forces, resultant between two forces, resolution

7/29/2019 forces, resultant between two forces, resolution

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Static 3rdsecondary Chapter One Forces--

Vectors

2

3

4

1

We can write : BA - AB And AC - CA ........

StaticIt deals with the bodies when they are at rest

---------------------------------------------------------------------------------------------------------------------

Introduction

Name Symbol Direction Magnitude

Line segment AB No Yes

Ray AB Yes No

Straight line AB No No

Vector AB Yes Yes

So , vector is a physical quantity which has both magnitude and direction .

Shapes of vectors

1stshape: we can write vector as ( a , b ) 2

ndshape: we can write vector as a i + b j

So for example : F= ( 2 , 3 ) means F=2 i + 3 j and vector ( -4 , -1 ) means -4 i j

This is cal led the position vector for any point

---------------------------------------------------------------------------------------------------------------------

Note :


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Static 3rdsecondary Chapter One Forces-2-

A B

CAB BC AC

is the same as :

AC BA BC ...... Why ?

AC AB BC So AB BC AC

A

M

OP

B

N

CQ

DExpress OC ,AB,QC and QN in terms of anda b

Addition of vectors

we can add vectors using the tr iangle law of additi on

---------------------------------------------------------------------------------------------------------------------

Example

OACB is a parallelogram , the points P , Q , M and N

are the midpoints of the sides

OA and OB a b

Answer

OC OA AC OA OB a b

AB AO OB -OA OB - a b

1 1QC OP OB

2 2 b

1 1QN QC CN

2 2 b a

---------------------------------------------------------------------------------------------------------------------

length of vectors Norm

2 2

2 2

if A x , y Then A x y is the length of vector A .

And it is denoted by norm A

Example : If A 3 ,-4 Then the norm length of A = 3 4 5

---------------------------------------------------------------------------------------------------------------------


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1F

2FR

2 2 2

1 2 1 2

2 2

1 2 1 2

2

1 2

R F F 2F F cos

R F F 2F F Cos

Where : R : The resultant force

: The angle between the two forces

Where 0 180

The direction of the resultant :

F SinTan = , Where is the

F F Cos

angle between R and the first force

o

1 1 2 2 2 1

o2 1 2 1 2

1 2 2 2

1 21 2

2

When = 90 : If R F F F Cos 0 Or If R F F F Cos 0

Proof

F Sin F F Cos F F CosTan = Cot = Cot 90 =F F Cos F Sin F Sin

F F Cos0 = F F Cos 0

F Sin

Chapter oneforces

Definition: A force is an action of a body on another body to change or try to change the

state of the 2nd

body :

Each force is denoted by an arrow ( )

The uni t of force:

Forces have more than one unit : we can say NewtonOrKg . wtOrgm . wtOrDyne

Where 5Dyne 10 newton B

---------------------------------------------------------------------------------------------------------------------

Resultant force R

The resultant force of two or more forces is a single force has the same effects as the two or

more forces .

How to find the resultant force between Twoforces

Rule:

Very important remark


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1F

2

F

2

2

2F

R1F

R 2 F Cos2

1F 2F F

1F 2F

2F

1FR

oThen 90

2 2 2 2 2

1 2 1 2

21

1

R F F R F F

FTan , where is the measure angle between R and F

F

1F 2F

1F 2F

R

1F 2F

1F 2Fo180

o180

Special Cases

(1) If and are equal in magnitude where :

Where: : The angle between the two forces2

---------------------------------------------------------------------------------------------------------------------

(2) If :

---------------------------------------------------------------------------------------------------------------------(3)If the two forces and have the same line action and the same direction , o0

Then:

1 2R F F Maximum magnitude of R and R has the same line action and in the same

direction.

---------------------------------------------------------------------------------------------------------------------

(4)If the two forces and have the same line action But in opposite directions

Then:

1 2R F F Minimum magnitude of R and R has the same line action and in the

direction of the greatest force.

---------------------------------------------------------------------------------------------------------------------


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o30

F

4 3F

o

30

o

for Two equal forces :

R 2 F Cos 4 3 2 F Cos 30

23

4 3 2 F F 4 kg .wt 2

15N

36 N R

Example (1)

Two forces their magnitude are 16 and 14 Newton acting at a particle and the measure angle

between them is o60 . find their resultant in magnitude and direction .

Answer

1F=16 N

2F=14 N o60

2 22 2 2 o

1 2 1 2

o

2

o

1 2

1 o

Resultant :

R F F 2F F Cos R 16 14 2 16 14 Cos60 26N

Direction :

F Sin 14 Sin60Tan Tan 0.5271

F F Cos 16 14Cos60Tan 0.5271 27 48'

---------------------------------------------------------------------------------------------------------------------

Example (2)

Two orthogonal forces acting at a particle , their magnitude are 36 and 15 gm . wt , find the

magnitude and direction of their resultant

Answer

1F=36 gm.wt 2F=15 gm.wt

2 22 2 2

1 2

2

1

1 o

for Two Orthogonal forces :

R F F R 36 15 39 gm.wt

Direction:

F 15Tan Tan 0.4167

F 36

Tan 0.4167 29 37'

---------------------------------------------------------------------------------------------------------------------

Example (3)

The magnitude of the resultant of two equal forces is 4 3 Kg . wt , if the measure of the

angle between the resultant and one of the forces is 30 , Then find the two forces .

Answer

1F=

2F=F o30

2

R = 4 3


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Example (4)

o

1 2

o

If the angle between the two forces F and F is 90 , Then their resultant is 2 37 Dyne.

And if the angle is 60 , Then their resultant is 2 43 Dyne , find the magnitude of eachforce , find also the largest and the smallest resultant of these two forces .

Answer

o

22 2 2 2 2 2 2

1 2 1 2 1 2

(1) for Two Perpendicular forces " 90 " :

R F F 2 37 F F F F 148 (1)

o

22 2 2 2 21 2 1 2 1 2 1 2

2 2 2 2

1 2 1 2 1 2 1 2

1 2 1 2

1

2

( 2 ) When " 60 " :

R F F 2F F Cos 2 43 F F 2F F Cos60

1172 F F 2F F F F F F 172 ( 2 )

2

Substitute (1) in (2) : 148 F F 172 F F 172 148 24

24F 3

F

Substitute (3) in (1)

2

2 2 2

2 2 22

2 2

4 2 4 2

2 2 2 2

2 2

2 2

2 2

2 2

2 2

1 2

1 2

:

24 576 F 148 F 148 " F "

F F

576 F 148F F 148F 576 0

F 144 F 4 0

F 144 Or F 4

F 12 Dyne F 2 Dyne

F 2 Dyne F 12 Dyne

Then the maximum resultant is R F F 12 2 14 dyne

Then the minimum 1 2resultant is R F F 12 2 2 dyne

---------------------------------------------------------------------------------------------------------------------


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1 o o

2 22 2 2

1 2 1 2

22 22Tan Tan 77 11' Cos 77 11' 0.22162

5 5

R F F 2F F Cos R 10 9 2 10 9 0.22162 14.86 N

15 kg .wt

18 kg .wt

o30

o

1 2

2 2 2

1 2 1 2

2 2 o

o

2

o

1 2

1

F 15kg .wt F 18kg .wt 120

Resultant :

R F F 2F F Cos

R 15 18 2 15 18 Cos120 16.7kg .wt

Direction :

F Sin 18 Sin120Tan Tan 2.598

F F Cos 15 18Cos120

Tan 2.598

o68 56'

o o o

2 22 2 2

1 2 1 2

2

1 2

3 -4 3Tan 36 52' Add 180 143 7' Cos And Sin

4 5 5

-4R F F 2F F Cos R 12 15 2 12 15 9 Newton

5

315

F sin 95Tan ( undifined ) m( ) 90-4F F cos 12 12

12 155

Example (5)

Find the resultant of two forces 10 and 9 Newton act at a point and the angle between them

Is where22

Tan

5

Answer

1F=10 N 2F=9 N and 0 180

---------------------------------------------------------------------------------------------------------------------

Example (6)

Two forces of magnitude 12 and 15 Newton are applied to a point and the tangent of the3

angle between them is , Find the resultant of the two forces and the measure of its angle4

of inclination on t

he first force .

Answer

1F=12 N

2F=15 N and must be 0 180

---------------------------------------------------------------------------------------------------------------------

Example (7)

o

Two forces act at a point , the first force is 15 kg . wt and acts towards the east , the second

force is 18 kg . wt and makes an angle 30 west of north . find their resultant and direction .Answer


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F kg .wt

4kg.wt

o120

o

1 2

2 2 2

1 2 1 2

2 2 o

2 2 2 2

1 2

o o

1 2

F 4kg .wt F F kg .wt 120

( a ) When R F :

R F F 2F F Cos

F 16 F 2 4 F Cos120

F 16 F 4F 4F 16 F F 16 F 4 kg .wt

( b )When R F : F F and F 4 kg .wt

90 Cot 90 0 F F Cos 0

o

o

2

1 2

oo o o

o

F 4 Cos120 0

1

F 4 0 F 2 kg .wt2

( c ) When 45 :

F SinTan

F F Cos

4Sin120Tan 45 F 4Cos120 4 Sin120

F 4Cos120

F 2 2 3 F 2 2 3 kg .wt

( d ) When R bisects the angle between the two forces :

Then the two for

ces are equal to each other F 4 kg .wt

R

1F F kg.wt

o30

R4 kg .wt

Example (8)

The magnitude of two forces are 4 and F kg. wt. act at a point , and the angle between them

Is 120 . find F in each of the following cases :

(a) R = F ( where R is the resultant of the two forces )(b) R F

(c) The angle between R and F is 45.

(d) R bisects the angle between the two forces.Answer

---------------------------------------------------------------------------------------------------------------------


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1F

o120

R2F

2FR 3

o60

o

2 2 2 o

1 2 1 2

2 2 21 2 1 2

o

2

22 2 o

1 2 1 2

2 2 2

1 2 1 2

( 1) When 120 :

R F F 2F F Cos120

R F F F F (1)

( 2 ) When F is in the opposite direction " 60 ":

R 3 F F 2F F Cos60

3R F F F F ( 2 )

Substitute (1) in (2)

2 2 2 2

1 2 1 2 1 2 1 2

2 2 2 2 2 2

1 2 1 2 1 2 1 2 1 1 2 222 2

1 1 2 2 1 2 1 2

o o o

3 F F F F F F F F

3F 3F 3F F F F F F 2F 4F F 2F 0 " 2" F 2F F F 0 F F 0 F F

The resultant in the first case to the resultant in the second case 60 30 90

o60

o60

o30o

30

Example (9)

The difference in magnitude between two forces , acting at a point is 9 gm. Wt . and their

Resultant is 21 gm. wt. Find the two forces and the angle between them , if their resultant is

perpendicular on the smallest one.Answer

Let the two forces be F and ( F 9 ) gm wt.

,

1 2

2 2 2

2 2 2 2 2 2

R F Cot 0 F F Cos 0 F F 9 Cos 0

FCos (1)

F 9

And R 21 21 F ( F 9 ) 2F( F 9 )Cos 2

From ( 1) and ( 2 ) and by substitution

441 F ( F 9 ) 2F 441 F F 18F 81 2F 18F 81

18F 360

F 20 gm. wt

The magnitude of the two forces are 20 , 29 gm. wt.

F 20Cos 133 36`

F 9 29

---------------------------------------------------------------------------------------------------------------------

Example (10)

1 2

o

2

F and F are two forces act at a point , their resultant is R gm . wt when the angle between

them is 120 and their resultant becomes R 3 gm . wt When F becomes in the opposite

direction , Prove that1 2

F F and the resultant in the first case is perpendicular to the

resultant in the second case .

Answer


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Example (11)oIf three forces act at a particle are F , 2F , F' , The angle between F and 2F is 60 , The

maximum and the minimum values of the resultant of the three forces are 26 and 12 newtons

respectively , Then find the magnitudes of F and F'

Answer

22 2 2 2 2 o

1 2 1 2

2 2 2 2 2

The resultant of the first two forces :

R F F 2F F Cos R F 2F 2F 2F Cos60

1R 5F 4F R 7F R 7F

2

The Maximum resultant of the Three forces :

R 7F F' 7F F' 26 (1)

The Mini

mum resultant of the Three forces :

R 7F F' 7F F' 12 ( 2 ) Or F' 7F 12 ( 3 )

By adding (1) and (2) :

19 7 19 7

2 7F 38 F N Then F' 26 7 7N 7 7

And By adding (1) and (3) :

26 192F' 38 F' 19 N Then F 7N

7

---------------------------------------------------------------------------------------------------------------------


11/22




Static 3rdsecondary Chapter One Forces--

Example (12)

Two forces of magnitude 3f , 2 f Newton act at a point Find the measure of the angle

between the two forces if their resultant is perpendicularto the smaller force . if f = 15 ,Then

find the magnitude of the resultant.Answer

o

1 2

2

1 2

Let the measure of the angle between the two forces be and the measure of the angle

between the resultant and the smaller force be .

So F 3F F 2F 90

F sin 2F sinAnd Tan

F F cos 3F 2F cos

But When 9

0 Cot 0 3F 2F cos 0

33 2cos 0 Cos 150

2

o

1 2

2 2 2 2 2 2

1 2 1 2

when F 15 F 15 3 F 30 150

3R F F 2F F cos 3F 4F 2 3F 2F F

2

R F 15 Newton

---------------------------------------------------------------------------------------------------------------------


12/22





F

F Cos

F Sin

o30

32 N

1F

2Fo

1

o

2

F 32Cos30 16 3 N

F 32Sin30 16 N

o45

6 gm . wt

1F

2Fo

1

o

2

F 6 Cos45 3 2 N

F 6 Sin45 3 2 N

80 N

1F

2Fo

2

o

1

F 80Cos60 40 N

F -80 Sin60 -40 3 N

o60

Resolution of a force into two perpendicular Components

Any Inclined force can be resolved into two Components

horizontal force and vertical force

---------------------------------------------------------------------------------------------------------------------

Example (1)

Resolve a force of magnitude 32 N and direction 30 north of east .

Answer

Note : we can wr ite the above force as ( 32 N , 30) OrF =16 3 i +16 j

---------------------------------------------------------------------------------------------------------------------

Example (2)

Find each of the Xcomponent and YComponent of o1R 6 gm.wt , 45 .Answer

Note : we can wr ite it as vector component as 1F =3 2 i +3 2 j---------------------------------------------------------------------------------------------------------------------

Example (3)

Find each of the Xcomponent and YComponent of o2R 80 N , 150 .Answer

Note : we can wr ite it as vector component as 1F =-40 3 i +40 j


13/22





2 2R X Y

Where : X : Sum of the forces in the X - direction

Y : Sum of the forces in the Y - direction

The direction of the resultant :

YTan = , Where is the Polar angle

X

2

90

180

2703

2

0

When:

x 0And y 0 , Then lies in the first quadrant

x 0Andy 0 , Then lies in the second quadrant

180

x 0Andy 0 , Then lies in the third quadrant

180

x 0And y 0 , Then li

es in the fourth quadrant

360

Example (4)

Find each of the Xcomponent and YComponent of o3R 48 N , 240 .Answer

---------------------------------------------------------------------------------------------------------------------

Af ter we knew how to resolve a force ,now we have to f ind the resul tant of the resolved forces

The resultant of Coplanar forces meeting at a pointmore than 2 forces

Rule:

---------------------------------------------------------------------------------------------------------------------

Very important note : Arrangements of quadrants :

1stquad2n quad

4t quad.3rdquad++++++

- - - - - - - - -

-

-

-

-

+

+

+

+


14/22





o5 2Cos45

o5 2 Sin 45

o2 3Cos30

o2 3 Sin 30

Example (1)

---------------------------------------------------------------------------------------------------------------------

Example (2)

---------------------------------------------------------------------------------------------------------------------

Example (3)

---------------------------------------------------------------------------------------------------------------------

o

o

2 2 2 2

o st

In the opposite figure :

X 8 5 2 Cos45 13 N

Y 3 5 2 Sin45 8 N

R 13i 8 j

And R X Y (13 ) ( 8 ) 233 15.264 Newton.

Y 8Tan , 31 36` " Agreed as X and Y are in 1 Quadrant"

X 13

o

o

2 2 2 2

o

o


X 2 3 Cos30 3 0 N

Y -4 3 2 3 Sin30 -5 3 N

R -5 3 j

And R X Y (0 ) ( -5 3 ) 5 3 Newton.

Y -5 3Tan , 90 " Refused "

X 0

Or 270 "agreed as for negative sign of y - axis"

o o o

o o o

2 2 2 2

o


X 7 3 Cos30 3 3 Cos30 6 Cos60 3 N

Y 7 3 Sin30 3 3 Sin30 6 Sin60 2 3 0 N

R 3i

And R X Y ( 3 ) (0 ) 3 Newton.

Y 0Tan 0

X 3


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1F 40 N

o60

22

1 st

X 40 40Cos60 40Sin30 40

Y 40 Cos30 40Cos30 40 3

R 40 40 3 6400 80

40 3Tan 3 Tan 3 60 " Agreed as X and Y are in 1 Quadrant"

40

The magnitude of R is 80 Newton and acts in the direction of t

he 2nd force

2F 40 N

o30 o

30

3F 40 N

1 o

X 60Cos30 88Cos 30 14 3

Y 60 88 sin30 60 sin30 14

R 588 196 784 28 gm .wt

14 1Tan

14 3 31

Tan 30 " refused" 3

Or 180 30 210 " as x 0 And y 0"

The magnitude of R is 28 gm .wt .and acts in the direc

tion of 2nd force

1F 60 gm . wt

o30 o30

2F 88 gm . wt 3F 60 gm . wt

Examples

Example (1)

Three coplanar forces act at a point. The magnitude of each equals 40 Newton .The angleBetween the first and the second force equals the angle between the second and the third force

And each of them is of measure 60 .Find the magnitude of the resultant and prove that this

Resultant acts in the direction of the second force.

Answer

---------------------------------------------------------------------------------------------------------------------

Example (2)

Three coplanar forces of magnitude 60 , 88 and 60 gm .wt. act at a point. The first towards

North , the second makes 30south of west and the third makes 30south of east. Find theirResultant.Answer

---------------------------------------------------------------------------------------------------------------------


16/22





1 o

X 5 10cos60 20cos60 15 3 cos30 2.5

5 3Y 10 sin60 15 3 sin30 20 sin60

2

25 75R 25 5 Newton

4 4

5 3 2Tan 3 Tan 3 -602 5

But X 0 , Y 0 "So lies inthe second quad."

180 60 120

R is of magni

tude 5 Newton and acts in a direction inclined to the 1st with angle of 120

1F 5 N

o60

2F 10 N

o60 o30

3F 15 3 N

o30

o30

4F 20 N

22

1 o

X 9 6 cos60 6 3 cos30 3 kg.wt

Y 6 sin60 6 3 sin30 9 3 -3 3 kg.wt

R 3 -3 3 6 Newton

-3 3Tan 3 Tan 3 -60

3

But X 0 , Y 0 "So lies inthe fourth quad."

360 60 300

R is of magnitude 6 Newton and

acts in a direction inclined to the 1st with angle of 300

1F 9 N

o60

2F 6 N

o60 o30

3F 6 3 N

o30

4F 9 3 N

Example (3)

Four coplanar forces of magnitude 5 , 10 , 15 3 and 20 Newton .act at a point .the measure of

The angles between the first and the second is 60 ,and between second and the third90 and

Between the third and the fourth is 150 .Find the magnitude and the direction of their resultant.Answer

---------------------------------------------------------------------------------------------------------------------

Example (4)

Four coplanar forces of magnitude 9 , 6 , 6 3 and 9 3 kg . wt act at a point .the measure of

The angles between the first and the second is 60 ,and between second and the third90 andBetween the third and the fourth is 120 .Find the magnitude and the direction of their resultant.

Answer

---------------------------------------------------------------------------------------------------------------------


17/22





1F 25 N

o30

2F 20 N

o60 o60

4F 20 N

2F 2 3 kg.wt

o60

1F 20 kg .wt

o60 o30

3F 12 3 kg .wt

o30

o30

4F 36 kg.wt

A B

C

D

H

3F 40 N

2 2

1

st

X 25 20Cos30 20 Sin60 25 N

Y 40 20Sin30 20Cos60 60 N

R 25 60 4225 65 Newton

60 12 12Tan Tan

25 5 5

67 23` " Agreed" as X and Y are in 1 Quadrant"

The resultant acts in the direction 67 23` wit

h AB

1

2

3

4

F is of magnitude 20 Kg. wt. and 60 north of the east .

F is of magnitude 2 3 kg .wt due north .

F is of magnitude 12 3 kg .wt 30 north of west .

F is of magnitude 36 Kg .wt. 30 east of s outh .

2 2

o

X 20Cos60 36 Sin 30 12 3 Cos 30 10 kg. wt.

Y 2 3 20 sin60 12 3 sin30 36Cos 30 0 kg. wt.

R 10 0 10 kg. wt.

0Tan 0 0

10

The resultant is 10 kg. wt.and acts due east.

Example (5)

Four coplanar forces of magnitude 25 , 20 , 40 , 20 Newton, act at A and in the directions

AB ,AC ,AD and AH where m( BAC ) 30 ,m( BAD ) 90 and m( BAH ) 150 .

Pr ove that the resultant of these forces is 65 Newton and detrmine its direction.

Answer

---------------------------------------------------------------------------------------------------------------------

Example (6)

The following coplanar forces act at a point :

Find their resultant and its direction.

Answer

---------------------------------------------------------------------------------------------------------------------


18/22





X 9 4 2 cos 45 5 2 cos 45 0

Y 6 4 2 sin45 5 2 sin45 5 0

R 0 " The body is in Equlibrium"

2F 6 N

o45

3F 4 2 N

5F 5 N

ABCD is a rectangle , AB andAD are perpendicular

In ABC : AB 8 cm ,BC 6 cm and m( B ) 90

by using Pythagora's theorem AC 64 36 10 cm

8 4 6 3Cos And Sin

10 5 10 5

4X 2 5cos 2 5 6 kg .wt.

5

Y

3

3 5 sin 3 5 6 kg .wt.5

R 36 36 6 2 kg .wt.

6Tan 1 45

6

R makes 45 with AB and its magnitude 6 2 kg .wt.

1F 9 No

45o

45

4F 5 2 N

A B

CD

8cm

6cm

1F 2 kg.wt

2F 5 kg .wt3F 3 kg .wt

Example (7)

Five coplanar forces act at a point of magnitude 9 , 6 ,4 2 ,5 2 and 5 Newton act due to

east , north , western north , western south and south respectively .Find the magnitude and

direction of Their resultant.Answer

---------------------------------------------------------------------------------------------------------------------

Example (8)

ABCD is a rectangle where AB = 8 cm , BC = 6 cm. Three forces 2 , 5 , 3 kg .wt. act at point

(A) Along the directions AB ,AC ,AD respectively. Find their resultant and its inclination with

AB

Answer

---------------------------------------------------------------------------------------------------------------------


19/22





8cm

8cm

A B

CD

AB and AD are perpendicular

In ADE : AD 8 cm ,DE 6 cm and m( D ) 90

AE 64 36 10 cm.

let m( DEA ) m( EAB )

8 4 6 3

Sin And Cos10 5 10 5

X 2 4 2 Cos45 10Cos 12

Y 4 10 sin 4 2 sin45 16

R 144 256 400

16 4

20 gm .wt. & Tan12 3

R makes 53 7' with AB and its magnitude 20 gm .wt.

E6cm 2cm

8cm

o45

M

A

BC

D

E F

4

14

5

23

o

60

o60o60o60

o60

o60

2 2

o

X 4 Cos60 4Cos60 5 2Cos60 3Cos60 2

Y Sin60 4Sin60 2Sin60 3Sin60 0

R 2 0 2 gm .wt

0Tan 180 " As x 0 , y 0 x axis"

2

R 2 gm .wt and acts along MD.

2

4 2

104

Example (9)

ABCD is a square whose side is 8 cm long. E CD such that DE = 6 cm . The forces 2 , 4 2,

10 , 4 gm .wt. act at a point (A) along the directions AB , AC , AE , AD respectively .Find

their Resultant and its inclination with AB .

Answer

---------------------------------------------------------------------------------------------------------------------

Example (10)ABCDEF is a uniform hexagon , M is the point of intersection of its diagonals , The forces 4 , 1

4 , 5 , 2 , 3 gm. wt. act at point M along the directions MA,MB,MC,MD,ME and M F

Respectively. Find the resultant and its direction.

Answer

---------------------------------------------------------------------------------------------------------------------


20/22





2 2

o

X 6 3 4Cos 30 4Cos 30 2 3 N

Y 4 sin30 4 sin30 0 N

R 2 3 0 2 3 N

0Tan 0 0 " As x 0 and y 0 x axis"

2 3

The resultant is 2 3 N and acts along BC

o60 o60

o

30o

30

o30 o30

A

B C

6 3 N

4 N

4 N

o4 Cos30

o4 Cos30

o4 Sin 30

o4 Sin 30

A

B C

D

EF

o30

22

o st

o

X 6 2 3 Cos30 6 Cos60 2 3 Cos30 15

Y 2 3 Sin30 6 Sin60 2 3 Sin30 3 3

R 15 3 3 6 7 N

3 3Tan 19 6' 24'' " As x 0 , y 0 1 quad."

15R 6 7 N and inclines by 19 6'24'' along AD.

o30o30

o30

6 N

6 N

2 3 N

2 3 N

Example (11)

ABCDEF is a regular hexagon , forces of magnitude 6 , 2 3 , 6 ,2 3 Newtons

act alongAB ,AC ,AD,AERespectively. Find the magnitude and direction of the resultant .

Answer

---------------------------------------------------------------------------------------------------------------------Example (12)

oABC is an isosceles triangle , m( BAC ) 120 , forces of magnitudes 4 , 6 3 , 4 newtons act at a

point A along directions parallel to AB , BC , CA respectively , find the resultant of these

forces .

Answer

---------------------------------------------------------------------------------------------------------------------


21/22





6 2 gm .wt .

o45

1F

8 2 gm .wt

o45

2F

1

1 1

2

2 2

The resultant is 7 gm.wt and acts towards east means:

X 7 and Y 0

X F 6 2 cos45 8 2 cos45

7 F 2 F 9 gm.wt.

Y 6 2 sin45 8 2 sin45 F

0 6 8 F 0 F 14 gm. wt.

A

B C

o60

o60

o60

8 N

8 N

12 N

o60

22

o

2nd o o o

X 8 12Cos60 8Cos60 2 N

Y 12 Sin60 8Sin60 2 3 N

R 2 2 3 4 N

2 3Tan 60

2" As x 0 , y 0 2 quad." 180 60 120

R 4 N and acts along CA.

Example (13)Three forces of magnitude 8 , 8 , 12 newtons act at a particle and their direction are parallel to the sides

of an equilateral triangle taken in the same cyclic order , find the magnitude and the direction of their

resultant .

Answer

---------------------------------------------------------------------------------------------------------------------Example (14)

Four coplanar forces1 2F ,6 2 , 8 2 , F gm . wt. act at a point due east , eastern north,

Western north and south respectively .Find1 2F ,F if the resultant equals 7 gm .wt. towards east.

Answer

---------------------------------------------------------------------------------------------------------------------


22/22




7 N

o60

1F

o30 o30

2F

o60o30

4 3 N

o60

11 3 N

1 2

1 2

2 1

1 2

1 2

2 1

As the resultant acts towards South " OY` "

X 0 , Y -8

X 7 F cos60 F cos60 4 3 cos30 0

1 17 F F 6 0

2 2

F F 2 (1)

Y F sin60 F sin60 4 3 sin30 11 3 0

3 38 F F 2 3 11 3

2 2

F F 18 2

From(1) and (2) 1 2: F 8 , F 10

3 2 Newton

o45

F

4 2 Newton

o60

o

o

22 2 2 2

2

1 o

X F 3 2 cos45 2 3 Sin60

X F 3 3 X F

Y 3 2 sin45 2 3Cos60 3

Y 3 3 3 Y 3

And R X Y 3 2 F 9

F 18 9 9 F 3 Newton

Y 3And Tan 1 Tan 1 45 " agreed"

6 Newton

Example (15)

Five forces1 2

7 ,F ,F ,4 3 ,11 3 Newton act at a point, the angle between the first and

Second is 60 and between the second and the third is 60 and between the third and the fourth

Is 90 and between the forth and the fifth is 60 .Find 1 2F ,F if their resultant equals 8 Newton

Towards south.

Answer

---------------------------------------------------------------------------------------------------------------------Example (16)

o

Coplanar forces F , 3 2 , 2 3 and 3 Newton are applied at a point , such that the first

force acts in direction East and the measure of the angle between the first and the second

force is 45 and between o

o

the second and the third force is 105 and between the third and

the fourth force is 120 , if the magnitude of the resultant of the forces is equal to 3 2

Newton , Then find the value of F and the measure of the angle between the line of action of

their resultant and the first force .

Answer

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forces, resultant between two forces, resolution