2

Then from 1 and 2 : F R the body is in Equilibruim

Then the

friction is not limiting and not about to move

Movement

Example (1)

A wooden block of weight 6 kg.wt. is placed on a horizontal table, and is connected by a string

passing over a smooth pulley at the edge, to a weight of magnitude 1.5 kg.wt. which is hanging

freely. Given that the block is in equilibrium, find the force of friction and the normal reaction.If the coefficient of friction between the block and the table is

1

3, state whether or not the body

is about to move.

Answer

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Example (2)

A wooden block of weight 2 kg.wt rests in equilibrium on a plane, inclined at o30 to the

horizontal, under the action of a force whose magnitude is 2.5 kg.wt and whose direction is thatof the line of greatest slope upwards. If the coefficient of friction is 0.9, find the force of friction.

State whether or not the motion is about to begin.

Answer

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6


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16

R

FFF Cos

F

F

F F

The body is about to move Then the friction is limiting F R

1

F R 14

3a F F Cos F F

5

4 and R 16 F Sin R 16 F

5

3 1 4 3 1 from 1 : F 16 F F 4 F

5 4 5 5 5

2

2

3 1 F F 4 F 5 kg.wt

5 5

4b R' R 1 where R 16 5 12 kg.wt

5

1 1 R' 12 1 3 17 kg.wt and also Tan

4 4

F

F Sin

4

3

5

R'

Example (3)

A body of weight 16 kg.wt is placed on a horizontal rough plane. and the friction coefficient

between them is1

4

. Find:

a The least force acting on the body inclined to the horizontal plane at an angle whose

3 Cos is so that the motion is about to start.

5

b The magnitude and the direction of the resultant reaction.

Answer

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26

o60

R

FF

7

8

1 2

2 2 2

F 1 2 1 2

2 2 o

F



F F F 2F F Cos

F 7 8 2 7 8 Cos60 13 gm.wt

F

And R 26 gm.wt And the body is about to move friction is limting

13 1F R 13 26 Tan Angle of friction

26 2

F

F F

First case: The body is about to slide Then the friction is limiting F R

F R So, F 30 Cos and R 60 30 Sin

30 Cos 60 30 Sin divide by 30

Cos 2 Sin 1

Second case:

F

o

F 60 Cos and R 60 60 Sin

60 Cos 60 60 Sin divide by 60

Cos 1 Sin 2

From 1 and 2 : 2 Sin 1 Sin divide by

2 Sin 1 Sin 2 Sin 1

1Sin 30

2

2 Sin 2From 1 :

Cos

o

o

Sin30 3

Cos30 3

60

R

FF 30 Cos

30

60

R

FF60 Cos

60

30 Sin

60 Sin

Example (4)

A body of mass 60 kg is placed on a horizontal rough plane, a force of 30 kg.wt acted on it in a

direction inclined to the horizontal at an angle so that it is about to slide, and then a force

of a 60

kg.wt acted in the opposite direction of the first force, so that it is about to slide also,

find the coefficient of friction and find the angle .

Answer

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Example (5)

A body of mass 26 gm is placed on a horizontal rough plane, the body is about to move under

the action of the of two forces of magnitudes 7 and 8 gm.wt acting horizontally on the body and

the angle betw oeen them is 60 , find the angle of friction between the body and the plane.

Answer


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6

o120

R

FF

2

4

1 2

2 2 2

F 1 2 1 2

2 2 oF



F F F 2F F Cos

F 4 2 2 4 2 Cos120 2 3 Newto

st

F

1 o

nd

n

And R 6 Newton

1 case : If the body is in equilibrium friction is not limting

Angle of friction Tan

And the body is in equilibrium F R 2 3 6Tan

2 3 2 3Tan Tan 30

6 6

2 case : If the b

o

F

F

2 2 2

F 1 2 1 2

2 2 2 o

2 2

ody is about to move friction is limting

Angle of friction Tan Tan45 1

And the body is about to move F R

where R 6 Newton F 1 6 6 Newton

And F F F 2F F Cos

6 4 F 2 4 F Cos120 36 16

2

2 2

2

2 2

2

o

1

o2 1

F 4F

4 96F 4F 20 0 then by using formula: F

2

F 2 1 6 Newton

F Sin 4Sin120 2To get the direction of this force: Tan

F F Cos 22 1 6 4Cos120

1 o 12

Tan 35 16' with F 2

Example (6)

A body of weight 6 Newton rests on a rough horizontal plane. Two forces 2 and 4 Newton act

horizontally on the body, the angle between them being o120 . If the body is in equilibrium,

prove that the angle of friction is not less than o30 . If o45 and the two forces act in theirprevious direction and the 4 Newton force remains constant, find the least value for the other

force to move the body and determine the direction of motion.

Answer


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R

o30

o30

112

o150 Sin30

FF

150

o150 Cos30

oF

o

F

first case: F 112 150 Sin30 38 gm.wt downward

3And R 150 Cos30 75 3 gm.wt and5

3 R 75 3 45 g.wt F R

5

The body is not about to move

Then the friction is not limiting

Second case:

When the body

o

F F

o

is about to moveThen the friction is limiting

F R 45 gm.wt where F F 150 Sin30

F 45 150 Sin30 120

The force must increase to 120 gm.wt

in order to make the body about to move

Movement

R

o30

o30

F

o150 Sin30

FF

150o150 Cos30

Movement

Example (7)oA body of 150 gm.wt is placed on a rough plane inclines at angle 30 to the horizontal and the

3friction coefficient between them is . A force of 112 gm.wt acts on the body parallel to the5

line of the greatest slope upwards and the body is still in equilibrium, find the magnitude and

the direction of the force of friction in this case, and determine is it the limiting magnitude or

not. Also mention the change that should happen to the magnitude of the force so that the body

is about to move.

Answer

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R

o30

o30

o30

oT Sin30

T oT Cos30

o40 Sin30

FF

40

o40 Cos30

F

F

o o

F

F

The body is about to move Then the friction is limiting F R

1 F R 1

4

The least force means that the body is about to move down

So, F T Cos30 40 Sin30

F

o o

320 T

2

and R 40 Cos30 T Sin30

1 R 20 3 T

2

3 1 1from 1 : 20 T 20 3 T

2 4 2

3 1 3 120 T 5 3 T T T 20 5 3

2 8 2 8

3 1 20 5 3T 20 5 3 T T 15.3 Newton

2 8 3 1

2 8

Movement

Example (8)

A body of weight 40 Newtons rests on a rough plane which is inclined to the horizontal at an

angle o30 . The body is pulled up by a string which makes an angle o30 with the plane. The

string lies in the vertical plane which contains the body and the line of greatest slope. If the

coefficient of friction is1

4. Find the least force in the string which prevents the load from

moving down .

Answer

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Static 3rdsecondary Chapter OneFr iction--

R

F

W Sin

FF

W

130 Cos

Movement

12

13

5

F F

st

F

5R 130 Cos 130 50 Newton

13And the body is about to move the friction is limiting

2F R F 50 20 Newton

5

1 case: If movement is downwards

12F 130 Sin F 130 20 100 Newton

13

"This is the

nd

F

least force needed"

2 case: If movement is upwards

12F F 130 Sin 20 130 140 Newton

13

"This is the greatest force needed"

R

F

W Sin

FF

W

130 Cos

Movement

Example (9)

A 130 Newton weight is placed on a rough plane which is inclined to the horizontal by an angle

whose cosine is5

13

. A force is applied to the weight parallel to the line of greatest slope

upwards. If the coefficient of friction between the weight and the plane is2

5, then find the limits

between which the applied force lies, so as to make the weight about to move.

Answer

limits which the force apply means to get the least and the largest force which will make the

body about to move up and down.


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R

20 Sin

FF

20

Movement

1F

R

FF

20

Movement

20 Cos

2F

20 Sin

20 Cos

F

1 F F

1 F 1

the body is about to move the friction is limiting

F R

3R 20 Cos 20 12 Newton

5

4F 20 Sin F 20 F

5

F 16 F 16 R F 16 12

st1 case: I f movement is downwards

2 2 2

2 F

2 F 2 2

1 2

1

3 4 4R 20 Cos F Sin 20 F 12 F

5 5 5

F Cos 20 Sin F

3 3 4F 16 F 16 R F 16 12 F

5 5 5

3 4When F F : 16 12 16 12 16 12

5 5

48 36 64 4816 12

5 5 5 5

nd2 case: I f movement is downwards

2

2 2

2

1 2

48 88 320 5 48 88 32 0 8

5 5 5

6 11 32 0 3 4 2 1 0

4 1 1refused Or F F 16 12 10 Newton

3 2 2

2F Cos

2F Sin

5

3

4

Example (10)

A body whose weight is 20 Newton is placed on a rough plane inclined to the horizontal by an

angle whose tangent is4

3

. If1

F is the least force when applied along the line of greatest slope

upwards, the body is about to move downwards. 2F is the least force if applied horizontally,

the body is about to move downwards. If 1 2F F then find the coefficient of friction of the

rough plane and the magnitude of any of the two forces.

Answer


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38

FF Cos

F Sin

R

FF

4

3

5

st

nd

1 case: Inclined plane The body is moving on an inclined plane

1 by angle whose tangent is

51 1

Tan Tan5 5

2 case: horiz

under its own weight onl y

F

F

F

ontal plane

the body is by a force F inclined by angle

F R

3 4So, F F Cos F and R 38 F Sin 38 F 5 5

3 1 4So, F R F 38 F Multiply

5 5 5

about to move

F

by 5

4 4 193F 38 F 3F F 38 F 38 F 10 Newton

5 5 5

3Then from 1 : F 10 6 Newton and from 2 : R 30 Newton

5

Example (11)

A body of weight 38 Newton is about to move under its own weight when placed on a rough

plane inclined to the horizontal at an angle whose tangent is1

5

. If this body is placed on a

horizontal plane which is as rough as the inclined plane, and is acted on by a force inclined to

the horizontal at an angle whose sine is4

5so that the body is about to move. Find the force and

the normal reaction.

Answer

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14/20





R

o30

o30

F

o2 Sin30

FF

2

o2 Cos30

Movement

st

o

o

nd

1 case: Inclined plane The body is moving on an inclined plane

by angle 30

3Tan30

3

2 case: When a force acted on the p

under its own weight onl y

o

F

o o

F

o o

F

lane

the body is by a force F

inclined by angle 60 to the horizontal

F R

3where F F Cos30 2 Sin30 F 1

2

1and R 2Cos30 F Sin30 3 F

2

3 3 1So, F R F 1 3 F

2 3 2

about to move

F

3 3 3 3 2 3F 1 1 F F F 2 F 2 F 3 kg.wt

2 6 2 6 3

3 1Then from 1 : F 3 1 kg.wt 2 2

o30

o30

oF Sin30

oF Cos30

Example (12)

A 2 kg.wt is placed on a horizontal rough plane. The plane is tilted gradually, so the weight is

about to slide down when the inclination of the plane is o30 to the horizontal. If the weight is

then attached to a string which is pulled in a direction inclined at o60 to the horizontal so thatthe body is about to move upwards. Given that the string is in the vertical plane through the

line of greatest slope, calculate the tension in the string and the friction force.

Answer


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R

o60

o60

o3Sin60

FF

3

o3Cos60

Movement

st

o

o

nd

1 case: Inclined plane The body is sliding on an inclined plane

by angle 30

3

Tan30 3

2 case: when the angle of the plane increas

under its weight

o

o

F

F

o

F

ed to be 60

3 3Before the force acts: F 3Sin60 kg.wt

2After the force acted:

To find the least force which prevents the body

to move down movement is down

F R 1

3where F 3Sin60 F

o

rd

F

o o

F

o o

3F

2

3and R 3Cos60 kg.wt

2

3 3 3 3from 1 F F 3 kg.wt

2 3 2

3 case: when the force is horizontal

F R 1

3 3 1where F 3Sin60 F Cos 60 F

2 2

3 3and R F Sin60 Cos60 F

2 2

3 3 1from 1

2 2

3 3 3 3 3 3 1

F F F F3 2 2 2 3 2

F 3 kg.wt

oF Sin60

R

o60

o60

o3Sin60

FF

3

o3Cos60

o60

F

oF Cos60

R

o60

o60

o3Sin60

FF

3

o3Cos60

Movement

F

Example (13)

A body of 3 kg.wt is placed on a rough plane. when the plane is is inclined to the horizontal at

an angle o30 , then it is about to slide down. If the inclination of the plane to the horizontal

increased to become o60 , find the force of friction. Then find the magnitude of the least forceacting in the direction of the line of the greatest slope that prevents the body to move

downwards. And if this force is replaced by another horizontal force, prove that its magnitude

is equal to the first force.

Answer


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R

F

W Sin

FF

W

W Cos

Movement

F

The body is moving on an inclined plane by angle

Tan Tan Tan

To find the least force to make the body about to move up:

Then the friction is limiting F

under its own weight only

F

R 1

where F F W Sin and R W Cos

Then from 1 : F W Sin Tan W Cos

SinF W Sin

Cos

W Cos

2 2 2

W Sin

F 2W Sin

To get the resultant in case the body is limiting:

R' R 1 R' W Cos 1 Tan W Cos Sec W Cos Sec

1 W Cos W

Cos

Example (14)

When a weight W is placed on a rough plane inclined at an angleto the horizontal, it is found

that the weight is about to slide down. Prove that the least force along the line of greatest slope

which makes the weight about to move upwards is equal to 2W Sin. Prove also that theresultant reaction is equal to W.

Answer

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17/20





R

3Sin

FF

3

3Cos

Movement

Any body is about to move by its weight when

the angle of inclination the angle of friction

In this problem, the two bodies will move when ,

1then we must put the body of smaller friction

3

b

1elow the body of the greater friction

2

4Cos

4

4Sin

T

T

When the two bodies are about to move

st

F

1 body

1

3

F 3Sin T

R 3Cos

13Sin T 3Cos

3

T 3Sin Cos 1

nd

F

2 body

1

2

F 4Sin T

R 4Cos

14Sin T 4Cos

2

T 2Cos 4Sin 2

Then from 1 and 2 : 3Sin Cos 2Cos 4Sin

37Sin 3Cos Divide by Cos 7Tan 3 Tan

7

Example (15)

Two bodies of weights 3 and 4 kg.wt are placed on a plane inclined to the horizontal at angle

the two bodies are connected with a light string coincide to the line of the greatest slope, the

coefficie

1 1nts of friction between the bodies and the plane respectively are and .

3 2

If the inclination of the plane increased gradually. Determine which body must placed below the

other so that the two bodies start to move together, give reason.

3Then prove that Tan when the two bodies are about to slide together.

7

Answer

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R

F Cos

W SinF

F

W Cos

Movement

W

F

F

We can solve this problem by two methods

The least tension required to move the rock upward

Tan and R W Cos F Sin

F F Cos W SinThe body is about to move F R

F Cos W Sin W Cos

F irst method :

F Sin Tan

SinF Cos W Sin W Cos F Sin

Cos

After multiplying both side by Cos

F Cos Cos W Sin Cos W Cos Sin F Sin Sin

F Cos Cos F Sin Sin W Cos Sin W Sin Cos

F Cos Cos Sin Sin W Cos Sin Sin Cos

1

o

F Cos W Sin

W Sin F Then the minimum tension occurs when Cos is maximum

Cos

The maximum value of Cos 1 Cos 1 0

1000 Sin35When 35 and W 1000 F 574 kg.wt

1

Another solutio

F

o o o

o

when appears, you may use R' instead of F and RThen we can use Lami's rule between the three force R' , F' ,W

F W R'

Sin 180 Sin 90 Sin 90

F W

Sin Sin 90

W SinF W F

Sin Cos

n

CosThen continue................

F

F Sin

R

F Cos

W Sin

FF

W Cos

Movement

W

R'

F

F Sin

90

Example (16) (* Excell ent* )

A rock of weight Wis placed on a rough road inclined to the horizontal at angle , if a horse

pulled the rock upward by a string which makes an angle with the road, so that the rock is

about to move, given that the angle of friction between the rock and the road is . Thenprove that the least tension of the string to make the rock about to move upward occurs when

, then find the magnitude of this force when o35 , and the mass of the rock is1000 kg.

Answer


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R

F

W Sin

FF

W Cos

Movement

W

F

F

R W Cos Tan F W Sin F

The body is about to move

F R W Sin F W Cos Tan

SinW Sin F W Cos

Cos

SinF W Sin W CosCos

Multiply both sides by Cos :

F Cos W Sin Cos W Cos Sin F Co

F

o

s W Sin

W Sin F F W Sin Sec

Cos

when appears on an inclined plane, you may use R' instead of F and R

Then we can use Lami's rule under the three force R' , F' ,W

F

Sin 180

Another solution

o o

o

W R'

Sin 90 Sin 90

F W F W

Cos Sin CosSin 180

W Sin F F W Sin Sec

Cos

R

F

W Sin

FF

W Cos

Movement

W

R'

Example (17)

A body whose weight is W is placed on a rough plane inclined at angle to the horizontal

and the angle of friction is . A force Facts on the body parallel to the plane to prevent the

body from slipping. Prove that F W Sin Sec Answer

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20/20




R

W

500 Sin

500

500 Cos

4

5

3

W 175 25

WFF

Movement

st

F

F

1 case: When the least weight attached is T W 175 25 200 gm.wt

The least weight means that the movement is down

The body is in equilibrium, F R

3Where R 500 Cos 500 300 gm.wt

5

And F 500 Sin W 5

F

nd

F

400 200 200 gm.wt

5

F 200 2

R 300 3

2 case: To get the maximum weight

The maximum weight means that the movement is up

2The body is in equilibrium, F R where

3

3And R 500 Cos 500 300 g

5

F

F

m.wt

And F T 25 500 Sin4

T 25 500 F T 25 4005

2T 25 400 300 T 25 600

3

T 575 gm.wt

R

W

500 Sin

500

500 Cos

W T 25

W

FF

Movement

Example (18)

A body of 500 gm.wt is placed on a rough plane which is inclined to the horizontal by an angle

of measure , such that4

Tan

3

, the body is then attached to a string passing over a smooth

pulley at the top of the plane and a scale pan of 25 gm mass is attached to the other end of the

string, if the least weight to be added to the plane to keep the body in equilibrium is 175 gm.wt.Find the coefficient of friction, then prove that the maximum weight that can be added to the

pan without disturbing equilibrium is 575 gm.wt.

Answer

chapter 1 friction

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