Chapter 2: Force Vectors - Civil Engineering Department …civil.emu.edu.tr/courses/civl211/LECTURE-2.pdfAddition of a System of Concurrent Coplanar (2D) Forces Resolve vectors into

Engineering Mechanics: Statics

Chapter 2: Force Vectors

Objectives

To show how to add forces and resolve them into components using the Parallelogram Law.

To express force and position in Cartesian vector form and explain how to determine the vector’s magnitude and direction.

Chapter Outline

Scalars and Vectors

Vector Operations

Vector Addition of Forces

Addition of System of Coplanar Forces

Vector operation

Cartesian Vectors

Addition and Subtraction of

Cartesian Vectors

Position Vectors

Force Vector Directed along a Line

Dot Product

Number

Any positive or negative value.

e.g.: 5, -3.07, π...

Scalar

A quantity characterized by a positive or negative number and has a unit.

e.g.: mass (5.1 kg), volume (0.028 m3)...

Review of Scalars and Vectors

Scalars and Vectors

Vector A quantity that has both magnitude and direction

measured from a given axis.

e.g.: Position (3.4 km North), force (8.73 N towards right side), ...

– Represented by a letter with an arrow over it such as or bold such as A

– Magnitude is shown by or simply A

A

A

Scalars and Vectors

Scalars

Vectors

Characteristics

VALUE

(POSITIVE

OR

NEGATIVE)

&

UNIT

(both togather called

MAGNITUDE)

VALUE

(POSITIVE

OR

NEGATIVE)

UNIT

(both togather called

MAGNITUDE)

&

DIRECTION

Application

mass, speed, volume, time

velocity, acceleration force

Scalars and Vectors

Example

Magnitude of Vector = 4 units

Direction of Vector = 20° measured

counterclockwise from the horizontal axis

Sense of Vector = Upward and to the right

The point O is called tail

of the vector and the point

P is called the tip or head

Vector Operations

aA

Multiplication and Division of a Vector by a Scalar

- Product of vector A and scalar a = aA

- Magnitude =

- If a is positive (+), sense of aA is the same as sense of A

- If a is negative (-), sense of

aA, it is opposite to the

sense of A

Vector Operations

Multiplication and Division of a Vector by a Scalar - Negative of a vector is found by multiplying the

vector by ( -1 )

- Law of multiplication applies

e.g.: A/a = ( 1/a ) A, if a≠0

Vector Operations

Vector Addition Addition of two vectors A and B gives a

resultant vector R by the parallelogram law

special case of parallelogram law is triangle construction (see the next slide)

Commutative rule

e.g.: R = A + B = B + A

Vector Addition

Parallelogram Law

Triangular Construction

Vector Operations

Parallelogram Law description

- To resolve a force into components along

two axes directed from the tail of the force

- Start at the head, constructing lines parallel

to the axes


Vector Operations

Vector Addition

- Special case: Vectors A and B are collinear (both have the same line of action). Triangular shape does not form.

Vector Operations

Vector Subtraction

Special case of addition

e.g.: R’ = A – B = A + ( - B )

Rules of Vector Addition Applies

Vector Operations

Resolution of Vector

Any vector can be resolved into two components by the parallelogram law

The two components A and B are drawn such that they extend from the tail of R

Resolution of a vector is breaking up a vector into its components.

It is the reverse application of the parallelogram law.

Vector Operations


When two or more forces are added,

successive applications of the

parallelogram law is carried out to find

the resultant

e.g.: Forces F1, F2 and F3 acts at a point O

- First, find resultant of

F1 + F2

- Resultant,

FR = ( F1 + F2 ) + F3


Method 1: (graphical)

parallelogram Law OR Establishing triangles

Method 2:

Trigonometry

Parallelogram Law - Make a sketch showing the vector addition using the

parallelogram law

- Two “component” forces add according to the

parallelogram law yielding a resultant force.

- Resultant force is shown by the diagonal of the

parallelogram

- The components are the sides of the parallelogram

- Label all the known and unknown force

magnitudes and angles.

- Identify the two unknowns.

Procedure for Analysis:

Trigonometry

- Redraw half portion of the parallelogram

- Magnitude of the resultant force can be determined

by the law of COSINE

- Direction of the resultant force can be determined

by the law of SINE


Magnitude of the two components can be

determined by the law of COSINE.

Direction of the two components can be

determined by the law of SINE.


Example:

The screw eye is subjected to two forces F1

and F2. Determine the

magnitude and direction

of the resultant force.


Solution

Parallelogram Law Unknown:

magnitude of FR and

angle θ



FR= Magnitude of the resultant force

(determined by the law of cosine) θ= angle (determined by the law of sine)

ϕ = direction of the resultant force = θ +15°

Half portion of the parallelogram

Solution

Trigonometry Law of Cosine

N

NNNNFR

6.212

4226.0 300002250010000

115cos 150 100 2 150 10022

-


Solution

Trigonometry Law of Sines

8.39

9063.0 6.212

150sin

sin115

N 212.6

θ sin

N 150

θ

θN

N


Solution

Trigonometry Direction Φ of FR measured from the horizontal

354.761

1539.7613

=

+=ϕ


q

q

p

p

365 N

β 38.5°

The p and q axes are not perpendicular to each other. If the component of 365 N force along pp

axis is 237.5 N, determine the other component along qq axis and the unknown angle β.

Example:

Vector Operations

q

q

p

p

β 38.5°



Solution:

Vector Operations

Establish a paralelogram or a triangle. Given are: resultant and component along pp axis.

q

q

p

p

β 38.5°



Solution:

Vector Operations

38.5°


q

q

p

p

β 38.5°



Solution:

Vector Operations

38.5°

β


∑FFR =

Addition of a System of Concurrent Coplanar (2D) Forces

Concurrent at a point:


Cartesian Vector Notation

- Cartesian unit vectors i and j are used to designate the x and y directions

- Unit vectors i and j have dimensionless magnitude of unity ( = 1 )

Cartesian Vector Notation F = Fx i + Fy j F’ = F’x i + F’y (-j)

F’ = F’x i – F’y j



Resolve vectors into components using x and y axes system.

Each component of the vector will have a magnitude and a direction.

The directions are based on the x and y axes. The “unit vectors” i and j are used to designate x and y axes.

F = Fx i + Fy j

x and y axes are always perpendicular to each other. Together, they can be

directed at any inclination.

Cartesian Vector Notation

jFiFF yx

+=

F

Vector components may be expressed as

products of the unit vectors with the scalar

magnitudes of the vector components.

Fx and Fy are referred to as the scalar

components of

Coplanar Force Resultants To determine resultant of several coplanar forces:

- Resolve force into x and y components

- Addition of the respective components using scalar algebra

- Resultant force is found using the parallelogram law


Coplanar Force Resultants - Positive scalars = sense of direction

along the positive coordinate axes

- Negative scalars = sense of direction along the negative coordinate axes

- Magnitude of FR can be found by Pythagorean Theorem

RyRxR FFF 22 +=


Coplanar Force Resultants - Direction angle θ (orientation of the force)

can be found by trigonometry

Rx

Ry

F

F1tanθ =



Example: Add (sum) the given three

coplanar forces


Step 1: to resolve each force into

its x-y components.

Step 3: find the magnitude and

the angle of the resultant vector.

Step 2: to add all the x components

together and add all the y

components together.

These two totals become the resultant

vector.

Cartesian Vector Method (solution by three steps)


Example: Add (sum) the given three

coplanar forces

Cartesian vector notation

F1 = F1x i + F1y j

F2 = - F2x i + F2y j

F3 = F3x i – F3y j


Step 1

Vector resultant is therefore

FR = F1 + F2 + F3

= F1x i + F1y j - F2x i + F2y j + F3x i – F3yj

= (F1x - F2x + F3x)i + (F1y + F2y – F3y)j

= (FRx)i + (FRy)j


Step 2

( )( )↑→

+

+


Coplanar Force Resultants FRx = (F1x - F2x + F3x)

FRy = (F1y + F2y – F3y)

In all cases,

FRx = ∑Fx

FRy = ∑Fy

* Take note of the sign conventions

Step 3

Scalar Notation:

Rx

Ry

RyRxR

F

F

FFF

1

22

tanθ =

+=


The magnitude and the angle of the resultant vector

WAYS OF FINDING THE COMPONENTS OF FORCES IN 2D

1- By an ANGLE measured from any axis

(Cosine of this angle gives the component of that axis)



2- By right angle TRIANGLE of known sides



3- By COORDINATES or known lengths.


(18, 10)

(0, 0)

10 cm

18 cm

Example:

The end of the boom O is subjected to three

concurrent and coplanar forces. Determine

the magnitude and orientation of the

resultant force.

3 35˚


Solution

N788.324

N5

3200N35cos250F

:FF

N606.416

N5

4200N35sin250N400F

:FF

Ry

yRy

Rx

xRx

Σ

Σ

35˚

324.788 N

416.606 N


Solution

Resultant Force

From vector addition,

Direction angle θ is

N

NNFR

250.528

788.324 606.41622

0598.1429402.37180

9402.37

N 606.416

N 788.324tan 1

θ

324.788 N

416.616 N


142.0598°

Example:


Example:


R

R

Documents

Chapter 2: Force Vectors - Civil Engineering Department …civil.emu.edu.tr/courses/civl211/LECTURE-2.pdfAddition of a System of Concurrent Coplanar (2D) Forces Resolve vectors into