Chapter 0
Math Review
Please review the following math subjects
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Exponents
Scientific Notation and Powers of 10
Algebra
Solving Quadratic Equations
Quadratic Formula
Solving Two Equations with Two Unknown Variables
Direct, Inverse, and Inverse-Square Relationships
Graph of Proportionality Relationship
Data-Driven Problems
Linearizing the Data
Find slope and Intercept
Logarithmic and Exponential Functions
Areas and Volumes
Plane Geometry and Trigonometry
Chapter 1
1.1
Models, Measurements, and Vectors
Introduction: Physics is based on experimental observations
● Observe the phenomena of nature
● Discover patterns → principles and theories
● With principles and theories → make predictions
● With predictions → carry out new observations
Theories having very broad impact → Physical Laws
What does physicists do?
● Ask appropriate questions
● Design experiments to answer these questions
● Draw conclusions from experimental observations
Example: “The Leaning Tower of Pisa” and Galileo Galilei
The Characteristics of Physical Laws
● fundamental
● true, no contradiction
● universal
● simple (in mathematical express)
● absolute (affected by nothing)
● eternal (forever true)
● omnipotent (everything must comply with them)
1.2
Idealized Models: A simplified version of a physical
system that would be too complicated to analyze
without simplification
Example: The projectile motion of a cannon shell can
be studied as that of a point particle.
1.3
Physical Quantity and Units
A physical quantity is used to quantify a physical measurement.
Example: length, weight, pressure, etc.
A physical quantity consists of a number and a unit.
Units define certain standards for the measurements.
Basic units in the SI unit system:
length: meter (m)
mass: kilogram (kg)
time: second (s)
1.4
Unit Conversion and Dimensional Consistency
Conversion Examples:
speed limit
mile
1600m
m
70
 70 
 31
hour
3600s
s
men’s 100 m
world record
m
)
100m
mile
mile

 23.5
9.58s (9.58s ) /(3600 s )
hour
hour
(100m) /(1600
Dimensional Consistency: Check the units of any physical quantity
you have just calculated to find out if or not it is consistent with the
units this physical quantity should have.
For example, the density should be in units of kg/m3. If not, something
Is likely wrong.
1.5 Precision and Significant Figures
How precise can you measure a physical quantity?
For example, using a meter stick to measure lengths, you may get
0.7880 m or 0.3575 m. These quantities have 4 significant figures.
In the sense of significant figures, 0.788 m ≠ 0.7880 m.
● Multiplication and Division
2.4 × 2.30 = 5.5 (≠ 5.52 or 5.520)
● Addition and Subtraction
2.4 + 2.320 = 4.7 (≠ 4.720)
1.24×106 + 3.23×105 = 1.24×106 + 0.323×106 = 1.56×106
1.24×106 + 3.23×104 = 1.24×106 + 0.0323×106 = 1.27×106
1.24×106 + 3.23×103 = 1.24×106 + 0.00323×106 = 1.24×106
1.6 Estimates and Orders of Magnitude
How many metric tons of gold can US$100B buy?
● gold price is currently about $1200 per oz
● 1 oz (Troy Ounce) is about 30 g (31.1 g, to be more precise)
● 1 metric ton is equivalent to 1000 kg, which is 106 g.
The amount of gold US$100B can buy:
($100×109)/($1200/oz) ≈ 83.3×106 oz
= (83.3×106 oz) ×(31.1 g/oz)/(106 g/metric ton)
= 2.59×103 metric tons
about 2600 metric tons, or, on the order of 2 kT as an estimate
of the order of magnitude.
1.3
Vectors and Vector Addition
A scalar quantity has a magnitude only.
A vector quantity has both a magnitude and a direction.
Vector Example: Displacement
y
Point 1
Point 2
x
The Magnitude of a Vector: A = | |
1.7
Vectors and Vector Addition
Vector Addition: Adding Vectors Graphically
+
Two vectors
and
=
+
=
How about subtraction?
=
-
-
=
+(- )
-
1.8
Components of Vectors
Vector Calculus Using Vector Components
The angle q is measured ccw from the +x-axis.
Example 1.6 on P. 17-18
How far did Raoul walk?
(a) on the east leg of the trip
Ax = A cos(35°) = 500 cos(35°)
= 410 m
(a) on the north leg of the trip
Ay = A sin(35°) = 500 sin(35°)
= 287 m
How far did Maria walk?
(c) on the east leg of the trip
Bx = B cos(180°+55°) = 700 cos(235°)
= - 402 m
Or, Bx = - B cos(55°) = - 700 cos(55°)
= - 402 m
(d) on the north leg of the trip
By = B sin(180°+55°) = 700 sin(235°)
= - 573 m
Or, By = - B sin(55°) = - 700 sin(55°)
= - 573 m
Vector Addition
Using Vector Components
=
+
Rx = Ax + Bx
By
Ry = Ay + By
R = (Rx 2 + Ry 2)1/2
q = tan-1 (Ry/Rx)
Bx
Adding Vectors Using Components
Example 1.7: Vector has a magnitude of 50 cm
and direction of 30º, and vector has a magnitude
of 35 cm and direction 110º (both angles measured
ccw from ). What is the resultant vector ?
Page 19
Rx = Ax + Bx = 50 cos(30°) + 35 cos(110°) = 43.3 + (-12.0) = 31.3 cm
Ry = Ay + By = 50 sin(30°) + 35 sin (110°) = 25.0 + 32.9 = 57.9 cm
R = (Rx 2 + Ry 2)1/2 = (31.32 + 57.92)1/2 = 66 cm
q = tan-1 (Ry/Rx) = tan-1 (57.9/31.3) = 62°.