chapter 2 - force vector_2

8/18/2019 Chapter 2 - Force Vector_2

1/44

Luqman Nul Hakim Bin Juwara

Jabatan Kejuruteraan Mekanikal

Politeknik Muadzam Shah

[email protected]

DJJ3053

ENGINEERING MECHANICS

CHAPTER 2 :

FORCE VECTOR


2/44

Force Vector

This topic introduces the differentiation between scalars and vector, vector operations, and analysis of force vectorin plane and space.

Learning Outcomes (LO)

Upon completion of this topic, students should be able to :

2.1 Distinguish scalars and vectors.

2.1.1 Differentiate between scalars and vectors

2.1.2 Distinguish between free vectors, sliding vectors, fixed vectors.

2.2 Explain vectors and vector operations.

2.2.1 Calculate addition of vectors

2.2.2 Calculate subtraction of vectors

2.2.3 Determine resolution of vectors

2.3 Explain rectangular components.

2.3.1 Explain two forces acting on a particle

2.4 Determine the resultant force of coplanar forces by addition

2.4.1 Explain scalar notation

2.4.2 Explain Cartesian vector notation

2.4.3 Determine coplanar forces and resultant force using scalar notation and Cartesian notation

2.5 Explain position vectors and x, y, z coordinates

2.5.1 Explain position vectors and x, y, z coordinate

2.6 Analyze the force vector directed along the line

2.6.1 Explain the force vector directed along the line

2.6.2 Determine the force vector directed along the line


3/44

Force Vector

Scalars A quantity characterized by a positive or negative number

Example :

1. Length2. Mass

3. Time

4. Pressure5. Temperature

Symbol : A


4/44

Force Vector

Vector A quantity that has both a magnitude & a direction

Example :

1. Force2. Position

3. Moment

4. Velocity

Symbol : or A

A


5/44

Free Vector

May be moved anywhere in space provided it maintains thesame direction & magnitude.

Types Of Vectors


6/44

Sliding Vector

May be applied at any point along its line/path of action.

Types Of Vectors


7/44

Fixed Vector

Must remain at the same point of application.

Types Of Vectors


8/44

Fy

Fx

F

∑F=Fx + Fy θ

2 Forces Acting On A Particle


9/44

Resolution of Vectors

• Parallelogram Law

A

B

R R = A + B

• Triangle Rule (Trigonometry): Head-to-tail

A BR

A

B

R = A + B


10/44

Vector Addition

Vector Subtraction


11/44

The ring shown in figure shown below is subjected to two forces, F1

and F2. If it is required that the resultant force have a magnitude of1kN and be directed vertically downward, determine the magnitudeof F1 and F2 provided

Example 1

Ans:

N F

N F

446

653

2

1

30


12/44

Solution:

N F

N F

N F

N F

446

130sin

1000

20sin

653

130sin1000

30sin

2

2

1

1


13/44

The vertical force F acts downward at A on the

2-membered frame. Determine the magnitudes of the twocomponents F directed along the axes of AB and AC.

Set F = 500N.

Example 2

Ans:

N F

N F

AC

AB

366

448


14/44

June 2015 Session : 1(b)

Resolve the 30 N forces into component along the u and v

axes as shown in figure below? (6 marks)

Ans:

N53.15

N96.21

v

u

F

F


15/44

The screw eye is subjected to two forces, F1 and F2.Determine the magnitude and direction of theresultant force.

Example 3

Ans:

8.54

213

N F R


16/44

•

Parallelogram LawUnknown: magnitude of FR and angle θ

Solution


17/44

Solution•

Trigonometry LawLaw of Cosines

Law of Sines

N N

N N N N F R

2136.2124226.0300002250010000

115cos150100215010022

8.39

9063.06.212

150sin

115sin

6.212

sin

150

N

N

N N


18/44

DirectionΦ of FR measured from the horizontal

8.54

158.39

Solution


19/44

Determine the magnitude of the resultant force FR = F1 + F2

and its direction, measured counterclockwise from the positivex axis.

Example 4

Ans:

108

867

N F R


20/44

Determine the magnitude of the resultant force FR = F1 + F2

and its direction, measured counterclockwise from the positivex axis.

Example 5

Ans:

353

2.393

N F R


21/44

sin

and cos

F F

F F

F F F

y

x

y x

j F i F F y x

Scalar Notation

Cartesian Vector Notation


22/44

Coplanar Force & Resultant Force

j F i F F

j F i F F

j F i F F

y x

y x

y x

333

222

111

Cartesian vector notation:

j F i F F F F F

Ry Rx

R

321

y y y Ry

x x x Rx

F F F F

F F F F

321

321

Scalar notation:


23/44

Rx

Ry

Ry Rx R

F

F

F F F

1-

22

tanDirection;

;Magnitude


24/44

Example 6

The link is subjected to two forces F1 and F2. Determine

the magnitude and orientation of the resultant force.

Ans:

9.67

629

N F R


25/44

Solution

N N N F

F F

N N N F

F F

Ry

y Ry

Rx

x Rx

8.58245cos40030sin600

:

8.23645sin40030cos600

:

N N N F R 6298.5828.23622

9.678.236

8.582tan 1

N

N

Scalar Notation


26/44

Solution

Cartesian Vector Notation

F1 = { 600cos30°i + 600sin30° j } NF2 = { -400sin45°i + 400cos45° j } N

Thus,

FR = F1 + F2 = (600cos30ºN - 400sin45ºN)i

+ (600sin30ºN + 400cos45ºN) j = {236.8i + 582.8 j}N

The magnitude and direction of FR are determined in the same manneras before.


27/44

Example 7

Determine the magnitude of the resultant force and its

direction, measured clockwise from the positive x axis

Ans:

5.46

8.97

N F R


28/44

Example 8

Determine the magnitude of the resultant force and its

direction, measured counterclockwise from positive x axis

Ans:

253

546

N F R


29/44

June 2012 Session : 1(a)

Refer to the figure, determine the angle of for

connecting member A to the plate so that the resultantforce of FA and FB is directed horizontally to the right. Also,what is the magnitude of the resultant force? (10 marks)

Ans:

kN4.10

9.54

R F


30/44

June 2012 Session : 1(b)

Based on the figure, if and , express the

force in the Cartesian form and determine the magnitudeof resultant force acting on the eyebolt and its directionfrom the positive x-axis (15 marks)

Ans:

04.3

kN67.8

R F

30 kN6T


31/44

June 2014 Session : 1(b) ii

Determine the magnitudeand direction of the resultantforce of the three forces

(11 marks)

Ans:

49.83

lb463

R F


32/44

June 2015 Session : 1(c)

The link in figure below is subjected to three forces F1,F2 and F3. Determine the magnitude FR and direction

is measured as counterclockwise from the positive x axis.Apply the resultant force as a Cartesian vector.

(13 marks)

Ans:

8.39

2024

ji F R


33/44

3 FORCES ACTING ON A PARTICLE

Right-HandedCoordinate System

x

z

y

Rectangular Componentsof a Vector


34/44

Cartesian Vector

A = Axi + A y j + AZ k

Magnitude of a Cartesian Vector:

2 2 2 z y x A A A A

Direction of a Cartesian Vector:

A

A x cos A

A y cos

A

A z cos


35/44

uA = A / A = ( Ax/ A)i + ( A y / A) j + ( AZ / A)k Unit Vector

222

z y x A A A A where magnitude of A,

Position Vectors: r = xi + y j + zk Coordinates: (x,y,z)


36/44

Force Vector Directed along a Line

F = F u = F ( r/r )


37/44

Example 9

With a force of 350 N, acting on the support A, as

Cartesian Vector, determine its direction

Ans:

149

107

6.64

300100150

N k ji F


38/44

Solution

m623

)5.75.1()02()03(

k ji

k jir

N

N

F

r

r

300k -100j-150i

k 7

6- j

7

2-i

7

3350

uF

k 7

6- j

7

2-i

7

3r u

m7m)6(m2m3 222

1497

6cos

1077

2cos

6.64

7

3cos

1

1

1


39/44

Example 10

The roof is supported by cables as shown below. If the

cables exert force FAB = 100N and FAB = 120N on the wallhook at A as shown, determine the magnitude of theresultant force acting at A

Ans:

N F R 217


40/44

Solution

N

N r

N

r

AB

AB AB

AB

AB

70.7k -i7.70

k 66.5

4-i

66.5

4100

r 100F

m66.544

m4k -4i

4m)k -(00)j-(0i)0m4(r

22


41/44

Solution

N

N r

N

r

AC

AC AC

AC

AC

80k -40j80i

k 64- j

62i

64120r 120F

m6424

m4k -2j4i

4m)k -(00)j-(2mi)0m4(r

222

N

N N N

R

AC AB R

217

7.150407.150F

150.7k -40j-i7.150 80k -40j80i70.7k -70.7i

FFF

222


42/44

Example 11

Two tractors pull on the tree with the forces shown.

Represent each force as a Cartesian vector and thendetermine the magnitude and coordinate direction anglesof the resultant force

Ans:

154

5.92

3.64

lb236

R F


43/44

Example 12

The tower is held in place by three cables. If the force of

each cable acting on the tower is shown, determine themagnitude and coordinate direction angles of resultantforce. Take x = 20m, y = 15m

Ans:

168

6.90

6.77

5.1

kN F R


44/44

QUESTION

&

ANSWER SESSION

Thank You

For Listening

chapter 2 - force vector_2

Documents