V. Limits of Measurement 1. Accuracy and Precision

Accuracy - a measure of how close a measurement is to the true value of the quantity being measured.

Example: Accuracy Who is more accurate when measuring a book that has a true length of 17.0cm? Susan: 17.0cm, 16.0cm, 18.0cm, 15.0cm Amy: 15.5cm, 15.0cm, 15.2cm, 15.3cm

Precision – a measure of how close a series of measurements are to one another. A measure of how exact a measurement is.

Example: Precision Who is more precise when measuring the same 17.0cm book? Susan: 17.0cm, 16.0cm, 18.0cm, 15.0cm Amy: 15.5cm, 15.0cm, 15.2cm, 15.3cm

Example: Evaluate whether the following are precise, accurate or both. Accurate Not Precise Not Accurate Precise Accurate Precise

2. Significant Figures The significant figures in a measurement include all of the digits that are known, plus one last digit that is estimated.

8 Exact Numbers An exact number is obtained when you count objects or use a defined relationship. - Counting objects are always exact 2 soccer balls 4 pizzas - Exact relationships, predefined values, not measured 1 foot = 12 inches 1 meter = 100 cm For instance is 1 foot = inches? No 1 ft is EXACTLY 12 inches.

9 Solution A. Exact numbers are obtained by 2. counting 3. definition B. Measured numbers are obtained by 1. using a measuring tool

2.1 Uncertainty in Measurement cm

2.1. Uncertainty in Measurement A measurement always has some degree of uncertainty.

2.1 Uncertainty in Measurement Different people estimate differently. Record all certain numbers and one estimated number.

2.1 Measurement and Significant Figures Every experimental measurement has a degree of uncertainty. The volume, V, at right is certain in the 10’s place, 10mL<V<20mL The 1’s digit is also certain, 17mL<V<18mL A best guess is needed for the tenths place. Chapter Two 13

14 What is the Length? We can see the markings between cm We can’t see the markings between the.6-.7 We must guess between.6 &.7 We record 1.67 cm as our measurement The last digit an 7 was our guess...stop there

Learning Check What is the length of the wooden stick? 1) 4.5 cm 2) 4.54 cm 3) cm

cm or 3 (2.2/8) ?

Below are two measurements of the mass of the same object. The same quantity is being described at two different levels of precision or certainty.

2.2 Significant Figures Significant figures are the meaningful figures in our measurements and they allow us to generate meaningful conclusions Numbers recorded in a measurement are significant. –All the certain numbers plus first estimated number e.g cm We need to be able to combine data and still produce meaningful information There are rules about combining data that depend on how many significant figures we start with………

2.3 Rules for Counting Significant Figures 1.Nonzero integers always count as significant figures has 4 significant figures 23.3has 3 significant figures

Rules for Counting Significant Figures 2.Zeros a.Leading zeros - never count significant figures b.Captive zeros - always count significant figures c.Trailing zeros - count only if the number is written with a decimal point significant figure significant figures significant figures

Rules for Counting Significant Figures 3.Exact numbers - unlimited significant figures Not obtained by measurement  Determined by counting: 3 apples  Determined by definition: 1 in. = 2.54 cm

Practice Rule #1 Zeros  All digits count Leading 0’s don’t Trailing 0’s do 0’s count in decimal form 0’s don’t count w/o decimal All digits count 0’s between digits count as well as trailing in decimal form

How Many Significant Figures? , x x x minutes in an hour 500 laps in the race

One convention about trailing zero

2.3 Rules for Counting Significant Figures All non-zero digits are significant. Zeros between two non-zero digits are significant. Leading zeros are not significant. Trailing zeros in a number containing a decimal point are significant. trailing zeros in a number not containing a decimal point can be ambiguous. (scientific notation is the solution)

- Round off to 1,2,3,4 significant figures

2.4 Scientific notation Standard decimal notationScientific notation 22× ×10 2 4, ×10 3 −53,000−5.300× × ×10 − ×10 −9

Chapter Two 28 Two examples of converting standard notation to scientific notation are shown below.

Chapter Two 29 Two examples of converting scientific notation back to standard notation are shown below.

Scientific notation is helpful for indicating how many significant figures are present in a number that has zeros at the end but to the left of a decimal point. The distance from the Earth to the Sun is 150,000,000 km. Written in standard notation this number could have anywhere from 2 to 9 significant figures. Scientific notation can indicate how many digits are significant. Writing 150,000,000 as 1.5 x 10 8 indicates 2 and writing it as x 10 8 indicates 4. Scientific notation can make doing arithmetic easier.

How many sig figs?

2.5 Rules for Multiplication and Division I measure the sides of a rectangle, using a ruler to the nearest 0.1cm, as 4.5cm and 9.3cm What does a calculator tell me the area is? What is the range of areas that my measurements might indicate (consider the range of lengths that my original measurements might cover)?

Rules for Multiplication and Division The number of significant figures in the result is the same as in the measurement with the smallest number of significant figures.

2.6 Rules for Addition and Subtraction The number of significant figures in the result is the same as in the measurement with the smallest number of decimal places.

2.7 Rules for Combined Units Multiplication / Division –When you Multiply or Divide measurements you must carry out the same operation with the units as you do with the numbers 50 cm x 150 cm = 7500 cm2 20 m / 5 s = 4 m/s or 4 ms-1 16m / 4m = 4 Addition / Subtraction –When you Add or Subtract measurements they must be in the same units and the units remain the same 50 cm cm = 200 cm 20 m/s – 15 m/s = 5 m/s

32.27  1.54 =  =  =  10 6  =     =  Calculate the following:

= = = – = Look for the last important digit Calculate the following:

Mixed Order of Operation  = ( )  ( ) = = = = =  = =

Calculate the following. Give your answer to the correct number of significant figures and use the correct units 11.7 km x km = 12 mm x 34 mm x mm = m / 7 s = 108 kg / 550 m3 = 23.2 L + 14 L = 55.3 s s = cm – 4.2 cm = km – km =

B.2)Sig Figs in Addition/Subtraction Express the result with the same number of decimal places as the number in the operation with the least decimal places. Ex: 2.33 cm cm 5.3 cm (Result is rounded to one decimal place)

B.3) Sig Figs in Multiplication/Division Express the answer with the same sig figs as the factor with the least sig figs. Ex: 3.22 cm x 2.0 cm 6.4 cm 2 (Result is rounded to two sig figs)

2330 cm cm 2330 cm cm 2330 cm More example

B.4) Constant and Counting Numbers Constant number have infinite sig. figs. Counting numbers have infinite sig figs. Ex: 3 apples Eg. π= ……

practice 1.Calculate Volume of sphere with 2. Perimeter of the big circle

Try the following = (8.71 x )/0.056 = 13m 4.91m 2

IV Dimension Analysis – some simple rules

Example 1 - impossible: 40m + 20m/s or 12.5 s - 20m 2 - Can Do:50.0m m=70.6m and 40m/s +11m/s =51m/s

Example 2 The above expression yields: a)4.5 m kg b)4.5 g km c)A or B d)Impossible to evaluate (dimensionally invalid)