mechanics lecture 2

8/9/2019 Mechanics Lecture 2

1/20

Force vectors

Scalars and vectors

2D and 3D force systems


2/20

Scalars and vectors

Scalar any positive or negative physical quantity that

can be completely specified by its magnitude.

Examples: mass lenght time.

Vector any physical quantity that requires both a

magnitudeand a directionfor its complete description.

Examples: force position moment.


3/20

Scalars and vectors

!raphical representation of a vector by an arro".

Magnitudeof the vector given by the length of the arro".

Directionof the vector#s line of action given by the angle bet"een

the vector and a fixed axis. Sense of direction of the vector given by the head of the tip of the

arro".

Direction$agnitude

Sense


4/20

%ector operations

$ultiplication and division of a vector by a scalar

&f a vector is multiplied by a positive scalar its

magnitude is increased by that amount.

'hen multiplied by a negative scalar it "ill alsochange the directional sense of the vector.

V

VV2

V.50


5/20

%ector addition

(ll vector quantities obey the parallelogram la" of addition.

)rocedure description:

*irst +oin the tails of the components at a point so that it ma,es

them concurrent.

-ring to intersection the 2 lines dra" as parallel lines to eachindividual vector to form the ad+acent sides of a parallelogram.

he diagonal of this parallelogram "ill give the resultant vector.

1V1V

1V

2V2V 2V

R

21 VVR +=


6/20

%ector addition

Triangle rule a special case of the parallelogram la"

he vectors are being added in a head-to-tail fashion by

connecting the head of the first "ith the tail of the second.

he resultant extends from the tail of the first vector to the headof the second.

&t is easy to see that vector addition is commutative.

1V

2VR

1V 2VR

1V2V

21 VVR +=

12 VVR +=


7/20

%ector addition

(s a special case if t"o vectors are collinear i.e. both have

the same line of action the parallelogram la" reduces to an

algebraicor scalar addition.

1V 2V

R

21 VVR +=


8/20

(pplication

he truc, is to be to"ed using t"o ropes. Determine the

magnitude of forces acting on each rope in order to

develop a resultant force of /01 directed along the

positive axis x. Set the angle as 014.


9/20

%ector substraction

5an be defined as a special case of addition so the rules of

vector addition also apply to vector substraction.

( )21 VVR +=

1V

1V

1V2V 2V

2V

R

R

)arallelogram la"

%ector substraction

riangle construction


10/20

Vector addition of forces

5ommon problems in statics:

finding the resultant force ,no"ing its components

resolving a ,no"n force into 2 components

he parallelogram la" must be

used to determine the resultantof the t"o forces acting on the

hoo,.


11/20

Finding the resultant force

6ne can use either parallelogram la" or triangle rule.

6ne has to apply the la" of cosines or the la" of sines to the

triangle in order to obtain the magnitude of the resultant force and

its direction.

2F

1F

1F 2F

RF

21 FFFR +=


12/20

(pplication

&f 7314 and 70 , determine the

magnitude of the resultant force acting

on the eyebolt and its direction

measured cloc,"ise from the posite x

axis.


13/20

Finding the components of a force

( force is to be resolved in 2 components along the axes of a 2D

coordinate system.

&n order to determine the magnitude of each component a

parallelogram is constructed first by dra"ing lines starting from the tip of

force parallel to each axis.

he force components are established by simply +oining the tail of the

force to the intersection points on the individual axes.

he parallelogram can then be reduced to a triangle and the la" of

sines can be applied to determine the un,no"n magnitude of the

components.

u

v

F

u

v

F vF

uF

F


14/20

Application

8esolve the hori9ontal 11 into components acting

along the t"o axis and determine the magnitudes of

these components.


15/20

Addition of a system of coplanar forces

'hen a force is resolved into 2 components along the

x and y axis are called rectangular components.

Scalar notation

F

x

y

xF

yF

cosFFx =

sinFFy =


16/20


5artesian vector notation

x

y

xF

yFi

j

jFiFF yx +=


17/20


&n order to determine the resultant of several coplanar forces

one has to resolve each force into its x and y components

and then to add the respective components using scalar

algebra.

1F

3F

2F jFiFF yx += 111

jFiFF yx += 222

jFiFF yx = 333


18/20


he vector resultant is:

( ) ( )

yyyRy

xxxRx

RyRx

yyyxxxR

FFFF

FFFF

jFiF

jFFFiFFFFFFF

321

321

321321321

+=

+=

+=

+++=++=


19/20


'e can represent the components of the resultant force of any

number of coplanar forces symbolically by the algebraic sum of

the x and y components of all forces:

he magnitude of the resultant force "ill be given by:

he direction of the resultant force "ill be given by:

Rx

Ry

F

Farctg=

22RyRxR FFF +=

==

yRy

xRx

FFFF

RF

x

y


20/20

Application

Determine the magnitude of and and its direction

so that the resultant force is directed along the

positive x axis and has a magnitude of ;201 .

AF

mechanics lecture 2

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