physics chapter 1 2014
TRANSCRIPT
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1. 1.
1.1 1.1
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.
)
: (), (), (), (), (),
(), ().
)
: ( 1),
( 2), (), (), (),
(), () ().
) .
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2
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1.1 ()
Quantity Symbol SI Unit Symbol
Length l metre m
Time t second s
Temperature T/ kelvin K
Electric current I ampere A
Amount of substance N mole mol
Luminous Intensity candela cd
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Derived quantity Symbol Formulae Unit
Velocity v s/t m s-1
Volume V l w t M3
1.2 .
Acceleration a v/t m s-2
Density m/V kg m-3
Momentum p m v kg m s-1
Force F m a kg m s-2 @ N
Work W F s kg m2 s-2 @ J
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Unit Prefixes
It is used for presenting larger and smaller values.for presenting larger and smaller values. Table 1.3 shows all the unit prefixes.
Prefix Value Symbol
tera 1012 T
giga 109 G
mega 106 M
kilo 103
kdeci 10-1 d
centi 10-2 c
milli 10-3 m
micro 10-6
nano 10-9 n
pico 10-12 p
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:
Solve the following problems of unit conversion
a. 30 mm2 = ? m2 b. 865 m h-1 = ? m s-1
c. 300 g cm-3 = ? kg m-3 d. 2.4 x 10-5 cm3 = ? m3
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Solution :Solution :
(a) 30 mm2 = ? m2
( ) ( )232 m10mm1 =262 m10mm1 =
25262 m103.0orm1030mm30 =
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()865 m h
-1
= ? m s-1
11stst methodmethod
=
h1
m10865hkm865
31
=
s3600m10865hkm865
3
1
11 sm240hkm865 =
22ndnd methodmethod
=
s3600
h1
km1
m1000
h1
km865hkm865 1
=
s3600
h1
km1
m1000
h1
km865
hkm865
1
11 sm240hkm865 =
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(c) 300 g cm-3
= ? kg m-3
() 2.4 x 10-5 cm3 = ? m3
( )
=332-
33-
3
3-
m10
cm1
g1
kg10
cm1
g300cmg300
-353 mkg103.0cmg300 =
( )
311
65
325
m104.2
10104.2
10104.2
=
==
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1.21.2 Scalars and Vectors
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a) Define scalar and vector quantities.
b) Perform vector addition and subtraction
operations graphically.
c eso ve vec or n o wo perpen cu arcomponents (x and y axes).
d) Illustrate unit vectors ( ) in Cartesian
coordinate.
, ,i j k
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) ():
) ()
)cos()cos( ABBABA ==
rr
:
.
)sin()sin( ABBABA == rr
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ScalarScalarquantity is defined as a quantity withquantity with
magnitudemagnitude only.
e.g. mass, time, temperature, pressure, electriccurrent, work, energy and etc.
Mathematics operational : ordinary algebra
VectorVector quantity is defined as a quantity with bothquantity with both
magnitude & direction.magnitude & direction.
e.g. displacement, velocity, acceleration, force,
momentum, electric field, magnetic field and etc.
Mathematics operational : vector algebra
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1.6 () .
Vector A LengthLength of an arrow magnitudemagnitude of vector A
displacement velocity acceleration
DirectionDirection of arrow directiondirection of vector A
15
.
sr vr ar
s av
vv =r
aa =r
s (bold)v (bold)
a (bold)
Table 1.6Table 1.6
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r
Qr
QP
rr
=
16
k ,
kk , .
kk , .
Akr
Akr
r
r
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..
ParallelogramParallelogram TriangleTriangle
:
Addition of Vectors
Brr
+
17
ParallelogramParallelogram TriangleTriangle
rr
r
r
Brr +
O
r
r
Brr +
O
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Triangle of vectors method:
a)Use a suitable scale to draw vector A.
b)From the head of vector A draw a line to
represent the vector B.c) Complete the triangle. Draw a line from the tail of
vector A to the head of vector B to represent the
18
vec or .
BBrrrr
+=+ Commutative RuleCommutative Rule
Br
r
B
rr
+O
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If there are more than 2 vectors therefore
Use vector polygon and associative rule. E.g. RQPrrr
++
rQr
r
19
r
Qr
r QPrr
+
RQPRQP
rrrrrr
++=++ Associative RuleAssociative Rule
RQPrrr
++
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:.
.
:
BABArrrr
+=+
( ) AAA
rrr
+=+
numberrealare,
20
BABArrrr
+=+ 2
r
r
Brr
+
O BArr
+2
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Br
2
Brr
22 +
BABA rrrr 22 +=+
21
r
2O
BABA rrrr 222 +=+
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r
( ) ( ) AAArrr
312 =+=+
r
22
r
3
AAAA rrrr 12 +=+r
2r+
=( ) AAA
rrr
1212 +=+
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:1.2.4
rCr
DCrr
r
DCDCrrrr
+=
23
ParallelogramParallelogram TriangleTriangle
O OCr
Dr
Crr
Cr
r
DCrr
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Vectors subtraction can be used
to determine the velocity of one object
relative to another object i.e. to determine
the relative velocity.
to determine the change in velocity of a
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.
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:Resolving a Vector
Dr
yDr
y :
Dr
yDr
y
25
Dr
0x
Dx cos= DD cos=
Dy
sin= DDy sin=
Dr
0x
= sinDx = sinDDx
= cosDy = cosDDy
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D:
D:
( ) ( )2y2
x DDDD +=orr
yD= = y
D1
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D
xD xD
jDiDD yx +=
r
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Example 1 : y
45o
)( N30F2r
)( N10F1r
x20
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The figure above shows three forces F1, F2 and F3 acted on a particle
O. Calculate the magnitude and direction of the resultant force on
particle O.
)( N40F3r
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Solution :Solution :
O
y
x45o30o
20
1F
r
y1F
r
x1Fr
x3Fr
x2Fr
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3Fr
2Fr
y2Fr y3
Fr
++== 321r FFFFF
rrrrr
+= yxr FFFrrr
x3x2x1x FFFFrrrr
++=y3y2y1y FFFF
rrrr
++=
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Solution :Solution :
Vector x-component y-component
o20FF 1x1 cos=1F
r
r
o2010Fx1 cos=N9.40=x1F
o20FF 1y1 sin=o
2010Fy1 sin=N3.42=y1F
o= o4530F si=
29
3Fr
2F N21.2=x2F N21.2=y2Fo3040Fx3 cos=
N34.6=3F
o3040Fy3 sin=N20.0=y3F
VectorVector
sumsum
( )34.621.29.40 ++= xFN4.00= xF
( ) ( )20.021.23.42 ++= yFN37.8= yF
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y
Solution :Solution :The magnitude of the resultant force is
( ) ( )22 += yxr FFF
N38.0=rF
( ) ( )22 37.84.00 +=rF
o264
30
xO
and its direction is
=
x
y
F
F 1tan
( )iseanticlockwaxis-xpositivefrom264or84.0 oo
=
= 4.00
37.8tan 1
rFr
y
Fr
xF
r
84.0
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(3)
3
m2)3,4,(),,(),,( === kjizyxsr
y/m
31
sr
2
4x/m
z/m
0
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Scalar (dot) productScalar (dot) product The physical meaning of the scalar productphysical meaning of the scalar product can be explained by
considering two vectors and as shown in figure 1.3a.
r r
r
r
r
Figure 1.3aFigure 1.3a
32
Figure 1.3b shows the projection of vector onto the direction of
vector .
Figure 1.3c shows the projection of vector onto the direction of
vector .
r
r
r
r
r
cosFigure 1.3bFigure 1.3b
r
r
cos
Figure 1.3cFigure 1.3c
ABABArrrr
toparallelofcomponent=
BABBA
rrrr
toparallelofcomponent=
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From the figure 1.3b, the scalar product can be defined as
meanwhile from the figure 1.3c,
where
( )BABA cos= rr
vectorsobetween twangle:
( )ABAB cos= rr
33
.
The angle ranges from 0 to 180 . When
The scalar product obeys the commutative law of multiplicationcommutative law of multiplication i.e.
oo 900
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Calculate the and the angle between vectors and for thefollowing problems.
a)
Example 1 : rrr
r
kjiA 32 +=r
kjiB 52 +=
r
35
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SolutionSolution ::a)
The magnitude of the vectors:
( )( ) ( )( ) ( )( ) kkjjiiBA 531221 ++= rr
1522 =BA
rr
19= rr
36
The angle ,
( ) ( ) ( ) 14321222
=++=A( ) ( ) ( ) 30512 222 =++=B
ABBA cos=
rr
=
=
3014
19coscos 11
AB
BA
rr
o
158=
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Vector (cross) productVector (cross) product
Consider two vectors :
In general, the vector product is defined as
and its magnitudemagnitude is given by
krjqipB ++=r
kzjyixA ++=
r
CBArrr
=
ABBACBA sinsin === rrrrr
37
where
The angle ranges from 0 to 180 so the vector product always
positivepositive value.
Vector product is a vector quantityvector quantity.
The direction of vector is determined by
vectorsobetween twangle:
RIGHTRIGHT--HAND RULEHAND RULE
Cr
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For example: How to use right hand rule :
Point the 4 fingers to the direction of the 1st vector.
Swept the 4 fingers from the 1st vector towards the 2nd vector.
The thumb shows the direction of the vector product.
Cr
rCBArrr
=
38
Direction of the vector product always perpendicular to the
plane containing the vectors and .
r
Br
r
Cr
CABrrr
=
rrrr but ( )ABBArrrr =
Br
)(Cr
r
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The vector product of the unit vectors are shown below :
x
y
kj
i
ijkkj
==
kijji ==
jkiik ==
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Example of vector product is a magnetic force on the straighta magnetic force on the straight
conductor carrying current places in magnetic fieldconductor carrying current places in magnetic field where the
expression is given by
z0 === kkjjii
0in == o2
0siii
0in == o2 0sjjj
0in == o2 0skkk
BlIFrrr
=
IlBF sin=
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The vector product can also be expressed in determinant form as
11stst method :method :
rqp
zyx
kji
BA
= rr
rr
40
22ndnd method :method :
Note :Note :
The angle between two vectorsThe angle between two vectors can only be determined by
using the scalar (dot) product.scalar (dot) product.
zz =
( ) ( ) ( )kypxqjxrzpizqyrBA
++=
rr
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Given two vectors :
Determine
a) and its magnitude
b)
Example 2:
B
rr
Brr
kjiA 425 +=r
kjiB 5 ++=r
r r
41
c) the angle between vectors and .
B
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SolutionSolution ::a)
511
425
=kji
BArr
42
The magnitude,
( )( ) ( )( )( ) ( )( ) ( )( )( ) ( )( ) ( )( )( )kjiBA 121514551452 += rr
kjiBA 72141 +=
rr
( ) ( ) ( )kjiBA 25425410 ++= rr
2.26=BArr
( ) ( ) ( )222 72114 ++=BArr
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b)
c) The magnitude of vectors,
kjikjiBA 5425 +++= rr
23=Brr
( )( ) ( )( ) ( )( ) kkjjiiBA 541215 ++= rr
2025 +=BArr
222
43
Using the scalar (dot) productscalar (dot) product formula,
ABBA cos=
rr
=
=
2745
23coscos 11
AB
BA
rr
o
7.48=
==
( ) ( ) ( ) 27511 222 =++=B