1. BASIC MATH FOR PHYSICS
Unit I-Part A

2. SCIENCE is… the search for relationships that explain
and predict the behavior of the universe.

3. PHYSICS is… the science concerned with relationships between matter,
energy, and its
transformations.

4. There is no such thing as absolute certainty of a scientific claim.
The validity of a scientific conclusion is always limited by:
the experiment
design, equipment, etc...
the experimenter
human error, interpretation, etc...
our limited knowledge
ignorance, future discoveries, etc...

5. an experimentally confirmed explanation
Scientific Law
a statement describing a natural event
Scientific Theory
an experimentally confirmed explanation
for a natural event
Scientific Hypothesis
an educated guess (experimentally untested)

6. perception of “Communist” system natural resistance to change
developed in France in 1795
a.k.a. “SI” - International System of Units
The U.S. was (and still is) reluctant to “go metric.”
very costly to change
perception of “Communist” system
natural resistance to change
American pride

7. The SI unit of: length is the meter, m time is the second, s
mass is the kilogram, kg.
electric charge is the Coulomb, C
temperature is the degree Kelvin, K
an amount of a substance is the mole, mol
luminous intensity is the candle, cd

8. “Derived units” are combinations of these “fundamental units”
Examples include speed in m/s, area in m2,
force in kg.m/s2, acceleration in m/s2,
volume in m3, energy in kg.m2/s2

9. exa E
peta P
tera T
giga G
106 mega M
kilo k
102 hecto h
deka da
atto a
femto f
pico p
nano n
micro m
milli m
centi c
deci d

10. Accuracy % error = x 100% accepted - observed accepted
All measurements have some degree of uncertainty.
Precision
single measurement - exactness, definiteness
group of measurements - agreement, closeness together
Accuracy
closeness to the accepted value
accepted - observed
accepted
% error =
x 100%

11. Example of the differences between precision and accuracy for a set of measurements:
Four student lab groups performed data collection activities in order to determine the resistance of some unknown resistor (you will do this later in the course). Data from 4 trials are displayed below.
Suppose the accepted value for the resistance is 500.
Then we would classify each groups’ trials as:
Group 1: neither precise nor accurate
Group 2: precise, but not accurate
Group 3: accurate, but not precise
Group 4: both precise and accurate
Group
Trial 1
Trial 2
Trial 3
Trial 4
Avg 1 34
612 78
126 2
127
128 3 20
500 62
980 4
502
501
503
498

12. ALGEBRA & EQUATIONS
THE USE OF BASIC ALGEBRA REQUIRES ONLY A FEW FUNDAMENTAL RULES WHICH ARE USED OVER AND OVER TO REARRANGE AND SOLVE EQUATIONS.
(1) ANY NUMBER DIVIDED BY ITSELF IS EQUAL TO ONE.
(2) WHAT IS EVER DONE TO ONE SIDE OF AN EQUATION MUST BE DONE EQUALLY TO THE OTHER SIDE.
(3) ADDITIONS OR SUBTRACTIONS WHICH ARE ENCLOSED IN PARENTHESES ARE GENERALLY CARRIED OUT FIRST.
(4) WHEN VALUES IN PARENTHESES ARE MULTIPLIED OR DIVIDED BY A COMMON TERM EACH CAN BE MULTIPLIED OR DIVIDED SEPARATELY BEFORE ADDING OR SUBTRACTING THE GROUPED TERMS.

13. 10/10 =1 X /X = 1 Y /Y =1 10 + X + 5 = Y + 10 X + 15 = Y + 10 5 x (
ALGEBRA & EQUATIONS
RULE 1 – A VALUE DIVIDED BY ITSELF EQUALS 1
10/10 =1
X /X = 1
Y /Y =1
RULE 2 – OPERATE ON BOTH SIDES EQUALLY
10 +
X = Y
+ 10
X + 15 = Y + 10
IF WE ADD 10 TO THE
LEFT SIDE WE MUST
ADD 10 TO THE RIGHT
5 x (
X = Y )
5 x
X = Y
IF WE MULTIPLY THE LEFT
SIDE BY 5 WE MUST
MULTIPLY THE RIGHT BY 5
5X + 25 = 5Y

14. RULE 3 – OPERATION IN PARENTHESES ARE DONE FIRST
ALGEBRA & EQUATIONS
RULE 3 – OPERATION IN PARENTHESES ARE DONE FIRST
Y = ( ) ( X + 2)
Y = 9 ( X + 2 )
THE PARENTHESES
TERMS (5 + 5) ARE
ADDED FIRST
Y = 9 X + 18 2
S = T 2
S = T ( )
THE PARENTHESES
TERMS (22 – 7) ARE
SUBTRACTED FIRST
S = 225 T

15. RULE 4 – VALUES CAN BE DISTRIBUTED THROUGH TERMS IN PARENTHESES
ALGEBRA & EQUATIONS
RULE 4 – VALUES CAN BE DISTRIBUTED
THROUGH TERMS IN PARENTHESES
Y = 4 ( T + 15 )
Y = 4 T + 60
EACH TERM IN THE
PARENTHESES MUST
BE MULTIPLIED BY 4
Y =
( R x R )
- 3 R
+ 2 R
- 6
Y = ( R + 2 ) ( R - 3 )
Y = R R - 6 2
ALL TERMS MUST BE
MULTIPLIED BY
EACH OTHER THEN ADDED
Y = R R 2

16. SOLVING ALGEBRAIC EQUATIONS
SOLVING AN ALGEBRAIC EQUATION REQUIRES THAT THE UNKNOWN VARIABLE BE ISOLATED ON THE LEFT SIDE OF THE EQUAL SIGN IN THE NUMERATOR POSITION AND ALL OTHER TERMS BE PLACED ON THE RIGHT SIDE OF THE EQUAL SIGN.
THIS MOVEMENT OF TERMS FROM LEFT TO RIGHT AND FROM NUMERATOR TO DENOMINATION AND BACK, IS ACCOMPLISHED USING THE BASIC RULES OF ALGEBRA WHICH WERE PREVIOUSLY DISCUSSED.

17. SOLVING ALGEBRAIC EQUATIONS
THESE RULES CAN BE IMPLEMENTED PRACTICALLY USING SIMPLIFIED PROCEDURES (KEEP IN MIND THE REASON THAT THESE PROCEDURES WORK IS BECAUSE OF THE ALGEBRAIC RULES).
PROCEDURE 1 – WHEN A TERM WITH A PLUS OR MINUS
SIGN IS MOVED FROM ONE SIDE OF THE EQUATION
TO THE OTHER, THE SIGN IS CHANGED.
Y = 3X
- 5
N = 6 M
+ 4

18. SOLVING ALGEBRAIC EQUATIONS
PROCEDURE 2 – WHEN A TERM IS MOVED FROM
THE DENOMINATOR ACROSS AN EQUAL SIGN TO THE
OTHER SIDE OF THE EQUATION IT IS PLACED IN THE
NUMERATOR. LIKEWISE, WHEN A TERM IS MOVED
FROM NUMERATOR ON ONE SIDE IT IS PLACED IN THE
DENOMINATOR ON THE OTHER SIDE.
A C
B D =
----- = ---
F x K G
M N M B K =
F G x M
N x K
A C D =
x B

19. TRIGNOMETRY
TRIGNOMETRIC RELATIONSHIPS ARE BASED ON THE RIGHT TRIANGLE (A TRIANGLE CONTAINING A 900 ANGLE). THE MOST FUNDAMENTAL CONCEPT IS THE PYTHAGOREAN THEOREM (A2 + B2 = C2) WHERE A AND B ARE THE SHORTER SIDES (THE LEGS) OF THE TRIANGLE AND C IS THE LONGEST SIDE CALLED THE HYPOTENUSE.
RATIOS OF THE SIDES OF THE RIGHT TRIANGLE ARE GIVEN NAMES SUCH AS SINE, COSINE AND TANGENT. DEPENDING ON THE ANGLE BETWEEN A LEG (ONE OF THE SHORTER SIDES) AND THE HYPOTENUSE (THE LONGEST SIDE), THE RATIO OF SIDES FOR A PARTICULAR ANGLE ALWAYS HAS THE SAME VALUE NO MATTER WHAT SIZE THE TRIANGLE.

20. The Right Triangle C = the hypotenuse A A & B = the legs B C A
900 
A & B
= the legs
Pythagorean Theorem
A B = C 2 B
C = A + B 2
A RIGHT TRIANGLE
A = C B 2  
900 +
= 1800
B = C A 2

21. TRIG FUNCTIONS
THE RATIO OF THE SIDE OPPOSITE THE ANGLE AND THE HYPOTENUSE IS CALLED THE SINE OF THE ANGLE. THE SINE OF 30 0 FOR EXAMPLE IS ALWAYS ½ NO MATTER HOW LARGE OR SMALL THE TRIANGLE. THIS MEANS THAT THE OPPOSITE SIDE IS ALWAYS HALF AS LONG AS THE HYPOTENUSE IF THE ANGLE IS (30 0 COORESPONSES TO 1/12 OF A CIRCLE OR ONE SLICE OF A 12 SLICE PIZZA!)
THE RATIO OF THE SIDE ADJACENT TO THE ANGLE AND THE HYPOTENUSE IS CALLED THE COSINE. THE COSINE OF 60 0 IS ALWAYS ½ WHICH MEANS THIS TIME THE ADJACENT SIDE IS HALF AS LONG AS THE HYPOTENUSE. (60 0 REPRESENTS 1/6 OF A COMPLETE CIRCLE, ONE SLICE OF A 6 SLICE PIZZA)
THE RATIO OF THE SIDE ADJACENT TO THE ANGLE AND THE SIDE OPPOSITE THE ANGLE IS CALLED THE TANGENT. IF THE ADJACENT AND THE OPPOSITE SIDES ARE EQUAL, THE RATIO (TANGENT VALUE) IS 1.0 AND THE ANGLE IS 45 0 ( 45 0 IS 1/8 OF A FULL CIRCLE)

22. Fundamental Trigonometry
(SIDE RATIOS)
Sin = A / C  C C C A A A
Cos = B / C  
Tan  = A / B B B B
A RIGHT TRIANGLE

23. (the number of atoms in a drop of water)
Scientific Numbers
In science, we often encounter very large and very
small numbers. Using scientific numbers makes
working with these numbers easier
5,010,000,000,000,000,000,000
a very large number
(the number of atoms in a drop of water)
a very small number
(mass of a gold atom in grams)

24. Scientific numbers use powers of 102
100 = 10 x 10 = 10 3
1000 = 10 x 10 x 10 = 10 1 -1
0.10 = 1 / = 10 -2 2
0.01 = 1 / 100 = 1 / = 10 2
523 = 5.23 x 100 = 5.23 x 10 -2 2
= 5.23/100 = 5.23/ = 5.23 x 10

25. Scientific Numbers
RULE 1
As the decimal is moved to the left
The power of 10 increases one
value for each decimal place moved
Any number to the
Zero power = 1
450,000,000 = 450,000,000. x 10 8 2 3 1
450,000,000 = 450,000,000. x 10 8
4.5 x 10

26. Scientific Numbers 0.0000072 = 0.0000072 x 10 -6 -2 -3 -1
RULE 2
As the decimal is moved to the right
The power of 10 decreases one
value for each decimal place moved
Any number to the
Zero power = 1
= x 10 -6 -2 -3 -1
= x 10 -6
7.2 x 10

27. When scientific numbers are multiplied
RULE 3
When scientific numbers are multiplied
The powers of 10 are added
100 x 1000 = 100,000 2
100 = 10
1000 = 10 3 2 3
(2 + 3) 5
x = = = 100,000 2 3 5
(3 x 10 ) x ( 2 x ) = 6 x 10

28. When scientific numbers are divided The powers of 10 are subtracted
RULE 4
When scientific numbers are divided
The powers of 10 are subtracted
10000 / 100 = 100 4
10000 = 10 2
100 = 10 4 2
(4 - 2) 2
/ = = = 100 4 2 2
(5 x ) / (2 x ) = 2.5 x 10

29. Scientific Numbers 2 (100) = 10,000 2 100 = 10 4 2 2 (2 x 2)
RULE 5
When scientific numbers are raised to powers
The powers of 10 are multiplied 2
(100) = 10,000 2
100 = 10 4 2 2
(2 x 2)
( ) = = = 10,000 2
(3000) = 9,000,000 3 2 2
(2 x 3) 6
(3 x ) = x = 9 x 10

30. Scientific Numbers square root = 1/2 power cube root = 1/3 power 1/2
RULE 6
Roots of scientific numbers are treated as fractional
powers. The powers of 10 are multiplied
square root = 1/2 power
cube root = 1/3 power
1/2
10,000 = (10,000) 4
10,000 = 10
1/2
(1/2 x 4) 4
1/2
(10,000) = (10 ) = = 100
1/2
1/2 6
1/2 3 6
(9 x ) = x ( ) = x 10

31. Scientific Numbers 2 2.34 x 10 3 + 4.24 x 10 --------- 3 2
RULE 7
When scientific numbers are added or subtracted
The powers of 10 must be the same for each term. 2
Powers of 10 are
Different. Values
Cannot be added !
2.34 x 10 3
x 10 3
Move the decimal
And change the power
Of 10 2
2.34 x = x 10 3
0.234 x 10
Power are now the
Same and values
Can be added. 3
x 10
4.47 x 10 3

32. Order Of Magnitude Estimation
Is Equivalent To
Rounding Off Your Estimate To The Nearest Power Of 10

33. Order Of Magnitude Examples
The meter is the standard of measurement in this class
A Football Field Is About 102m Long (100m)
It’s not 10m, and it’s not 1000m
A Credit Card is About 10-3m thick (1mm)
It’s not 0.1mm (the thickness of a sheet of copy paper), and it’s not 10mm (width of your pinky)
The Period of a Backyard Swing Is About 100s
It’s not 0.1s, and it’s not 10s

34. Units NO UNIT == NO MEANING
Almost always, our measurements are in some physical unit
THEY ARE NOT JUST NUMBERS!!!!
THEY HAVE PHYSICAL MEANING
NO UNIT == NO MEANING

35. These are Called “Derived Units”
In Addition, We Use Combinations of Basic Units
These are Called “Derived Units”
meters per second
kilograms per meter3

36. Unit Conversion Using The Unit Factor Method
“Unit” Has Two Meanings:
“The” Unit (kg, s, m) or “Unit == ONE”
A Way To Convert Values From One Unit To Another
It Works Because If You Multiply Something Times 1 It Does Not Change The Value
Also called “Factor-Label” method

37. Unit Factors
Unit Factors Are Created From Conversion Factors Like 2.54 cm = 1 inch
We Can Make 2 Unit Factors From This

38. Using Unit Factors
Multiply The Value You Want To Convert By The Appropriate Unit Factor
Cancel Out Similar Units, Leaving The Desired Unit
Multiply Values and You Are Done!!
Example : How Many Inches Are In 25.4 Centimeters?

39. How Many Centimeters Are In Six Inches?
Using Unit Factors
In Your Notes
How Many Centimeters Are In Six Inches?

40. Using Unit Factors
In Your Notes
Express 24 cm in inches

41. Multiple Conversion Steps
You Can Also String Multiple Unit Factors Together
In Your Notes
How Many Seconds Are In 2 Years?

42. Dimensional Analysis
“Analyzing The Dimensions” Of An Equation To Be Sure The Units Match Up And Make Sense
Start With A Fancy, Schmanzy Equation
Replace Each Unknown With Its “Dimension”

43. Dimensional Analysis d = distance in meters = [m]
v = velocity in meters / second = [m]/[s]
a = acceleration in meters / second2 = [m]/[s]2
t = time in seconds = [s] or [s]2

44. Dimensional Analysis
[m] = [m]/[s] X [s] + ½ [m]/[s]2 X [s]2 Cancel Common Dimensions [m] = [m] + ½ [m] Ignore The ½ Factor [meters] = [meters] This Equation Is Dimensionally Consistent