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Lesson 1 Division of Science
BS Civil Engineering (Tarlac State University)
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CHAPTER 1
DIVISION OF SCIENCE
Mathematics







Arithmetic
Plane and Analytic Geometry
Statistics
Algebra
Trigonometry
Differential and Integral Calculus
Differential Equation
Logic
 Inductive
 Deductive
Physical Science





Physics
Geology
Astronomy
Chemistry
Meteorology
Life Science




Anatomy
Physiology
Microbiology
Pathology
Social Science





Sociology
Psychology
Political Science
Economics
Anthropology
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PHYSICS
Wave
 Acoustic “sound”
 Optics “light”
Electromagnetism
 Electricity
 Magnetism
Mechanics
 Motion
Life Physics
Thermodynamics
 Heat
 Fluid
Atomic Physics
SIGNIFICANT FIGURES
 All non-zero digits are significant.
Ex. numbers 1, 2, 3, 4, 5, 6, 7, 8 & 9
 Zeros in between non-zero digits are significant
Ex. 105 – 3
20008 – 5
 Zeros to the right of a non-zero digits in unexpressed decimal point are not
significant
Ex. 200 - 1
 Zeros at the right of a non-zero digit in an express decimal point are significant.
Ex. 200.00 = 5
 Zeros at left of a non-zero digit but to the right of a decimal point are not
significant.
Ex. 0.0000001 – 1
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Rounding Off
 When the number to be rounded off is less than 5 change that number to zero “0”
to retain the preceding number.
Ex. 12.34 – 12
 When the number to be rounded off is greater than or equivalent to 5 change that
number to zero “0” and add one to the preceding number.
Ex. 98.76 – 99
SCIENTIFIC NOTATION
 It simplifies the writing, reading and computation of very large and small number
thereby decreasing the risk of errors.
Exponential
 Uses the power of ten
Writing as:
Where:
N = any given number
a = number having single non-zero digit to the left of a decimal point and
two decimal places.
RULES IN SIGNIFICANT NUMBER IN SCIENTIFIC NOTATION
a. Determine “a” – by shifting the decimal point of the original number to the left or
right, until one digit is to the left of it.
b. Determine “b” – by counting the number of decimal places the point has moved,
if it has been to the left “b” is positive if to the right “b” is negative.
Ex.
123456
0.0009876
–
1.23 x 105
9.88 x 10-4
left - +
right - -
BASIC CONCEPTS IN PHYSICS
Matter = Anything that occupied space and has mass.
Mass = Quantity of matter
Force = It is capable of changing the condition of rest or motion of a certain body.
Physical Quantity = Any number used to describe a physical phenomenon.
Ex. Height 5’2 – Quantity
2 Ways of Describing a Physical Quantity
a. Direct – which means measuring
b. Indirect – calculating, gather information
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Operational Definitions = Definitions that gives of procedure of measuring the quantity.
BASIC MATHEMATICAL OPERATIONS
1. Algebraic Methods
 Solving an unknown of a given operation
 Trigonometric functions and law
 Analytical method – interprets through graphs
Units and Problem Solving
Important Terms:
Unit = a quantity in terms of which another quantity is used.
Standard = a unit established to have a precise definition of a unit.
Standard Unit = if a unit becomes officially accepted
System of Units = a group of standard units and their combinations
International System of Units = the modernized version of the metric system which
includes the base quantities and derived quantities
SI Base Units = represented by standards
SI Derived Units = other quantities that may be expressed in terms of combinations of
the base units
Meter (m) = the SI unit for length, 1/10,000,000 of the distance from the North Pole to
the Equator along the meridian running through Paris
Kilogram (kg) = The SI unit of mass, specific volume of water but is now referenced to a
specific material standard The mass of a cylinder of a platinum, iridium, alloy, kept at the
international Bureau of Weights and Measurements in Paris, France
Second (s) = the SI unit of time, time required for cesium 133/133 CS to undergo
9192631770 vibration
MKS System = meter-kilogram-second
CGS System = centimeter-gram-second, Gaussian System
FPS System = foot-pounds-second, British Engineering System
Liter (L) = the nonstandard unit of volume
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Dimensional Analysis = a procedure by which the dimensional consistency of any
equation may be checked
Unit Analysis = using units instead of symbols in dimension analysis
Density (ρ) = ratio of mass to the volume
Conversion Factor = equivalent statements expressed in the form of ratios
Exact Number = those without any uncertainty or error
Measured Number = obtained from measurement processes and so generally have some
degree of uncertainty or error
Significant Figures (sf) = the number of reliably known digits it contains
Important Equations:
Density
ρ = m/V
Where:ρ = density
m = mass
V = volume
PREFIXES
21
Zetta x 10
Exa x 1018
Peta x 1015
Tera x 1012
Giga x 109
Mega x 106
Kilo x 103
Hecto x 102
Deca x 101
Meter/Gram/Liter/Pasca
l x 100
1m =
1m =
1in =
1ft =
1 yd =
1 mi =
1 mi =
1 km =
100
100 cm
3.28 ft
2.54 cm
12 in
3 ft
5280 ft
1.609 km
0.62 mi
Zepto x 10-21
Atto x 10-18
Femto x 10-15
Pico x 10-12
Nano x 10-9
Micro x 10-6
Milli x 10-3
Centi x 10-2
Deci x 10-1
1 kg =
1 hr =
1 min =
1 mL =
1L =
1 kips =
2.2 lb
60 min
60 sec
1 cc
1000 ml
1000 lbs
212
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o
C
o
1
32
F
K = 273 + oC
R = 460 + oF
Example 1:
1.
40 km/hr ® ft/sec =
2.
500g ® lbs
3.
5 ft 5 in ® m
4.
4000 mi → km
5.
1 metric ton to kilograms
N = +Y
S(-x, y)
W = -x
A(x, y)
E = +x
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T(-x, -y)
C(x, -y)
S = -y
Note: The direction will just be represented at the angle from any two of the fixed axes
which the vector is enclosed.
Resolution of Vectors
- Graphical Method
- Rules in determining the magnitude and directions of “R”
1. Construct an accurate Cartesian plane.
2. Plot the first force using a given scale.
3. At the end of the first force construct another accurate C.P. and make sure that it is
parallel to the first C.P.
4. Plot the second force using the new Cartesian Plane and so on so forth.
5. Using an arrow, connect the tail of the first force to the head of the last force and
label it “R” for resultant
6. To determine the magnitude of R, simply measure the length of R using the given
scale.
7. To determine the direction of R, simply measure the angle with respect with the Y
- axis.
Resultant Vectors
- It is a vector whose effect is the same as all the component vectors put
together.
Case I: Two or more forces acting on an object in the same direction
Ex. Given
F1 = 10N due E
F2 = 20 N due E
Scale 10 N = 1 cm
F1 = 10N/10 = 1cm
F2 = 20 N/10 = 2cm
R = 3cm due E
R = 30N, due E
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Case II: Two or more forces acting on an object on opposite directions
F1 = 40N due E
F2 = 60N due W
Scale: 20N = 1cm
F1 = 40N/20N = 2cm
F2 = 60N/20N = 3cm
R = 1cm, due W
R = 20N, due W
Case III: Two forces acting on an object perpendicular to each other.
F1 = 30N due E
F2 = 40N due S
Scale: 10N = 1cm
F1 = 30N/10N = 3cm
F2 = 40N/10N = 4cm
R = 5cm, S37oE
R = 50N
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Case IV: Two or more forces acting on an object at random directions.
F1 = 40N due E
F2 = 40N due E
F3 = 40N NE
Scale: 20N = 1cm
F1 = 40N/20N = 2cm
F2 = 40N/20N = 2cm
F3 = 40N/20N = 2cm
R = 5.8cm, N76oE
R = 116N
Ex.
F1 = 50N 50o E of S = 50N, S50E
F2 = 50N 30o W of N=50N, N30W
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Scale: 20N = 1cm
F1 = 50N/20N = 2.5cm
F2 = 50N/20N = 2.5cm
R = 0.95cm, N48oE
R = 19N,
ANALYTICAL METHODS
Case I: Just add and follow the direction of the forces.
F1 = 10 N due E
F2 = 30 N due E
R = 40 N due E
Case II: You just subtract and follow the directions of greater value.
F1 = 40N
due E
F2 = 60N due W
R = 20N due W
Case III: Use Pythagorean Theorem and tangent.
Example:
F1 = 30N due E
F2 = 40N due S
= S 36.8 E
= 36.87 E of S
Case IV: Construct a Cartesian plane without a need of a scale, plot the given vectors in
the Cartesian plane and use component method or sine and cosine law.
Ex.1: sine and cosine law
F1 = 50N 50oE of S
F2 = 50N 30o W of N
50N
50
20
30
50N
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40
= 17.36 N
Forces
50N
50N
Direction
S50E
N30W
F
50Sin50 =
50Sin30 =
Solution:
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Inclined Axis
F1 = 50#,1:2 in Q4
F2 = 100#, 3:1 in Q1
F3 = 155#, 3:2 in Q2
y : x = rise is to run
x
y
100#
3
50#
1
155#
3
Forces #
50
100
155
1
2
2
Fx
Fy
-9.63
+201.48
3 DIMENSIONAL
Components:
Slope:
Direction:
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Given:
x=3
y=4
z=5
F = 100#
Req’d.: Components & Directions
Solution:
Coordinate = head - tail
x = right or left
y = upward or downward
z = forward or backward
Example:
From (0, 12, 0)
P = 280# ® (-4, 0, 6)
F = 210# ® (6, 0, 4)
T = 260# ® (-4, 0, -3)
Required:
a. Resultant
b. Components
c. Directions.
Y
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X
Z
1
Solution:
force
x
280
-4
260
-4
210
6
y
-12
-12
-12
z
6
-3
4
d
14
13
14
fm
20
20
15
fx
-80
-80
90
-70
84.04
fy
-240
-240
-180
-660
11.89
fz
120
-60
60
120
79.75
Unit Vectors = A unit vector is vector with magnitude of unit. Its purpose is only to show
direction in space.
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SUM OF UNIT OF VECTORS
A = Axi + Ayj
B = Bxi + Byj
Let C = SUM OF A & B
C =A+ B
= (Axi + Ayj) + (Bxi + Byj)
= (Ax + Bx)i + (Ay + By)j
C = Cxi + Cyj
Mag of C =
Cx2 + Cy2
Difference of Two Vectors
Let D = A – B
= (Axi + Ayj) – (Bxi + Byj)
= (Ax - Bx)i + (Ay – By)j
D = Dxi + Dyj
Mag of D =
Dx2 + Dy2
Products of Vectors
I. Scalar Product (Dot Product)
The Scalar product of 2 L Vector is Zero.
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A.B = AB Cos Æ
Cos 0 = 1
A.B = Ax Bx + Ay By + Az Bz
Note
=
ixi=1
jxj=1
kxk=1
ixj=0
ixk=0
kxj=0
II. Vector Product of 2 Vectors is zero when they are parallel.
A x B = AB sin Æ
AxB
AxB=
Ax
Bx
i
Ax
Bx
Ay
By
j
Ay
By
Az
Bz
k
Az
Bz
(Ay Bz – Az By)i + (Az Bx – Ax Bz – Ax Bz) jt
(Ax By – Ay Bx) k
Example:
Mag of A
Mag of B
Mag of C
Mag of D
Dot Product
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Cross Product =
= (-3k – 10j – 4i) – (-15i + 2j + 4k)
A · B = AB
Name:
Course/Year/Section:
A. Count the number of significant figures and place the answer before the number and
transform the following to Scientific Notation
1. 0.000567
–
2. 6705001
–
3. 0.00090350 –
4. 84.650
–
5. 0.01425001 –
B. Perform the following operations. Final answer should be in scientific notation.
1.
0.03451 x 250 – 670.8
=
705 + 96.20 ¸ (0.35 x 0.00065)
77237
2.
(15430.0 4x 0.052) – 600
=
0.00705 + 6208 ¸ (3550 x 0.00015)
3.
(3165 x 5.35) – 0.0002643
=
905000 + 0.0051 ¸ (0.46 x 0.0006708)
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C. Convert the following.
1. 78.8 in to m
2. 1.53 m to ft
3. 1.8 lbs to kg
4. 9500 mg to kg
5. 4520 cm/s to ft/s
6. 2.4g to cg
7. 45.2 oC to oF
8. 77 oF to C
9. 20 mi/hr to m/s
10. 1120 oF to oK
11. 97.75 oK to C
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12. -1 oC to K
13. 87.9 oF to C
14. \
15. 5oF to R
D.1.
F1 = 89N, 73o W of N
F2 = 77N, N 40o E
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Analytical
Forces
Direction
Fx
Fy
R=
2. F1 = 157# 46 E of S
F2 = 175# 56 E of N
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Analytical
Forces
Direction
Fx
Fy
R=
3. F1 = 155, N 68o E
F2 = 196N, N 72 W
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Analytical
Forces
Direction
Fx
Fy
R=
4.
F1 = 70N due W
F2 = 150N due N
F3 = 20N due E
F4 = 30N due S
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Analytical
Forces
Direction
Fx
Fy
R=
E.
From (3,6,-1)
A 100KN(-5,0,4)
B 200KN(4,0,-5)
C 300KN(2,0,1)
To
Y
X
Z
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Force
x
y
z
d
fm
fx
fy
fz
R=
F.
A = -3i + 5j – 6k
B = 7i + 8j – 3k
Mag of A
A=
Mag of B
B=
C=
Mag of C
C=
D=
Mag of D
D=
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Dot Product
A.B =
Cross Product
AXB =
Mag of AXB =
q=
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