PhysicsTopic #1

MEASUREMENT & MATHEMATICS

Scientific Method• Problem to Investigate• Observations• Hypothesis• Test Hypothesis• Theory• Test Theory• Scientific Law Mathematical proof

Measurement & Uncertainty

• 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 8.7-8.9cm

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 =

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 +/- 0.05 grams. The next day you weigh the returned diamond again, getting 8.09 grams. Is this your diamond?

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 + 0.05g= 8.22g)

* 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

ACCURACY- How close a measurement comes to the TRUE value

PRECISION-How close a SERIES of measurements are to ONE ANOTHER

PERCENT (%) ERROR- Absolute value of the theoretical minus the experimental, divided by the theoretical, multiplied by 100Theoretical - Experimental / Theoretical x 100

Accuracy, Precision, and Percent Error

Metric System

• Expanded & updated version of the metric system:

Systeme International d’Unites

Fundamental SI UnitsPhysical Quantity Name Abbreviation

Length meter mMasskilogram kgTime second sTemperature Kelvin K

Electric current ampere A

Amt of Substance mole mol

Luminous Intensity candelacd

Metric Systemkilokilo kk 10103 3 = 1000= 1000

hectohecto hh 10102 2 = 100= 100dekadeka dada 10101 1 = 10= 10

meter, liter, meter, liter, gram (Base)gram (Base)

m, l, gm, l, g 10100 0 = 1= 1

decideci dd 1010-1-1 = 0.1= 0.1centicenti cc 1010-2 -2 = 0.01= 0.01millimilli mm 1010-3 -3 = 0.001= 0.001

SI Prefixes

pico p 10-12

nano n 10-9

micro µ 10-6

milli m 10-3

centi c 10-2

kilo k 103

mega M 106

giga G 109

tera T 1012

Little GuysLittle Guys Big GuysBig Guys

Reference Table

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. =

Decimal placed between 1st and 2nd digit

Small Number : 0.00043 =

2.38 x 105

4.3 x 10-4

Scientific Notation Express the following numbers in

Scientific Notation

1. 3,5702. 0.00553. 98,784 x 104

4. 45

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

• 0.0034 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

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• Adjust final answer, so leading value is

between 1 and 10

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

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)• 0.0578 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• 8709.0 m 5 Significant Figures (8, 7, 0, 9, 0)

Significant Figures

• Non-zero integers • Always counted as significant figures

** How many significant figures are there in 3,456?

4 Significant Figures

Significant FiguresZEROS

* Leading Zeros- Never significant

0.0486 3 Significant Figures0.003 1 Significant Figure

Significant FiguresZEROS

* Captive zeros- Always significant

16.07 4 Significant Figures10.98 4 Significant Figures70.8 3 Significant Figures

Significant FiguresZEROS

* Trailing Zeros- Significant only if the number contains a decimal point

9.300 4 Significant Figures1.5000 5 Significant Figures

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!

Units for length, mass, and time (aswell as a few others), are regarded asbase SI units

These units are used in combination to define additional units for other important physical quantities, such as force and energy Derived Units

Derived Units

Derived UnitsDerived Unitswebsite

• Units that are created based on formulas and Units that are created based on formulas and equationsequations– VolumeVolume

– VV = length= length··widthwidth··height = mheight = m·m·m = m·m·m = m33

– AreaArea – A = length·width = m·m = mA = length·width = m·m = m22

– ForceForce• F = mass·acceleration = kg·m·sF = mass·acceleration = kg·m·s-2 -2 = Newton, N= Newton, N

– WorkWork• W = Force·distance = N·m = Joule, JW = Force·distance = N·m = Joule, J

– PressurePressure• P = Force/Area = N·mP = Force/Area = N·m-2-2 = Pascal, Pa = Pascal, Pa

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

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

Trigonometric Functions

• sin θ = opposite/hypotenuse• cos θ = adjacent/hypotenuse• tan θ = opposite/adjacent

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