Transcript Presentation Lesson 02 Language of Physics

Physics Lesson 2
Math - Language
of Physics
Eleanor Roosevelt High School
Chin-Sung Lin
Why Math?
Why Math?
 Ideas can be expressed in a concise way
 Ideas are easier to verify or to disapprove by
experiment
 Methods of math and experimentation led to the
enormous success of science
Why Math?
“How can it be that mathematics, being a product of
human thought which is independent of experience,
is so admirably appropriate to the objects of reality?”
~ Albert Einstein ~
Math – Language of Physics
 SI Units
 Scientific Notation
 Significant Figures
 Precision & Accuracy
 Graphing
 Order of Magnitude
 Scalar & Vectors
Math – Language of Physics
 Mathematic Analysis
 Trigonometry
 Equation Solving
SI Units
SI Units
What is SI Units?
 Système Internationale d’Unités
 Seven fundamental units
 Dozens of derived units have been created
SI Units
What is UI Units?
Quantity
Base Unit
Time
second (s)
Length
meter (m)
Mass
kilogram (kg)
Temperature
Kelvin (K)
Amount of a substance
mole (mol)
Electric current
Ampere (A)
Luminous intensity
candela (cd)
SI Units
Prefixes Used with SI Units
Prefix
Symbol
Scientific Notation
tera
giga
mega
kilo
deci
centi
milli
micro
nano
pico
T
G
M
k
d
c
m

n
p
1012
109
106
103
10-1
10-2
10-3
10-6
10-9
10-12
SI Units
Prefixes Used with SI Units
 What does the prefix symbol “k” mean?
kilo, 103
SI Units
Prefixes Used with SI Units
 What does the prefix symbol “m” mean?
milli, 10-3
SI Units
Prefixes Used with SI Units
 What does the prefix symbol “n” mean?
nano, 10-9
SI Units
Prefixes Used with SI Units
 What does the prefix symbol “M” mean?
mega, 106
SI Units
Prefixes Used with SI Units
 What does the prefix symbol “G” mean?
giga, 109
SI Units
Prefixes Used with SI Units
 What does the prefix symbol “” mean?
micro, 10-6
SI Units
Prefixes Used with SI Units
 What does the prefix symbol “T” mean?
tera, 1012
SI Units
Prefixes Used with SI Units
 What does the prefix symbol “c” mean?
centi, 10-2
SI Units
SI Unit Conversion
 1 MHz = __________________ Hz
106
SI Units
SI Unit Conversion
 1 kg = __________________ g
103
SI Units
SI Unit Conversion
 1 G Bytes = __________________ Bytes
109
SI Units
SI Unit Conversion
 2 mm = __________________ m
2 x 10-3
SI Units
SI Unit Conversion
 5 ns = __________________ s
5 x 10-9
SI Units
SI Unit Conversion
 4 cm = __________________ m
4 x 10-2
SI Units
SI Unit Conversion
 3 m = __________________ m
3 x 10-6
Scientific Notation
Scientific Notation
What is Scientific Notation?
 Shorthand for very large / small numbers
 In the form of a x 10 n, where n is an integer
and 1 ≤ |a| < 10
Scientific Notation
Scientific Notation & Standard Notation
 1.55 x 106 = 1.550000 x 106 = 1,550,000
for a positive exponent, move the decimal point 6
place to the right
 2.5 x 10-4 = 0.00025
for a negative exponent, move the decimal point 4
place to the left
Scientific Notation
Calculating with Scientific Notation
 (2.5 x 103)(4.0 x 105) = (2.5 x 4.0)(103 x 105)
Rearrange factors
= 10.0 x 103+5
Multiply
= 10.0 x 108
Add exponents
= 1.0 x 10 x 108
Write 10.0 as 1.0 x 10
= 1.0 x 109
Add exponents
Scientific Notation
Using Calculator to Calculate Scientific Notation
 (2.5 x 103)(4.0 x 105) =
Step 1:
( )
Change to scientific notation mode
Step 2:
Calculate in scientific notation
Significant Figures
Significant Figures
What is Significant Figures?
 The result of any measurement is an
approximation
 Include all known digits and one reliably
estimated digit
Significant Figures
Zeros & Significant Figures
 Each nonzero digit is significant
 A zero may be significant depending on its location
 A zero between two significant digits is significant
 All final zeros of a number that appear to the right
of the decimal point and to the right of a nonzero
digit are also significant
 Zeros that simply act as placeholders in a number
are not significant
Significant Figures
Addition / Subtraction & Significant Figures
 the number of digits to the right of the decimal in
sum or difference should not exceed the least
number of digits to the right of the decimal in the
terms
 (1.11 x 10-4 kg) + (2.22222 x 10-4 kg)
= 3.33222 x 10-4 kg
= 3.33 kg
Significant Figures
Multiplication / Division & Significant Figures
 to round the results to the number of significant
digits that is equal to the least number of significant
digits among the quantities involved
 (1.1 x 10-4 kg) * (2.222 x 103 m/s2)
= 2.4442 x 10-1 kg m/s2
= 2.4x 10-1 kg m/s2
Precision & Accuracy
Precision & Accuracy
Precision
 The degree of exactness of a measurement
 Precision depends on the tools and methods
 The significant digits show its precision
Precision & Accuracy
Accuracy
 The degree of agreement of a measurement with an
accepted value obtained through computations or
other competent measurement
 Accuracy describes how well two descriptions of a
quantity agree with each other
Precision & Accuracy
Comparison of Precision & Accuracy
Precision & Accuracy
Comparison of Precision & Accuracy
Precision & Accuracy
Comparison of Precision & Accuracy
Precision & Accuracy
Comparison of Precision & Accuracy
Precision & Accuracy
Percent Error
Percent Error =
|accepted – measured|
accepted
x 100%
Precision & Accuracy
Percent Error
Two experiments:
(A) measured value = 125, accepted value = 100
(B) measured value = 100, accepted value = 75
Which one has higher percent error?
Graphing
Graphing
Scatter-Plot
 Showing Measured Data
Distance (meter)
Distance of the moving car
120
100
80
60
40
20
0
Distance
0
5
10
Time (second)
15
Graphing
Line Graph
 Showing Trend
Helicopter Motion
Graphing
Bar Graph
 Compare Nonnumeric Categories
No. of Students Each Day
Date
9/5/06
9/6/06
9/7/06
9/8/06
46
47
48
49
50
No. of Students
51
52
Graphing
Circle Graph
 Showing Percentage
Science Students of different Science
Subjects
Physic
s
Living
Envirome
nt
40%
Chemstry
20%
Earth
Science
30%
Order of Magnitude
Order of Magnitude
Definition
 Describe the size of a measurement rather than
its actual value
 The order of magnitude of a measurement is the
power of 10 closest to its value
Order of Magnitude
Example
 The order of magnitude of 1024 m (1.024 x 103
m) is 103
 The order of magnitude of 9600 m (9.6 x 103 m)
is 104
Scalars & Vectors
Scalars & Vectors
Comparison of Scalars & Vectors
Physical Quantities
Scalars
Vectors
Magnitude
Magnitude
Direction
Scalars & Vectors
Comparison of Scalars & Vectors
Physical Quantities
Scalars
Magnitude
Vectors
3 m/s
Magnitude
North
Direction
Scalars & Vectors
Comparison of Scalars & Vectors
Physical Quantities
Scalars
Magnitude
Vectors
3 m/s
Magnitude
60o
Direction
Scalars & Vectors
Examples of Scalars & Vectors
Physical Quantities
Scalars
Vectors
distance
Displacement
speed
Velocity
Mass
Force
Scalars & Vectors
Vector Representation
 An arrow is used to represent the magnitude and
direction of a vector quantity
 Magnitude: the length of the arrow
 Direction: the direction of the arrow
Head
Tail
Direction
Mathematical Analysis
Mathematical Analysis
Line of Best Fit
 To analyze a graph, draw a line/curve of best fit
which passes through or near the graphed data
 Describe data and to predict where new data will
appear
Distance vs. Time
60
50
40
30
Distance
20
10
0
0
1
2
3
4
5
6
Mathematical Analysis
Linear Relationship
 y = mx + b
where b is the y-intercept and m is the slope
y
Slope = m
b
x
Mathematical Analysis
Quadratic Relationship
 y = ax2 + bx + c
where c is the y-intercept
y
c
x
Mathematical Analysis
Inverse Relationship
y=a/x
y
x
Mathematical Analysis
Inverse Square Law Relationship
 y = a / x2
y
x
Equation Solving
Equation Solving
Solve Linear Equations
 3x + 7 = 8x – 3
Equation Solving
Solve Linear Equations
 3x + 7 = 8x – 3
7 + 3 = 8x – 3x
10 = 5x
5x = 10
x=2
Equation Solving
Solve Simple Rational Equations
3/8=9/x
Equation Solving
Solve Simple Rational Equations
3/8=9/x
x = 8 (9 / 3)
x = 24
Equation Solving
Solve Simple Quadratic Equations
 20 = 5 / x2
Equation Solving
Solve Simple Quadratic Equations
 20 = 5 / x2
x2 = 5 / 20 = 1 / 4
x=1/2
(most of the time, only pick the positive value)
Equation Solving
Solve Rational Equations
 1/R = 1/20 + 1/30
Equation Solving
Solve Rational Equations
 1/R = 1/20 + 1/30
1/R = 5/60 = 1/12
R = 12
Equation Solving
Solve Radical Equations
 3 = 2π√L/10
Equation Solving
Solve Radical Equations
 3 = 2π√L/10
3/(2π) = √L/10
(3/2π)2 = L/10
9/(4π2) = L/10
L = 45/2π2
Equation Solving
Solve Equations of Variables
 d = ½ gt2 ,
solve for t
Equation Solving
Solve Equations of Variables
 vf2 = vi2 + 2gd,
solve for d
Equation Solving
Solve Equations of Variables
 Fe = kq1q2/d2 , solve for d
Equation Solving
Solve Equations of Variables
 L = L0 √ 1 – v2 / c2 , solve for v
Trigonometry
Trigonometry
Trigonometric Ratios
 In ΔABC, BC is the leg opposite A, and AC is
the leg adjacent to
A. The hypotenuse is AB
B
sin A = a / c
cos A = b / c
c
a
tan A = a / b
A
b
C
Trigonometry
Trigonometric Ratios of Special Right Triangles
 30-60-90 & 45-45-90 special right triangles
B
B
2
A
30o
√3
√2
1
1
C
A
45o
1
C
Trigonometry
Calculation Using Trigonometric Ratio
 Identify the known angle/side & the unknown side
 Establish the trigonometric ratio between the
known side and unknown side using the angle
 Solve for the unknown
B
sin A = BC / AB
10
sin 30o = x / 10
x = 10 sin
30o =
5
A
30o
x
C
Trigonometry
Calculation Using Trigonometric Ratio
 Identify the known angle/side & the unknown side
 Establish the trigonometric ratio between the
known side and unknown side using the angle
 Solve for the unknown
B
cos A = AC / AB
10
cos 30o = x / 10
x = 10 cos
30o =
8.66
A
30o
x
C
Trigonometry
Calculation Using Trigonometric Ratio
 Identify the known angle/side & the unknown side
 Establish the trigonometric ratio between the
known side and unknown side using the angle
 Solve for the unknown
B
sin A = BC / AB
x
sin 30o = 10 / x
x = 10 / sin
30o =
20
A
30o
10
C
Trigonometry
Calculation of angles
 Identify the known side & the unknown angle
 Establish the trigonometric ratio between the
known sides and unknown angle
 Solve for the unknown using inverse
trigonometric function
B
tan θ = BC / AC
10
tan θ = 10 / 15 = 2 / 3
x=
tan-1
(2 / 3) =
33.7o
A
θ
15
C
The End