chapter 2_ scalar and vector.pdf

7/25/2019 chapter 2_ scalar and vector.pdf

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Scalar and Vector

1

CHAPTER 2

PHY 130


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LESSON OUTCOMESAfter completing this chapter, you should be able to

O state the definition of scalar and vector quantities and

give examples of each.

O state the differences between scalar & vector quantities.O determine the components of a given vectors

O find the resultant of two or more vectors.

O calculate vector addition & subtraction by using

geometrical method & unit vector method / resolvingmethod.

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OScalarquantity : quantity with magnitude only.

O e.g. mass, time, temperature, pressure, electric

current, work, energy and etc.

O Mathematics operational : ordinary algebra

O Vector quantity:quantity with both magnitude

direction.

O e.g. displacement, velocity, acceleration, force,

momentum, electric field, magnetic field and etc.

O Mathematics operational : vector algebra

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VE TORS

s

4

O Written form (notation) of vectors.

O Notation of magnitude of vectors.

Vector ALengthof an arrowmagnitudeof vector A

displacement velocity acceleration

v

a

s av

aa

s (bold) v (bold) a (bold)

Directionof arrow directionof vector A


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P

5

O Two vectorsequal if both magnitude and directionare the same.

O If vector A is multiplied by a scalar quantity kO Then, vector A is

O if k= +ve, the vector is in the same directionas vector A.

O if k = - ve, the vector is in the opposite directionof vector A.

Q

QP

Ak

Ak

A

A


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6


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Addition of Vectors

O There are two methods involved in addition of vectors graphically i.e.

OParallelogram

O Triangle

O For example :

7

Parallelogram Triangle

B

A

B

A

BA

O

BA

B

A

BA

O


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Subtraction of Vectors

O For example :

8

Parallelogram Triangle

DC

O

DC

O

D

DCDC

C

D

DC

C

D

DC


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Resolving a Vector

9

yD

xD

0x

y

D

Dx cos DDx

cos

D

Dysin DDy sin


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10


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11


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12


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O The magnitude of vector D:

O Direction of vectorD:

O VectorDin terms of unit vectors written as

13

or


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14

A car moves at a velocity of 50 m s-1in a direction north 30

east. Calculate the component of the velocity

a) due north. b) due east.

Solution :

Example :

N

EW

S

Nv

Ev

v30

60

a)

b)

30vvN cos

1sm43.3 Nv

3050vN cos

or60vvN sin

6050vN sin

30vvE sin

1sm25 Ev

3050vE sinor

60vvE cos

6050vE cos


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15

A particle S experienced a force of 100 N as shown in figure above. Determine

the x-component and the y-component of the force.

Solution :

Example :

120

F

Sx

12060

F

Sx

y

y

F

xF

Vector x-component y-component

60FFx cos

N50xF

60100Fx cos

orF

120FFx cos

N50x

F

120100Fx cos

60FFy sin

N86.6yF

60100Fy sin

or120FFy sin

N86.6

yF

120100Fy sin


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16

The figure above shows three forces F1, F2and F3 acted on a particle O.

Calculate the magnitude and direction of the resultant force on particle O.

Example : y

45o

O

)( N30F2

)( N10F1

30o x

)( N40F3

20


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Solution :

O

y

x

3F

45

o

30o

20

1F

y1F

2F

y2F

x1F

y3F

x3F

321r FFFFF

yxr FFF

x3x2x1x FFFF

y3y2y1y FFFF

x2F


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Solution :

Vector x-component y-component

20FF 1x1 cos

1F

3F

2F

2010Fx1 cosN9.40x1F

20FF 1y1 sin2010Fy1 sin

N3.42y1F4530Fx2 cos

N21.2x2F

4530Fy2 sinN21.2y2F

3040Fx3 cos

N34.6x3F

3040Fy3 sin

N20.0

y3F

Vector

sum

34.621.29.40 xFN4.00 xF

20.021.23.42 yFN37.8 yF


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19

y

xO

Solution :

The magnitude of the resultant force is

and its direction is

22 yxr FFF

N38.0rF

22 37.84.00 rF

x

y

F

F

1tan

iseanticlockwaxis-xpositivefrom264or84.0

4.00

37.8tan

1

rF

yF

xF

84.0

264


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Unit Vectors

O notations

O has a magnitude of one with no units,

O for any positional vector A, its unit vector is given by

O Unit vector for 3 dimension axes :

20

A

a

cba ,,

A

a

A

)(@- boldjjaxisy

)(@- boldiiaxisx

)(@- boldkkaxisz

1a


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ab

m542 kjia

21

Two vectors are given as:

Calculate

a) the vector and its magnitude,

b) the vector and its magnitude,

c) the vector and its magnitude.

Solution :

a)

The magnitude,

Example :

ba

m87 kjib

ibaba xxx572

jbaba yyy

484

m645 kjiba

kbaba zzz 615

m8.78645222

ba

ba

2


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22

b)

The magnitude,

c)

The magnitude,

iabab xxx927

jabab yyy

1248

m4129 kjiab

kabab zzz 451

m15.54129 222 ab

ibaba xxx 372222

jbaba yyy 084222

m1132 kiba

kbaba zzz 1115222

m11.41132

22 ba

chapter 2_ scalar and vector.pdf

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