Measurement Why are precise measurements and calculations essential to a study of physics?

Measurement & Precision The precision of a measurement depends on the instrument used to measure it. For example, how long is this block?

Measurement & Precision Imagine you have a piece of string that is exactly 1 foot long. Now imagine you were to use that string to measure the length of your pencil. How precise could you be about the length of the pencil? Since the pencil is less than 1 foot, we must be dealing with a fraction of a foot. But what fraction can we reliably estimate as the length of the pencil?

Measurement & Precision Suppose the pencil is slightly over half the 1 foot string. You guess, “Well it must be about 7 inches, so I’ll say 7/12 of a foot.” Here’s the problem: If you convert 7/12 to a decimal, you get 0.583. Can you reliably say, without a doubt, that the pencil is 0.583 and not 0.584 or 0.582? You can’t. The string didn’t allow you to distinguish between those lengths… you didn’t have enough precision. So, what can you estimate, reliably?

Measurement & Precision Basically, you have one degree of freedom… one decimal place of freedom. So, the only fractions you can use are tenths! You can only reliably estimate that the pencil is 0.6 ft long. It’s definitely more than 0.5 ft long and definitely less than 0.7 ft long. Thus, precision determines the number of significant figures we use to report measurements. In order to increase the precision of their measurements, physicists develop more-advanced instruments.

How big is the beetle? Measure between the head and the tail! Between 1.5 and 1.6 in Measured length: 1.54 in The 1 and 5 are known with certainty The last digit (4) is estimated between the two nearest fine division marks. Copyright © 1997-2005 by Fred Senese

How big is the penny? Measure the diameter. Between 1.9 and 2.0 cm Estimate the last digit. What diameter do you measure? How does that compare to your classmates? Is any measurement EXACT? Copyright © 1997-2005 by Fred Senese

What Length is Indicated by the Arrow?

Significant Figures Indicate precision of a measured value 1100 vs. 1100.0 Which is more precise? How can you tell? How precise is each number? Determining significant figures can be tricky. There are some very basic rules you need to know. Most importantly, you need to practice!

Counting Significant Figures The Digits Digits That Count Example # of Sig Figs Non-zero digits ALL          4.337 4 Leading zeros (zeros at the BEGINNING) NONE          0.00065 2 Captive zeros (zeros BETWEEN non-zero digits)          1.000023 7 Trailing zeros (zeros at the END) ONLY IF they follow a significant figure AND there is a decimal point in the number 89.00 but 8900 Leading, Captive AND Trailing Zeros Combine the rules above 0.003020 but 3020 3 Scientific Notation         7.78 x 103

Calculating With Sig Figs Type of Problem Example MULTIPLICATION OR DIVISION: Find the number that has the fewest sig figs. That's how many sig figs should be in your answer. 3.35 x 4.669 mL = 15.571115 mL rounded to 15.6 mL 3.35 has only 3 significant figures, so that's how many should be in the answer.  Round it off to 15.6 mL ADDITION OR SUBTRACTION: Find the number that has the fewest digits to the right of the decimal point. The answer must contain no more digits to the RIGHT of the decimal point than the number in the problem. 64.25 cm + 5.333 cm = 69.583 cm rounded to 69.58 cm 64.25 has only two digits to the right of the decimal, so that's how many should be to the right of the decimal in the answer. Drop the last digit so the answer is 69.58 cm.

Scientific Notation Number expressed as: 5.63 x 104, meaning Product of a number between 1 and 10 AND a power of 10 5.63 x 104, meaning 5.63 x 10 x 10 x 10 x 10 or 5.63 x 10,000 ALWAYS has only ONE nonzero digit to the left of the decimal point ONLY significant numbers are used in the first number First number can be positive or negative Power of 10 can be positive or negative

When to Use Scientific Notation Astronomically Large Numbers mass of planets, distance between stars Infinitesimally Small Numbers size of atoms, protons, electrons A number with “ambiguous” zeros 59,000 HOW PRECISE IS IT?

Exponent of Zero Means “1” 100 = 1 Powers of 10 Positive Exponents Exponent of Zero Means “1” 100 = 1

Exponent of Zero Means “1” 100 = 1 Powers of 10 Negative Exponents Exponent of Zero Means “1” 100 = 1

Converting From Standard to Scientific Notation Move decimal until it is behind the first sig fig Power of 10 is the # of spaces the decimal moved Decimal moves to the left, the exponent is positive Decimal moves to the right, the exponent is negative 428.5  4.285 x 102 (decimal moves 2 spots left) 0.0004285  4.285 x 10-4 (decimal moves 4 spots right)

Converting From Scientific to Standard Notation Move decimal point # of spaces the decimal moves is the power of 10 If exponent is positive, move decimal to the right If exponent is negative, move decimal to the left 4.285 x 102  428.5 (move decimal 2 spots right) 4.285 x 10-4  0.0004285 (decimal moves 4 spots left)

Systems of Measurement Why do we need a standardized system of measurement? Scientific community is global. An international “language” of measurement allows scientists to share, interpret, and compare experimental findings with other scientists, regardless of nationality or language barriers. By the 1700s, every country used its own system of weights and measures. England had three different systems just within its own borders!

Metric System & SI The first standardized system of measurement: the “Metric” system Developed in France in 1791 Named based on French word for “measure” based on the decimal (powers of 10) Systeme International d'Unites (International System of Units) Modernized version of the Metric System Abbreviated by the letters SI. Established in 1960, at the 11th General Conference on Weights and Measures. Units, definitions, and symbols were revised and simplified.

Components of the SI System In this course we will primarily use SI units. The SI system of measurement has 3 parts: base units derived units prefixes Unit: measure of the quantity that is defined to be exactly 1 Prefix: modifier that allows us to express multiples or fractions of a base unit As we progress through the course, we will introduce different base units and derived units.

SI: Base Units length meter m mass kilogram kg time second s Physical Quantity Unit Name Symbol length meter m mass kilogram kg time second s electric current ampere A temperature Kelvin K amount of substance mole mol luminous intensity candela cd

SI: Derived Units area square meter m2 volume cubic meter m3 speed Physical Quantity Unit Name Symbol area square meter m2 volume cubic meter m3 speed meter per second m/s acceleration second squared m/s2 weight, force newton N pressure pascal Pa energy, work joule J

Prefixes Prefix Symbol Numerical Multiplier Exponential Multiplier yotta Y 1,000,000,000,000,000,000,000,000 1024 zetta Z 1,000,000,000,000,000,000,000 1021 exa E 1,000,000,000,000,000,000 1018 peta P 1,000,000,000,000,000 1015 tera T 1,000,000,000,000 1012 giga G 1,000,000,000 109 mega M 1,000,000 106 kilo k 1,000 103 hecto h 100 102 deca da 10 101 no prefix means: 1

Prefixes Prefix Symbol Numerical Multiplier Exponential Multiplier no prefix means: 1 100 deci d 0.1 10¯1 centi c 0.01 10¯2 milli m 0.001 10¯3 micro 0.000001 10¯6 nano n 0.000000001 10¯9 pico p 0.000000000001 10¯12 femto f 0.000000000000001 10¯15 atto a 0.000000000000000001 10¯18 zepto z 0.000000000000000000001 10¯21 yocto y 0.000000000000000000000001 10¯24

Unit Conversions “Staircase” Factor-Label Method Type Visual Mathematical What to do… Move decimal point the same number of places as steps between unit prefixes Multiply measurement by conversion factor, a fraction that relates the original unit and the desired unit When to use… Converting between different prefixes between kilo and milli Converting between SI and non-SI units Converting between different prefixes beyond kilo and milli

“Staircase” Method Draw and label this staircase every time you need to use this method, or until you can do the conversions from memory

“Staircase” Method: Example Problem: convert 6.5 kilometers to meters Start out on the “kilo” step. To get to the meter (basic unit) step, we need to move three steps to the right. Move the decimal in 6.5 three steps to the right Answer: 6500 m

“Staircase” Method: Example Problem: convert 114.55 cm to km Start out on the “centi” step To get to the “kilo” step, move five steps to the left Move the decimal in 114.55 five steps the left Answer: 0.0011455 km

Factor-Label Method Multiply original measurement by conversion factor, a fraction that relates the original unit and the desired unit. Conversion factor is always equal to 1. Numerator and denominator should be equivalent measurements. When measurement is multiplied by conversion factor, original units should cancel

Factor-Label Method: Example Convert 6.5 km to m First, we need to find a conversion factor that relates km and m. We should know that 1 km and 1000 m are equivalent (there are 1000 m in 1 km) We start with km, so km needs to cancel when we multiply. So, km needs to be in the denominator

Factor-Label Method: Example Multiply original measurement by conversion factor and cancel units.

Factor-Label Method: Example Convert 3.5 hours to seconds If we don’t know how many seconds are in an hour, we’ll need more than one conversion factor in this problem