CHAPTER 1: Physical quantities and measurements (3 Hours)

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CHAPTER 1Physical quantities and
measurements(3 Hours)
CHAPTER 1 PHYSICAL QUANTITIES AND MEASUREMENTS
UNIT FIZIK KOLEJ MATRIKULASI MELAKA
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Learning Outcome
1.1 Physical Quantities and Units (1 hours)

• At the end of this chapter, students should be
  able to
• State basic quantities and their respective SI
  units length (m), time (s), mass (kg),
  electrical current (A), temperature (K), amount
  of substance (mol) and luminosity (cd).
• ( Emphasis on units in calculation)
• State derived quantities and their respective
  units and symbols velocity (m s-1), acceleration
  (m s-2), work (J), force (N), pressure (Pa),
  energy (J), power (W) and frequency (Hz).
• State and convert units with common SI prefixes.

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1.1 Physical Quantities and Units

• Physical quantity is defined as a .
• It can be categorized into 2 types
• Basic (base) quantity
• Derived quantity
• Basic quantity is defined as .
• ..
• Table 1.1 shows all the basic (base) quantities.

Quantity Symbol SI Unit Symbol
Length l metre m
Mass m . kg
Time t second s
Temperature T/? kelvin K
Electric current I ampere ..
Amount of substance . mole mol
Table 1.1
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• Derived quantity is defined as a quantity which
  can be expressed in term of base quantity.
• Table 1.2 shows some examples of derived quantity.

Derived quantity Symbol Formulae Unit
Velocity v s/t m s-1
Volume .. l ? w ? t m 3
Acceleration a v/t m s-2
Density ? m/V .
Momentum p kg m s-1
Force m ? a kg m s-2 _at_ N
Work W F ? s .. _at_ J
Pressure P F/A N m-2 _at_
Frequency f 1/T s-1 _at_ ..
Table 1.2
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1.1.1 Unit Prefixes

• It is used for presenting larger and smaller
  values.
• Table 1.3 shows all the unit prefixes.
• Examples
• 5740000 m 5740 km 5.74 Mm
• 0.00000233 s 2.33 ? 10?6 s 2.33 ?s

Prefix Multiple Symbol
tera ? 1012 T
giga ? . G
mega ? 106 M
kilo ? 103 ..
deci ? 10?1 d
centi ? 10?2 c
milli ? 10?3 m
micro ? 10?6
nano ? ,,,,,,, n
pico ? 10?12 p
Table 1.3
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Example 1.1
Solve the following problems of unit
conversion. a. 15 mm2 ? m2 b. 65 km h?1 ? m
s?1 c. 450 g cm?3 ? kg m?3 Solution a. 15
mm2 ? m2 b. 65 km h-1 ? m s-1 1st
method
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2nd method c. 450 g cm-3 ? kg
m-3
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Follow Up Exercise
1. A hall bulletin board has an area of 250 cm2.
What is this area in square meters ( m2 ) ?
2. The density of metal mercury is 13.6 g/cm3.
What is this density as expressed in kg/m3
3. A sheet of paper has length 27.95 cm, width
8.5 cm and thickness of 0.10 mm. What is the
volume of a sheet of paper in m3 ?

• Convert the following into its SI unit
• (a) 80 km h1 ? m s1
• (b) 450 g cm3 ? kg m3
• (c) 15 dm3 ? m3
• (d) 450 K ? C

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Learning Outcome
1.2 Scalars and Vectors (2 hours)

• At the end of this chapter, students should be
  able to
• a) Define scalar and vector quantities,
• b) Perform vector addition and subtraction
  operations graphically.
• c) Resolve vector into two perpendicular
  components (2-D)
• Components in the x and y axes.
• Components in the unit vectors in Cartesian
  coordinate.

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Learning Outcome
1.2 Scalars and Vectors

• At the end of this topic, students should be able
  to
• d) Define and use dot (scalar) product
• e) Define and use cross (vector) product
• Direction of cross product is determined by
  corkscrew method or right hand rule.

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1.2 Scalars and Vectors

• Scalar quantity is defined as a quantity with
  magnitude only.
• e.g. mass, time, temperature, pressure, electric
  current, work, energy and etc.
• Mathematics operational ordinary algebra
• Vector quantity is defined as a quantity with
  both magnitude direction.
• e.g. displacement, velocity, acceleration, force,
  momentum, electric field, magnetic field and etc.
• Mathematics operational vector algebra

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1.2.1 Vectors

• Table 1.4 shows written form (notation) of
  vectors.
• Notation of magnitude of vectors.

Vector A
Length of an arrow magnitude of vector A
Direction of arrow direction of vector A
displacement velocity acceleration



v (bold)
a (bold)
s (bold)
Table 1.4
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• Two vectors equal if both magnitude and direction
  are the same. (shown in figure 1.1)
• If vector A is multiplied by a scalar quantity k
• Then, vector A is
• if k ve, the vector is in the same direction
  as vector A.
• if k -ve, the vector is in the opposite
  direction of vector A.

Figure 1.1
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1.2.2 Direction of Vectors

• Can be represented by using
• Direction of compass, i.e east, west, north,
  south, north-east, north-west, south-east and
  south-west
• Angle with a reference line
• e.g. A boy throws a stone at a velocity of 20 m
  s-1, 50? above horizontal.

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• Cartesian coordinates
• 2-Dimension (2-D)

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• 3-Dimension (3-D)

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Unit vectors
A unit vector is a vector that has a magnitude of
1 with no units.
Are use to specify a given direction in space.
i , j k is used to represent unit
vectors pointing in the positive x, y z
directions.
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• Polar coordinates
• Denotes with or signs.

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1.2.3 Addition of Vectors

• There are two methods involved in addition of
  vectors graphically i.e.
• Parallelogram
• Triangle
• For example

Parallelogram Triangle

O
O
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• Triangle of vectors method
• Use a suitable scale to draw vector A.
• From the head of vector A draw a line to
  represent the vector B.
• Complete the triangle. Draw a line from the tail
  of vector A to the head of vector B to represent
  the vector A B.

Commutative Rule
O
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• If there are more than 2 vectors therefore
• Use vector polygon and associative rule. E.g.

Associative Rule
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• Distributive Rule
• a.
• b.
• For example
• Proof of case a let ?? 2

O
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O
?
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• Proof of case b let ?? 2 and ?? 1

?
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1.2.4 Subtraction of Vectors

• For example

Parallelogram Triangle

O
O
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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 moving
  object.
• Vector A has a magnitude of 8.00 units and 45?
  above the positive x axis. Vector B also has a
  magnitude of 8.00 units and is directed along the
  negative x axis. Using graphical methods and
  suitable scale to determine
• a) b)
• c) d)
• (Hint use 1 cm 2.00 units)

Exercise 1
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1.2.5 Resolving a Vector

• 1st method

• 2nd method

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• The magnitude of vector R
• Direction of vector R
• Vector R in terms of unit vectors written as

or
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Example 1.2
A car moves at a velocity of 50 m s-1 in a
direction north 30? east. Calculate the component
of the velocity a) due north. b) due
east. Solution
or
a) b)
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Example 1.3
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
Vector x-component y-component

or
or
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Example 1.4
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.
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Solution
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Solution
Vector x-component y-component




Vector sum
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Solution The magnitude of the resultant force
is and Its direction is 162? from
positive x-axis OR 18? above negative x-axis.
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Exercise 2

• Vector has components Ax 1.30 cm, Ay 2.25
  cm vector has components Bx 4.10 cm, By
  -3.75 cm. Determine
• the components of the vector sum ,
• the magnitude and direction of ,
• the components of the vector ,
• the magnitude and direction of . (Young
  freedman,pg.35,no.1.42)
• ANS. 5.40 cm, -1.50 cm 5.60 cm, 345? 2.80
  cm, -6.00 cm
• 6.62 cm, 295?
• For the vectors and in Figure 1.2,
  use the method of vector resolution to determine
  the magnitude and direction of
• the vector sum ,
• the vector sum ,
• the vector difference ,
• the vector difference .
• (Young freedman,pg.35,no.1.39)
• ANS. 11.1 m s-1, 77.6? U think
• 28.5 m s-1, 202? 28.5 m s-1, 22.2?

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Exercise 2

• Vector points in the negative x direction.
  Vector points at an angle of 30? above the
  positive x axis. Vector has a magnitude of 15
  m and points in a direction 40? below the
  positive x axis. Given that
  , determine the magnitudes of and .
• (Walker,pg.78,no. 65)
• ANS. 28 m 19 m
• Given three vectors P, Q and R as shown in Figure
  1.3.
• Calculate the resultant vector of P, Q and R.
• ANS. 49.4 m s?2 70.1? above x-axis

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1.2.6 Unit Vectors

• notations
• E.g. unit vector a a vector with a magnitude of
  1 unit in the direction of vector A.
• Unit vectors are dimensionless.
• Unit vector for 3 dimension axes

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• Vector can be written in term of unit vectors as
  
• Magnitude of vector,

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• E.g.

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Example 1.5

• Two vectors are given as
• Calculate
• the vector and its magnitude,
• the vector and its magnitude,
• the vector and its magnitude.
• Solution
• a)
• The magnitude,

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b) The magnitude, c) The magnitude,
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1.2.7 Multiplication of Vectors

• Scalar (dot) product
• The physical meaning of the scalar product can be
  explained by
• considering two vectors and as shown
  in Figure 1.4a.
• Figure 1.4b shows the projection of vector
  onto the direction of
• vector .
• Figure 1.4c shows the projection of vector
  onto the direction of
• vector .

Figure 1.4a
Figure 1.4b
Figure 1.4c
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• From the Figure 1.4b, the scalar product can be
  defined as
• meanwhile from the Figure 1.4c,
• where
• The scalar product is a scalar quantity.
• The angle ? ranges from 0? to 180 ?.
• When
• The scalar product obeys the commutative law of
  multiplication i.e.

scalar product is positive
scalar product is negative
scalar product is zero
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• Example of scalar product is work done by a
  constant force where the expression is given by
• The scalar product of the unit vectors are shown
  below

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Example 1.6
Calculate the and the angle ?
between vectors and for the following
problems. a) b) Solution a) The
magnitude of the vectors The angle ? ,
ANS.?3 99.4?
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Example 1.7
Referring to the vectors in Figure 1.5, a)
determine the scalar product between them. b)
express the resultant vector of C and D in unit
vector. Solution a) The angle between vectors C
and D is Therefore
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b) Vectors C and D in unit vector
are and Hence
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• Vector (cross) product
• Consider two vectors
• In general, the vector product is defined as
• and its magnitude is given by
• where
• The angle ? ranges from 0? to 180 ? so the
  vector product always positive value.
• Vector product is a vector quantity.
• The direction of vector is determined by

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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.
• Direction of the vector product
  always perpendicular
• to the plane containing the vectors and
  .

but
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• The vector product of the unit vectors are shown
  below
• Example of vector product is a magnetic force on
  the straight conductor carrying current places in
  magnetic field where the expression is given by

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• The vector product can also be expressed in
  determinant form as
• 1st method
• 2nd method
• Note
• The angle between two vectors can only be
  determined by using the scalar (dot) product.

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Example 1.8
Given two vectors Determine a) and its
magnitude b) c) the angle between vectors
and . Solution a) The magnitude,
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b) c) The magnitude of vectors,
.. . Using the scalar (dot)
product formula,
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Exercise 3

• If vector and vector ,
  determine
• a) , b) , c)
  .
• ANS.
• Three vectors are given as follow
• Calculate
• a) , b)
  , c) .
• ANS.
• If vector and
  vector ,
  determine
• a) the direction of
• b) the angle between and .
• ANS. U think, 92.8?

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