1. Topic #1 MEASUREMENT & MATHEMATICS
Physics
Topic #1
MEASUREMENT & MATHEMATICS

2. Scientific Method Problem to Investigate Observations Hypothesis
Test Hypothesis
Theory
Test Theory
Scientific Law  Mathematical proof
* Principal aim of all sciences (physics)  search for order
in our observations of the world around us
*Mathematical proof = statement that takes the form of a
relationship or equation between quantities (such as
Newton’s second law F=ma)

3. Measurement & Uncertainty
No measurement is absolutely precise
Estimated Uncertainty:
Width of a board is 8.8cm +/- 0.1cm
0.1cm represents the estimated uncertainty in the measurement
Actual width  between cm
Uncertainty associated with every measurement not being absolutely precise is due to---Limited accuracy of the measuring device; Inability to read beyond a fraction of the smallest division shown

4. Measurement & Uncertainty
Percent Uncertainty:
Ratio of the uncertainty to the measured value, x 100
Example:
Measurement = 8.8 cm
Uncertainty = 0.1 cm
Percent Uncertainty =
0.1/8.8 x 100% = 1%

5. Is the diamond yours?
A friend asks to borrow your precious diamond for a day to show her family. You are a bit worried, so you carefully have your diamond weighed on a scale which reads 8.17 grams. The scale’s accuracy is claimed to be +/ grams. The next day you weigh the returned diamond again, getting grams. Is this your diamond?

6. Scale Readings
- Measurements do not necessarily give the “true” value of the mass
- Each measurement could have a high or low by up to 0.05g
- Actual mass of your diamond  between 8.12g and 8.22g
Reasoning: (8.17g – 0.05g = 8.12g) (8.17g g= 8.22g)

7. * Actual mass your diamond
- Between 8.12g and 8.22g
* Actual mass of the returned diamond
- 8.09g +/- 0.05g  Between 8.04g and 8.14g
** These two ranges overlap  not a strong reason to doubt that the returned diamond is yours, at least based on the scale readings

8. Accuracy, Precision, and Percent Error
How close a measurement comes to the TRUE value
PRECISION-
How close a SERIES of measurements are to ONE ANOTHER
PRECISION = Refers to the repeatability of the measurement using a given instrument- Ex: You measure the width of a board many times  results = 8.81 cm, 8.85 cm, 8.78 cm, 8.82 cm ( you could say the measurements give a precision a bit better than 0.1 cm
ACCURACY = Refers to how close a measurement is to the true value
*** Estimated uncertainty – Meant to take into both precision and accuracy into account
PERCENT (%) ERROR- Absolute value of the theoretical minus the experimental, divided by the theoretical, multiplied by 100
Theoretical - Experimental / Theoretical x 100

9. Metric System Expanded & updated version of the metric system:
Systeme International d’Unites

10. Fundamental SI Units Physical Quantity Name Abbreviation
Length meter m
Mass kilogram kg
Time second s
Temperature Kelvin K
Electric current ampere A
Amt of Substance mole mol
Luminous Intensity candela cd

11. meter, liter, gram (Base)
Metric System
kilo k
103 = 1000
hecto h
102 = 100
deka da
101 = 10
meter, liter, gram (Base)
m, l, g
100 = 1
deci d
10-1 = 0.1
centi c
10-2 = 0.01
milli m
10-3 = 0.001
WEBSITE:
* We utilize prefixes in the metric system to indicate larger & smaller units

12. SI Prefixes
Little Guys
Big Guys

13. Reference Table

14. Scientific Notation
Alternative way to express very large or very small numbers
Number is expressed as the product of a number between 1 and 10 and the appropriate power of 10.
Large Number: 238,000. =
2.38 x 105
Decimal placed between 1st and 2nd digit
Small Number : =
4.3 x 10-4

15. Scientific Notation
Express the following numbers in Scientific Notation
1. 3,570
3. 98,784 x 104
4. 45
3.57 x x x 108

16. Scientific Notation “Scientific Notation” or “Powers of Ten”
Allows the number of significant figures to be clearly expressed
Example:
56, 800  5.68 x 104
 3.4 x 10-3
6.78 x 104  Number is known to an accuracy of 3 significant figures
6.780 x 104  Number is known to an accuracy of 4 significant figures

17. Scientific Notation Multiplying Numbers in Scientific Notation
Multiply leading values
Add exponents
Adjust final answer, so leading value is between 1 and 10
Dividing Numbers in Scientific Notation
Divide leading values
Subtract exponents

18. Scientific Notation
Adding & Subtracting Numbers in Scientific Notation
Adjust so exponents match
Then, add or subtract leading values only
Adjust final answer, so leading value is between 1 and 10

19. Significant Figures
All of the important/necessary or reliably known numbers
GUIDELINES
Non-zero digits  always significant
Zeros at the beginning of a number  Not significant (Decimal point holders)
m 3 Significant Figures (5, 7, 8)
Zeros within the number  Significant
108.7 m 4 Significant Figures (1, 0, 8, 7)
Zeros at the end of a number, after a decimal point  Significant
m 5 Significant Figures (8, 7, 0, 9, 0)

20. Significant Figures Non-zero integers
Always counted as significant figures
** How many significant figures are there in 3,456?
4 Significant Figures

21. Significant Figures ZEROS * Leading Zeros - Never significant

22. Significant Figures ZEROS * Captive zeros - Always significant

23. Significant Figures ZEROS * Trailing Zeros
- Significant only if the number contains a decimal point
 4 Significant Figures
 5 Significant Figures

24. Converting Units Physics problems require the use of the correct units
Conversion factors
Allow you to change from one unit of measurement to another
Ex: 1 foot = 12 inches
Converting units
Choose the appropriate conversion factor
Multiply by the conversion factor as a fraction
Make sure units cancel!
Keep in mind that if you do not use consistent units, troubles will arise. NASA dramatically illustrated the cost of such errors when it lost a spacecraft in A company supplied data to NASA based on British units (pounds) when NASA engineers expected metric units (newtons). Oops. That, alas, was the end of that space probe (and about $125 million and, one suspects, some engineer’s NASA career).

25. Derived Units Units for length, mass, and time (as
well as a few others), are regarded as
base SI units
These units are used in combination to
define additional units for other important physical quantities, such as force and energy  Derived Units

26. Derived Units website
Units that are created based on formulas and equations
Volume
V = length·width·height = m·m·m = m3
Area
A = length·width = m·m = m2
Force
F = mass·acceleration = kg·m·s-2 = Newton, N
Work
W = Force·distance = N·m = Joule, J
Pressure
P = Force/Area = N·m-2 = Pascal, Pa

27. Dimensional Analysis
Useful tool utilized to check the dimensional consistency of any equation to check whether calculations make sense
Length is represented by L
Mass is represented by M
Time is represented by T
For an equation to be valid, the left dimension must equal the right dimension

28. Trigonometry Pythagorean Theorem
Used to find the length of any side of a right triangle when you know the lengths of the other two sides
Right triangle  Triangle with a 90° angle
c2 = a2 + b2
c = Length of the hypotenuse
a, b, = Lengths of the legs
Relates hypotenuse to the legs of a right triangle

29. Trigonometric Functions
sin θ = opposite/hypotenuse
cos θ = adjacent/hypotenuse
tan θ = opposite/adjacent
SOH CAH TOA

30. Trigonometric Functions
If you know the ratio of lengths of 2 sides of a right triangle, you can use inverse functions to determine the angles of that triangle
θ = arcsin (opposite/hypotenuse)
θ = arccos (adjacent/hypotenuse)
θ = arctan (opposite/adjacent)
Often written: sin−1, cos−1, tan−1