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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.
2
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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
3
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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