# Vector_mechanics [Compatibility Mode]

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Vectors

Vector classifications:

- Fixedorboundvectors have well defined points of

application that cannot be changed without affecting

an analysis.

- Free vectors may be freely moved in space without

changing their effect on an analysis.

Equal vectors have the same magnitude and direction.

Negative vector of a given vector has the same magnitude

and the opposite direction.

Engineers Mechanics- Review of Vector Algebra/Applications

B

B

C

C

QPR

BPQQPR

cos2222 Law of cosines,

Law of sines,

P

C

R

B

Q

A sinsinsin

PQQP

Vector subtraction

Engineers Mechanics- Review of Vector Algebra/Applications

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Addition of three or more vectors through

repeated application of the triangle rule

The polygon rule for the addition of three or

more vectors.

SQPSQPSQP

Multiplication of a vector by a scalar

Engineers Mechanics- Review of Vector Algebra/Applications

Engineers Mechanics- Review of Vector Algebra/Applications

A quantity which has magnitude and

direction, but doesnt follow

parallelogram law, cannot be a vector.

Can you name such a quantity?

Think in terms of associative property!!

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Application: Resultant of Several Concurrent Forces Concurrent forces: set of forces which all

pass through the same point.

A set of concurrent forces applied to a

particle may be replaced by a single

resultant force which is the vector sum of the

applied forces.

Vector force components: two or more force

vectors which, together, have the same effect

as a single force vector.

Engineers Mechanics- Review of Vector Algebra/Applications

Rectangular Components of a Vector

cosFFx

Vector components may be expressed as products of

the unit vectors with the scalar magnitudes of the

vector components.

Fx andFy are referred to as the scalar components of

jFiFF yx

F

May resolve a vector into perpendicular components so

that the resulting parallelogram is a rectangle.

are referred to as rectangular vector components and

yx FFF

yx FF

and

Define perpendicularunit vectors which are

parallel to thex andy axes.ji

and

sinFFy

Engineers Mechanics- Review of Vector Algebra/Applications

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SQPR

Wish to find the resultant of 3 or more

concurrent forces,

jSQPiSQPjSiSjQiQjPiPjRiR

yyyxxx

yxyxyxyx

Resolve each force into rectangular components

x

xxxx

F

SQPR

The scalar components of the resultant are

equal to the sum of the corresponding scalar

components of the given forces.

y

yyyy

F

SQPR

x

yyx

R

RRRR

122 tan

To find the resultant magnitude and direction,

Engineers Mechanics- Review of Vector Algebra/Applications

Example-1

Two structural members A and B are bolted to

a bracket as shown. Knowing that both

members are in compression and that the force

is 20 kN in member A and 30 kN in member B,

determine, using trigonometry, the magnitude

and direction of the resultant of the forces

applied to the bracket by members A and B.

SOLUTION KEY

o Construct the force triangle

and apply the sine and

cosine rules.

Engineers Mechanics- Review of Vector Algebra/Applications

Example of a timber truss joint

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SOLUTION

Engineers Mechanics- Review of Vector Algebra/Applications

Example-1

A collar that can slide on a vertical rod is subjected to

the three forces shown. Determine (a) the value of the

angle for which the resultant of the three forces is

horizontal, (b) the corresponding magnitude of the

resultant.

SOLUTION KEY

o Since the resultant (R) is to be

horizontal, sum of the vertical

comp. of the forces, i.e., Ry = 0.Example of an Umbrella

Engineers Mechanics- Review of Vector Algebra/Applications

Example-2

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1 - 11

0y yR F

90 lb 70 lb sin 130 lb cos 0

13 cos 7 sin 9

213 1 sin 7 sin 9

2 2169 1 sin 49 sin 126 sin 81 2218 sin 126 sin 88 0

sin 0.40899

1.24

(a) Since R is to be horizontal,Ry = 0

Then,

Squaringboth sides:

or,

SOLUTION

Engineers Mechanics- Review of Vector Algebra/Applications

Example-2

1 -12

117.0 lbR or,

Engineers Mechanics- Review of Vector Algebra/Applications

Example-2

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Rectangular Components of a Vector in Space

1coscoscos222 zyx

Components of the vectorF

kji

FkjiF

kFjFiFF

FFFFFF

zyx

zyx

zyx

zzyyxx

coscoscos

coscoscos

coscoscos

is a unit vector along the line of action of

and are the direction

cosines for

F

F

zyx cosand,cos,cos222

zyx FFFF

Engineers Mechanics- Review of Vector Algebra/Applications

Application: Rectangular Components of a Force Vector in Space

222

zyx dddd

The magnitude of the force vector is

F and the direction of the force is

defined by the location of two

points,

222111 ,,and,, zyxNzyxM

d

FdF

d

FdF

d

FdF

kdjdidd

FF

zzdyydxxdkdjdid

NMd

zz

yy

xx

zyx

zyx

zyx

1

andjoiningvector

121212

d

d

d

d

d

d zz

y

yx

x cos;cos;cos

Engineers Mechanics- Review of Vector Algebra/Applications

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Scalar Product of Two Vectors The scalar productordot productbetween

two vectors P andQ is defined as

resultscalarcosPQQP

Scalar products:

- are commutative,

- are distributive,

- are not associative,

PQQP

2121 QPQPQQP

undefined SQP

Scalar products with Cartesian unit components,

000111 ikkjjikkjjii

kQjQiQkPjPiPQP zyxzyx

2222 PPPPPP

QPQPQPQP

zyx

zzyyxx

Engineers Mechanics- Review of Vector Algebra/Applications

Applications: Scalar Product of Two Force Vectors

Angle between two force vectors:

PQ

QPQPQP

QPQPQPPQQP

zzyyxx

zzyyxx

cos

cos

Projection of a force vector on a given axis:

OL

OL

PPQ

QP

PQQP

OLPPP

cos

cos

alongofprojectioncos

zzyyxx

OL

PPP

PP

coscoscos

- For an axis defined by a unit vector ():

Q

Q

Note:

Engineers Mechanics- Review of Vector Algebra/Applications

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Vector Product of Two Vectors

Vector product of two vectorsP andQ is defined

as the vectorVwhich satisfies the following

conditions:

1. Line of action ofVis perpendicular to plane

containingP andQ.

2. Magnitude ofVis

3. Direction ofVis obtained from the right-hand

rule.

sinQPV

Vector products:

- are not commutative,

- are distributive,

- are not associative,

QPPQ

2121 QPQPQQP

SQPSQP

Engineers Mechanics- Review of Vector Algebra/Applications

Vector Products: Rectangular Components

Vector products of Cartesian unit vectors,

0

0

0

kkikjjki

ijkjjkji

jikkijii

Vector products in terms of rectangular

coordinates

kQjQiQkPjPiPV zyxzyx

kQPQP

jQPQPiQPQP

xyyx

zxxzyzzy

zyx

zyx

QQQ

PPP

kji

Engineers Mechanics- Review of Vector Algebra/Applications

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Application: Moment of a Force About a Point Moment of a force produces a turning action on a

rigid body

The momentof a force F about O is defined as

FrMO

Engineers Mechanics- Review of Vector Algebra/Applications

r is the position vector of A from O

The moment vectorMO is perpendicular to the

plane containing O and the forceF.

Magnitude ofMO measures the tendency of the forceto cause rotation of the body about an axis alongMO.

dis the perpendicular distance of the line of action

of Force F from O. The sense of the moment may be

determined by the right-ha

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