measurement notes

Standards of MeasurementStandards of Measurement

Using a measuring tool to compare some Using a measuring tool to compare some dimension of an object to a standard.dimension of an object to a standard.

For example, at one time the For example, at one time the standard for length was the standard for length was the king’s foot. What are some king’s foot. What are some

problems with this standard?problems with this standard?

WHO CARES ABOUT MEASUREMENT?!SCENARIOS:1.Your friends ask you to go somewhere…

2.You want to go somewhere with your car…

3.You’re sick…

4.You’re hungry…

5.Buying a new outfit?

6.The sports field…

7.You start school at 7:45am but you’re leaving from your friends house…

8.A tree falls on the road… can you lift it?

9.You’re vacationing and your plane must take off…

UNITS OF MEASUREMENTUNITS OF MEASUREMENTUse SI units — based on the metric

system

Length

Mass

Volume

Time

Temperature

Types of Observations and Types of Observations and MeasurementsMeasurements

QUALITATIVEQUALITATIVE——changes in color and changes in color and physical state.physical state.

QUANTITATIVE MEASUREMENTS QUANTITATIVE MEASUREMENTS ------involve involve numbersnumbers

◦ UseUse SI unitsSI units — based on the — based on the metric systemmetric system

Stating a MeasurementStating a Measurement

In every measurement there is aIn every measurement there is a

Number Number followed by a…followed by a…

Unit Unit from a measuring devicefrom a measuring device

The number should also be as The number should also be as preciseprecise as the as the

measurement!measurement!

Some Tools for MeasurementSome Tools for Measurement

Which tool(s) Which tool(s) would you use to would you use to measure:measure:

A.A. temperature temperature

B.B. volume volume

C.C. time time

D.D. weight weight

How many stars are in the Universe?

300 Sextillion Stars

What is Scientific Notation?What is Scientific Notation?

Scientific notation = expressing Scientific notation = expressing really big numbers or really small really big numbers or really small numbers…numbers…

IN A CONCISE WAY!IN A CONCISE WAY!

Scientific notation consists Scientific notation consists of two parts:of two parts:

1. A number between 1 and 101. A number between 1 and 10

2.2.A power of A power of 1010N x 10N x 10xx

To change standard form to To change standard form to scientific notation…scientific notation…

1.1. Place the decimal point so that there is one Place the decimal point so that there is one non-zero digit to the left of the decimal point.non-zero digit to the left of the decimal point.

2.2. Count the number of decimal places the Count the number of decimal places the decimal point has “moved” from the original decimal point has “moved” from the original number. This will be the exponent of the 10.number. This will be the exponent of the 10.

33. If original number is less than 1 = negative . If original number is less than 1 = negative exponentexponent

If the original number is > 1 = positive exponentIf the original number is > 1 = positive exponent

ExamplesExamplesGiven: 289,800,000Given: 289,800,000

2.8989 x 109

Given: 0.000567Given: 0.0005675.67 x 10-4

To change scientific notation To change scientific notation to standard form…to standard form…Simply move the decimal point to Simply move the decimal point to

the right for positive exponent 10. the right for positive exponent 10.

Move the decimal point to the left Move the decimal point to the left for negative exponent 10.for negative exponent 10.

(Use zeros to fill in places.)(Use zeros to fill in places.)

ExampleExampleGiven: 5.093 x 10Given: 5.093 x 1066

5,093,0005,093,000

Given: 1.976 x 10Given: 1.976 x 10-4-4

0.00019760.0001976

Warm-Up: Express these numbers in Express these numbers in

Scientific Notation:Scientific Notation:

1)1) 405789405789

2)2) 0.0038720.003872

3)3) 30000000003000000000

4)4) 22

Metric PrefixesMetric Prefixes

Metric PrefixesMetric PrefixesKilo-Kilo- means 1000 of that unit means 1000 of that unit

◦ 1 kilometer (km) = 1000 meters (m)1 kilometer (km) = 1000 meters (m)

Centi-Centi- means 1/100 of that unit means 1/100 of that unit

◦ 1 meter (m) = 100 centimeters (cm)1 meter (m) = 100 centimeters (cm)

Milli-Milli- means 1/1000 of that unit means 1/1000 of that unit

◦ 1 Liter (L) = 1000 milliliters (mL)1 Liter (L) = 1000 milliliters (mL)

Metric Conversions“King Henry Died by Drinking Chocolate

Milk”K H D b D C M

1) 2400 g = _____ kg

2) 0.24 L = _____ mL

3) 0.3 hm = _____ m

4) 0.10 ms = _____ sec

Learning Check

1. 1000 m = 1 1. 1000 m = 1 ______ a) mm b) km c) dma) mm b) km c) dm

2. 0.001 g = 1 2. 0.001 g = 1 ___ ___ a) mg b) kg c) dga) mg b) kg c) dg

3. 0.1 L = 1 3. 0.1 L = 1 ______ a) mL b) cL c) dLa) mL b) cL c) dL

4. 0.01 m = 1 ___ 4. 0.01 m = 1 ___ a) mm b) cm c) dma) mm b) cm c) dm

Reading a Meter stickReading a Meter stick

First digit (known)First digit (known) = =

Second digit (known)Second digit (known) ==

Third digit (estimated) =Third digit (estimated) =

Length reportedLength reported ==

Known + Estimated DigitsKnown + Estimated Digits

• Known digitsKnown digits are 100% certain!are 100% certain!

• The third digit is estimated (uncertain)The third digit is estimated (uncertain)

• In the reported length, all three digits (ex. = In the reported length, all three digits (ex. = 2.75 cm) are significant 2.75 cm) are significant includingincluding the the estimated one. estimated one.

CERTAIN + ESTIMATED = Significant Digits!CERTAIN + ESTIMATED = Significant Digits!

Learning CheckLearning Check

What is the length of the line?What is the length of the line?

How does your answer compare with your How does your answer compare with your

neighbor’s answer? Why or why not?neighbor’s answer? Why or why not?

Measuring Volume

Make sure you’re at eye level.

Take measurement at the bottom of the concave meniscus.

Always estimate Always estimate ONEONE place past the smallest mark! place past the smallest mark!

Three Three targets with targets with three arrows three arrows each to each to shoot.shoot.

Can you hit the bull's-eye?Can you hit the bull's-eye?

Can you define accuracy and precision?Can you define accuracy and precision?

Accuracy Vs. Precision

Accuracy: How close a measured value is to the

accepted/true value.

Precision: How close a series of measurements are to one

another.◦ Note: they may NOT be accurate

Conversion FactorsExpresses an equal quantity of one unit to another.

1 in. = 2.54 cm or 1 inch per 2.54 cm.

: 1 in. and 2.54 cm 2.54 cm 1 in.

EXAMPLE: How many students are in this class?

Learning Check

Write conversion factors that relate Write conversion factors that relate each of the following pairs of units:each of the following pairs of units:

1. Liters and mL1. Liters and mL

2. Hours and minutes2. Hours and minutes

3. Meters and kilometers3. Meters and kilometers

How many minutes are in 2.5 hours?

Dimensional Analysis

Sample ProblemIf you have lived for 52,200 hours, If you have lived for 52,200 hours,

how long were you alive in years? how long were you alive in years? (Remember SIG FIGS!)(Remember SIG FIGS!)

Learning CheckLearning Check An adult human has 4.65 L of blood. How An adult human has 4.65 L of blood. How many gallons of blood is that?many gallons of blood is that?

Unit plan L qt gallon

Equalities:

1.06 qt = 1.0 L 1 gallon = 4 quarts

Setup:

You Try This One!You Try This One!If Jacob stands on Spencer’s If Jacob stands on Spencer’s shoulders, they are two and shoulders, they are two and a half yards high. How a half yards high. How many inches is that?many inches is that?

Learning Check How many seconds are in 1.4 days?

Unit plan: days hr min seconds

Wait a minute!

What is What is wrongwrong with the following with the following setup?setup?

1.4 day x 1.4 day x 1 day 1 day x x 60 min 60 min x x 60 sec60 sec

24 hr 1 hr 24 hr 1 hr 1 min1 min

Dealing with Two Units

If your pace on a treadmill is 65 If your pace on a treadmill is 65 meters per minute, what is your meters per minute, what is your speed in miles per hour?speed in miles per hour?

Sample ProblemThe United States uses 18,000,000 gallons of

gasoline per day. How many liters are used each minute?

Sample ProblemPotatoes sell 5 dollars for a 10 pound bag.

How much does one gram of potatoes cost?

What about Square and Cubic units?

Cubed Unit = CUBED NUMBER!Cubed Unit = CUBED NUMBER!

Best way: Square or cube the Best way: Square or cube the ENTIRE conversion factor.ENTIRE conversion factor.

Example: Convert 4.3 cmExample: Convert 4.3 cm33 to mm to mm33

Learning CheckLearning CheckA Nalgene water A Nalgene water

bottle holds bottle holds 1000 cm1000 cm33 of of dihydrogen dihydrogen monoxide. How monoxide. How many cubic many cubic meters is that?meters is that?

SolutionSolution1000 cm1000 cm33

Significant Figures

• The numbers reported in a measurement are limited by the measuring tool.

• Significant figures • All KNOWN Digits• One ESTIMATED Digit

Counting Significant Figures

RULE 1. All non-zero digits in a measured number are significant.

Sig Figs38.15 cm ___5.6 ft ___65.6 lb ___122.55 m ___

Leading Zeros

RULE 2. Leading zeros in decimal numbers

are NOT significant.

Sig Figs

0.008 mm ____

0.0156 oz ____

0.0042 lb ____

0.000262 mL ____

Sandwiched Zeros

RULE 3. Zeros between nonzero numbers are significant.

Sig Figs

50.8 mm _____

2001 min _____

0.702 lb _____

0.00405 m _____

Trailing Zeros

RULE 4. Trailing zeros in numbers without

decimals are NOT significant. Any final zero

used after a decimal point is significant.

Sig Figs

25,000 in. ____

200.30 yr ____

4.320 gal ____

25,005,000 g ____

Learning CheckIn which set(s) do both numbers contain the same number of significant figures?

1) 22.0 and 22.00

2) 400.0 and 40

3) 0.000015 and 150,000

Learning CheckLearning CheckState the number of significant figures in State the number of significant figures in each of the following:each of the following:

A. 0.030 mA. 0.030 m _______ _______

B. 4.050 LB. 4.050 L _______ _______

C. 0.0008 gC. 0.0008 g _______ _______

D. 3.00 mD. 3.00 m _______ _______

E. 2,080,000 beesE. 2,080,000 bees _______ _______

Rounding Significant Figures1) Round 2.0345 g to 3 sig figs.

2) Round 0.000455 to 2 sig figs.

Learning Check

A. Which answers contain 3 significant figures?1) 0.4760 2) 0.00476 3) 4760

B. All the zeros are significant in

1) 0.00307 2) 25.300 3) 2.050 x 103

C. 534,675 rounded to 3 significant figures is

1) 535 2) 535,000 3) 5.35 x 105

Significant Numbers in Calculations

• A calculated answer cannot be more precise than the measuring tool.

• A calculated answer must match the least precise measurement.

Adding and SubtractingThe answer = same number of decimal places as the measurement with the fewest decimal places.

25.2 one decimal place

+ 1.34 two decimal places 26.54answer 26.5 (rounded to one decimal place)

Learning Check In each calculation, round the answer to In each calculation, round the answer to the correct number of sig figs.the correct number of sig figs.

A. 235.05 + 19.6 + 2.1 = A. 235.05 + 19.6 + 2.1 =

1) 256.751) 256.75 2) 256.8 2) 256.8 3) 2573) 257

B. 58.925 - 18.2B. 58.925 - 18.2 ==

1) 40.7251) 40.725 2) 40.73 2) 40.73 3) 40.73) 40.7

Multiplying and Dividing

Answer = the same number of significant figures as the measurement with the fewest significant figures.

What’s wrong with this answer?

28.2 x 3.5 = 98.7

Learning CheckLearning Check A. 2.19 x 4.2 =A. 2.19 x 4.2 =

1) 91) 9 2) 9.2 2) 9.2 3) 9.1983) 9.198

B. 4.311 ÷ 0.07 =B. 4.311 ÷ 0.07 = 1)1) 61.5861.58 2) 62 2) 62 3) 60 3) 60

C. C. (2.54 x 0.0028) = (2.54 x 0.0028) = (0.0105 x 0.060)(0.0105 x 0.060)1) 11.31) 11.3 2) 112) 11 3) 0.041 3) 0.041

DENSITYDENSITY - an important and useful - an important and useful physical propertyphysical property

Density mass (g)volume (cm3)


Mercury

13.6 g/cm13.6 g/cm33 21.5 g/cm21.5 g/cm33

Aluminum

2.7 g/cm2.7 g/cm33

Platinum

ProblemProblem A piece of copper has a A piece of copper has a mass of 57.54 g. It is 9.36 cm long, mass of 57.54 g. It is 9.36 cm long, 7.23 cm wide, and 0.95 mm thick. 7.23 cm wide, and 0.95 mm thick. Calculate density (g/cmCalculate density (g/cm33).).



StrategyStrategy1. Get dimensions in common units.1. Get dimensions in common units.

2.2. Calculate volume in cubic Calculate volume in cubic centimeters.centimeters.

3. Calculate the density.3. Calculate the density.

SOLUTIONSOLUTION

1. Get dimensions in common units.1. Get dimensions in common units.

2.2. Calculate volume in cubic Calculate volume in cubic centimeters.centimeters.

3. Calculate the density.3. Calculate the density.

0.95 mm • 1cm

10 mm = 0.095 cm

57.54 g

6.4 cm3 = 9.0 g / cm3

(9.36 cm)(7.23 cm)(0.095 cm) = 6.4 cm(9.36 cm)(7.23 cm)(0.095 cm) = 6.4 cm33

Note only 2 significant figures in the answer!Note only 2 significant figures in the answer!

PROBLEM: Mercury (Hg) has a density of 13.6 PROBLEM: Mercury (Hg) has a density of 13.6 g/cmg/cm33. What is the mass of 95 mL of Hg in grams? . What is the mass of 95 mL of Hg in grams? In pounds? In pounds?

PROBLEM: Mercury (Hg) has a density of 13.6 PROBLEM: Mercury (Hg) has a density of 13.6 g/cmg/cm33. What is the mass of 95 mL of Hg in grams? . What is the mass of 95 mL of Hg in grams? In pounds? In pounds?

PROBLEM: Mercury (Hg) has a density of 13.6 g/cmPROBLEM: Mercury (Hg) has a density of 13.6 g/cm33. . What is the mass of 95 mL of Hg?What is the mass of 95 mL of Hg?


StrategyStrategy1.1. Use density to calc. mass (g) from Use density to calc. mass (g) from volume.volume.2.2. Convert mass (g) to mass (lb)Convert mass (g) to mass (lb)

Need to know conversion factorNeed to know conversion factor= 454 g / 1 lb= 454 g / 1 lb

First, note thatFirst, note that 1 cm1 cm33 = 1 mL = 1 mL



1.1. Convert volume to massConvert volume to mass

95 cm3 • 13.6 g

cm3 = 1.3 x 103 g

1.3 x 103 g • 1 lb

454 g = 2.8 lb

2.2. Convert mass (g) to mass (lb)Convert mass (g) to mass (lb)


Osmium is a very dense metal. What is its Osmium is a very dense metal. What is its

density in g/cmdensity in g/cm3 3 if 50.00 g of the metal if 50.00 g of the metal occupiesoccupies

a volume of 2.22cma volume of 2.22cm33??

1) 2.25 g/cm1) 2.25 g/cm33

2)2) 22.5 g/cm22.5 g/cm33

3)3) 111 g/cm111 g/cm33

Solution

2) Placing the mass and volume of the 2) Placing the mass and volume of the osmium metal into the density setup, we osmium metal into the density setup, we obtainobtain

D = D = massmass = = 50.00 g 50.00 g = = volumevolume 2.22 cm2.22 cm33

= 22.522522 g/cm= 22.522522 g/cm3 3 = 22.5 g/cm= 22.5 g/cm33

Volume DisplacementVolume Displacement

A solid displaces a matching volume of A solid displaces a matching volume of water when the solid is placed in water.water when the solid is placed in water.

33 mL33 mL25 mL 25 mL


What is the density (g/cmWhat is the density (g/cm33) of 48 g of a ) of 48 g of a metal if the metal raises the level of water in metal if the metal raises the level of water in a graduated cylinder from 25 mL to 33 mL? a graduated cylinder from 25 mL to 33 mL?

1) 0.2 g/ cm1) 0.2 g/ cm33 2) 6 g/m 2) 6 g/m33 3) 252 3) 252 g/cmg/cm33

33 mL33 mL 25 mL25 mL


Which diagram represents the liquid Which diagram represents the liquid layers in the cylinder?layers in the cylinder?

(K) Karo syrup (1.4 g/mL), (V) vegetable (K) Karo syrup (1.4 g/mL), (V) vegetable oil (0.91 g/mL,) (W) water (1.0 g/mL)oil (0.91 g/mL,) (W) water (1.0 g/mL)

1)1) 2) 2) 3) 3)

K

K

W

W

W

V

V

V

K

Learning CheckLearning Check The density of octane, a component of The density of octane, a component of gasoline, is 0.702 g/mL. What is the gasoline, is 0.702 g/mL. What is the mass, in kg, of 875 mL of octane?mass, in kg, of 875 mL of octane?

1) 0.614 kg1) 0.614 kg

2) 614 kg2) 614 kg

3) 1.25 kg3) 1.25 kg


If blood has a density of 1.05 g/mL, If blood has a density of 1.05 g/mL, how many liters of blood are donated how many liters of blood are donated if 575 g of blood are given?if 575 g of blood are given?

1) 1) 0.548 L0.548 L

2) 2) 1.25 L1.25 L

3) 3) 1.83 L1.83 L


A group of students collected 125 empty A group of students collected 125 empty aluminum cans to take to the recycling aluminum cans to take to the recycling center. If 21 cans make 1.0 pound of center. If 21 cans make 1.0 pound of aluminum, how many liters of aluminum aluminum, how many liters of aluminum (D=2.70 g/cm(D=2.70 g/cm33) are obtained from the ) are obtained from the cans?cans?

1) 1.0 L1) 1.0 L 2) 2.0 L2) 2.0 L 3) 4.0 L3) 4.0 L

measurement notes

Technology

original number

number of decimal places

asthe number

decimal point

measuring devicethe

ofscientific notation

bigscientific notation

standard form toto