Measurements

Chapter 3A. Measurement and Chapter 3A. Measurement and Significant FiguresSignificant Figures

A PowerPoint Presentation byA PowerPoint Presentation by

Paul E. Tippens, Professor of Paul E. Tippens, Professor of PhysicsPhysics

Southern Polytechnic State Southern Polytechnic State UniversityUniversity

A PowerPoint Presentation byA PowerPoint Presentation by

Paul E. Tippens, Professor of Paul E. Tippens, Professor of PhysicsPhysics

Southern Polytechnic State Southern Polytechnic State UniversityUniversity© 2007

NASANASA

PARCS is an atomic-clock mission scheduled to fly PARCS is an atomic-clock mission scheduled to fly on the International Space Station (ISS) in 2008. on the International Space Station (ISS) in 2008. The mission, funded by NASA, involves a laser-The mission, funded by NASA, involves a laser-cooled cesium atomic clock to improve the cooled cesium atomic clock to improve the accuracy of timekeeping on earth. accuracy of timekeeping on earth.

Objectives: After completing Objectives: After completing this module, you should be this module, you should be able to:able to:• Name and give the SI units of the seven

fundamental quantities.

• Write the base units for mass, length, and time in SI and USCU units.

• Convert one unit to another for the same quantity when given necessary definitions.

• Discuss and apply conventions for significant digits and precision of measurements.

• Name and give the SI units of the seven fundamental quantities.

• Write the base units for mass, length, and time in SI and USCU units.

• Convert one unit to another for the same quantity when given necessary definitions.

• Discuss and apply conventions for significant digits and precision of measurements.

Physical QuantitiesPhysical Quantities

A A physical quantityphysical quantity is a quantifiable is a quantifiable or assignable property ascribed to or assignable property ascribed to a particular phenomenon, body, a particular phenomenon, body, or substance. or substance.

TimeTimeElectriElectric c

ChargChargee

LengtLengthh

A A unitunit is a particular physical quantity is a particular physical quantity with which other quantities of the same with which other quantities of the same kind are compared in order to express kind are compared in order to express their value. their value.

Units of MeasureUnits of Measure

Measuring Measuring diameter of diameter of disk.disk.

A A metermeter is an is an established unit for established unit for measuring length.measuring length.

Based on definition, we Based on definition, we say the diameter is say the diameter is 0.12 0.12 mm or 12 centimeters. or 12 centimeters.

SI Unit of Measure for SI Unit of Measure for LengthLength

One One metermeter is the length of path is the length of path traveled by a light wave in a vacuum traveled by a light wave in a vacuum in a time interval of 1/299,792,458 in a time interval of 1/299,792,458 seconds.seconds.

1 m1 m

1 second

299,792,458t

SI Unit of Measure for SI Unit of Measure for MassMass

The The kilogramkilogram is the unit of is the unit of massmass - it is - it is equal to the mass of the international equal to the mass of the international prototype of the kilogram. prototype of the kilogram.

This standard is the only This standard is the only one that requires one that requires comparison to an artifact comparison to an artifact for its validity. A copy of for its validity. A copy of the standard is kept by the standard is kept by the International Bureau the International Bureau of Weights and Measures.of Weights and Measures.

SI Unit of Measure for SI Unit of Measure for TimeTime

The The secondsecond is the duration of 9 192 631 is the duration of 9 192 631 770 periods of the radiation 770 periods of the radiation corresponding to the transition between corresponding to the transition between the two hyperfine levels of the ground the two hyperfine levels of the ground state of the cesium 133 atom. state of the cesium 133 atom.

Cesium Fountain Cesium Fountain Atomic ClockAtomic Clock: The : The primary time and primary time and frequency frequency standard for the standard for the USA (NIST)USA (NIST)

Seven Fundamental UnitsSeven Fundamental Units

QuantityQuantity UnitUnit SymbolSymbol

LengthLength MeterMeter mm

MassMass KilogramKilogram kgkg

TimeTime SecondSecond SS

Electric CurrentElectric Current AmpereAmpere AA

TemperatureTemperature KelvinKelvin KK

Luminous IntensityLuminous Intensity CandelaCandela cdcd

Amount of Amount of SubstanceSubstance

MoleMole molmol

Website: Website: http://physics.nist.gov/cuu/index.htmlhttp://physics.nist.gov/cuu/index.html

Systems of UnitsSystems of Units

SI System:SI System: The international system of The international system of units established by the International units established by the International Committee on Weights and Measures. Committee on Weights and Measures. Such units are based on strict Such units are based on strict definitions and are the only definitions and are the only officialofficial units for physical quantities.units for physical quantities.US Customary Units (USCU):US Customary Units (USCU): Older units Older units still in common use by the United still in common use by the United States, but definitions must be based States, but definitions must be based on SI units.on SI units.

Units for MechanicsUnits for MechanicsInIn mechanics mechanics we use only three we use only three fundamental quantities: fundamental quantities: mass, length, and mass, length, and timetime. An additional quantity, . An additional quantity, force,force, is is derived from these three.derived from these three.

QuantityQuantity SI unitSI unit USCS unitUSCS unit

MassMass kilogram kilogram (kg)(kg)

slug (slug)slug (slug)

LengthLength meter (m)meter (m) foot (ft)foot (ft)

TimeTime second (s)second (s) second (s)second (s)

ForceForce newton (N)newton (N) pound (lb)pound (lb)

Procedure for Converting Procedure for Converting UnitsUnits

1. Write down quantity to be converted.

2. Define each unit in terms of desired unit.

3. For each definition, form two conversion factors, one being the reciprocal of the other.

4. Multiply the quantity to be converted by those factors that will cancel all but the desired units.

Example 1:Example 1: Convert Convert 12 in.12 in. to to centimeterscentimeters given that given that 1 in. = 2.54 1 in. = 2.54 cmcm..Step 1: Write down Step 1: Write down quantity to be quantity to be converted.converted.

12 in.12 in.

Step 2. Define each unit in terms of desired unit.

1 in. = 2.54 1 in. = 2.54 cmcm

Step 3. For each definition, form two conversion factors, one being the reciprocal of the other.

1 in.

2.54 cm

2.54 cm

1 in

Example 1 (Cont.):Example 1 (Cont.): Convert Convert 12 in.12 in. to to centimeterscentimeters given that 1 in. = given that 1 in. = 2.54 cm.2.54 cm.

From Step 3. or 1 in.

2.54 cm2.54 cm

1 in

2.54 cm12 in. 30.5 cm

1 in.

21 in. in.12 in. 4.72

2.54 cm cm

Wrong Wrong ChoiceChoice!!

Step 4. Multiply by those factors that will cancel all but the desired units. Treat unit symbols algebraically.

Correct Correct Answer!Answer!

Example 2:Example 2: Convert Convert 60 mi/h60 mi/h to units of to units of km/skm/s given given 1 mi. = 5280 ft1 mi. = 5280 ft and and 1 h = 1 h = 3600 s3600 s..Step 1: Write down Step 1: Write down quantity to be quantity to be converted.converted.

Step 2. Define each unit in terms of desired units.

mi60

h

Note: Note: Write units so that numerators Write units so that numerators and denominators of fractions are and denominators of fractions are clear.clear.

1 mi. = 5280 ft1 mi. = 5280 ft

1 h = 3600 s1 h = 3600 s

Ex. 2 (Cont):Ex. 2 (Cont): Convert Convert 60 mi/h60 mi/h to units of to units of km/skm/s given that given that 1 mi. = 5280 ft1 mi. = 5280 ft and and 1 h = 1 h = 3600 s3600 s..

Step 3. For each definition, form 2 conversion factors, one being the reciprocal of the other.

1 mi = 5280 ft1 mi = 5280 ft

1 h = 3600 1 h = 3600 ss

1 mi 5280 ft or

5280 ft 1 mi

1 h 3600 s or

3600 s 1 h

Step 3, shown here for clarity, can really Step 3, shown here for clarity, can really be done mentally and need not be be done mentally and need not be written down.written down.

Ex. 2 (Cont):Ex. 2 (Cont): Convert Convert 60 mi/h60 mi/h to units of to units of ft/sft/s given that given that 1 mi. = 5280 ft1 mi. = 5280 ft and and 1 h = 1 h = 3600 s3600 s..

Step 4. Choose Factors to cancel non-desired units.

mi 5280 ft 1 h60 88.0 m/s

h 1 mi 3600 s

Treating unit conversions algebraically helps to see if a definition is to be used as a multiplier or as a divider.

Uncertainty of Uncertainty of MeasurementMeasurement

All measurements are assumed to be All measurements are assumed to be approximate with the last digit approximate with the last digit

estimated.estimated.

0 1 2

The length in The length in ““cmcm” here is ” here is written as:written as:

1.43 cm1.43 cm

The last digit “The last digit “33” is estimated as ” is estimated as 0.3 of the interval between 3 and 0.3 of the interval between 3 and

4.4.

Estimated Measurements Estimated Measurements (Cont.)(Cont.)

0 1 2Length = 1.43 Length = 1.43 cmcm

The last digit is estimated, but is The last digit is estimated, but is significantsignificant. It tells us the actual length is . It tells us the actual length is between 1.40 cm and 1.50. It would not between 1.40 cm and 1.50. It would not be possible to estimate yet another digit, be possible to estimate yet another digit, such as 1.436.such as 1.436.This measurement of length can be given in three significant digits—the last is estimated.

Significant Digits and Significant Digits and NumbersNumbers

When writing numbers, zeros used When writing numbers, zeros used ONLY to help in locating the decimal ONLY to help in locating the decimal point are NOT significant—others are. point are NOT significant—others are. See examples.See examples.

0.0062 cm 0.0062 cm 2 significant 2 significant figuresfigures4.0500 cm 4.0500 cm 5 significant 5 significant

figuresfigures0.1061 cm 0.1061 cm 4 significant 4 significant figuresfigures50.0 cm 50.0 cm 3 significant 3 significant figuresfigures50,600 cm 50,600 cm 3 significant 3 significant figuresfigures

Rule 1. When approximate numbers are multiplied or divided, the number of significant digits in the final answer is the same as the number of significant digits in the least accurate of the factors.


245 N 6.97015 N/m

(3.22 m)(2.005 m)P ExamplExampl

e:e:

Least significant factor (45) has only Least significant factor (45) has only twotwo (2) digits so only (2) digits so only twotwo are justified in are justified in the answer.the answer.The appropriate The appropriate way to write the way to write the answer is:answer is:

P = 7.0 N/m2P = 7.0 N/m2

Rule 2. When approximate numbers are added or subtracted, the number of significant digits should equal the smallest number of decimal places of any term in the sum or difference.

Rule 2. When approximate numbers are added or subtracted, the number of significant digits should equal the smallest number of decimal places of any term in the sum or difference.Ex: Ex: 9.65 cm + 8.4 cm – 2.89 cm = 9.65 cm + 8.4 cm – 2.89 cm = 15.16 cm15.16 cmNote that the Note that the least preciseleast precise measure is measure is 8.4 cm8.4 cm. Thus, answer must be to . Thus, answer must be to nearest nearest tenthtenth of cm even though it of cm even though it requires 3 significant digits.requires 3 significant digits.The appropriate The appropriate way to write the way to write the answer is:answer is:

15.2 cm15.2 cm

Example 3.Example 3. Find the area of a metal Find the area of a metal plate that is 95.7 cm by 32 cm.plate that is 95.7 cm by 32 cm.

A = LW = (8.71 cm)(3.2 cm) = 27.872 A = LW = (8.71 cm)(3.2 cm) = 27.872 cmcm22

Only 2 digits Only 2 digits justified:justified:

A = 28 cm2A = 28 cm2

Example 4.Example 4. Find the perimeter of the Find the perimeter of the plate that is 95.7 cm long and 32 cm plate that is 95.7 cm long and 32 cm wide.wide.

p = 8.71 cm + 3.2 cm + 8.71 cm + p = 8.71 cm + 3.2 cm + 8.71 cm + 3.2 cm3.2 cm

Ans. to tenth of cm:Ans. to tenth of cm: p = 23.8 cmp = 23.8 cm

Rounding NumbersRounding NumbersRemember that significant figures Remember that significant figures apply to your apply to your reported resultreported result. . Rounding off your numbers in the Rounding off your numbers in the process can lead to errors.process can lead to errors.

Rule: Always retain at least one more significant figure in your calculations than the number you are entitled to report in the result.

Rule: Always retain at least one more significant figure in your calculations than the number you are entitled to report in the result.

With calculators, it is usually easier With calculators, it is usually easier to just keep all digits until you report to just keep all digits until you report the result.the result.

Rules for Rounding Rules for Rounding NumbersNumbers

Rule 1.Rule 1. If the remainder If the remainder beyond the last beyond the last digitdigit toto be reportedbe reported is less than 5, drop is less than 5, drop the last digit.the last digit.Rule 2.Rule 2. If the remainder is greater than If the remainder is greater than 5, increase the final digit by 1.5, increase the final digit by 1.

Rule 3.Rule 3. To prevent rounding bias, if the To prevent rounding bias, if the remainder is exactly 5, then round the remainder is exactly 5, then round the last digit to the last digit to the closest even numberclosest even number..

ExamplesExamplesRule 1. If the remainder Rule 1. If the remainder beyond the last beyond the last digitdigit to be reported is less than 5, drop to be reported is less than 5, drop the last digit. the last digit.

Round the following to 3 significant Round the following to 3 significant figures:figures:

4.994994.99499

0.09400.09403395,63295,632

0.02030.020322

becomes becomes 4.994.99


becomes becomes 95,60095,600


Rule 2. If the remainder is greater Rule 2. If the remainder is greater than 5, increase the final digit by 1. than 5, increase the final digit by 1.


ExamplesExamples

2.34522.3452

0.08750.08757723,650.023,650.0114.995024.99502



becomes becomes 23,70023,700


Rule 3. To prevent rounding bias, if the Rule 3. To prevent rounding bias, if the remainder is exactly 5, then round the remainder is exactly 5, then round the last digit to the last digit to the closest even numberclosest even number..


ExamplesExamples

3.77503.7750000.024450.024450096,65096,650005.09505.095000


becomes becomes 0.02440.0244becomes becomes 96,60096,600


Working with NumbersWorking with Numbers

Classroom work and Classroom work and laboratory work laboratory work should be treated should be treated differently. differently. In class, the Uncertainties in quantities are not usually known. Round to 3 significant figures in most cases.

In lab, we know the limitations of the measurements. We must not keep digits that are not justified.

Classroom Example:Classroom Example: A car traveling A car traveling initially at initially at 46 m/s46 m/s undergoes undergoes constant acceleration of constant acceleration of 2 m/s2 m/s22 for a for a time of time of 4.3 s4.3 s. Find total . Find total displacement, given formula.displacement, given formula.

210 2

2 212(46 m/s)(4.3 s) (2 m/s )(4.3 s)

197.8 m + 18.48 m 216.29 m

x v t at

For class work, we assume all given info For class work, we assume all given info is accurate to 3 significant figures.is accurate to 3 significant figures.

X = 217 m

X = 217 m

Laboratory Example:Laboratory Example: The length of a The length of a sheet of metal is measured as 233.3 sheet of metal is measured as 233.3 mm and the width is 9.3 mm. Find mm and the width is 9.3 mm. Find area.area.

Note that the precision of each Note that the precision of each measure is to the nearest tenth of a measure is to the nearest tenth of a millimeter. However, the length has millimeter. However, the length has four significant digits and the width four significant digits and the width has only three.has only three.How many significant digits are in the How many significant digits are in the product of length and width (area)?product of length and width (area)?

Two (9.3 has least significant Two (9.3 has least significant digits).digits).

Lab Example (Cont.):Lab Example (Cont.): The length of a The length of a sheet of metal is measured as sheet of metal is measured as 233.3 233.3 mmmm and the width is and the width is 9.3 mm9.3 mm. Find . Find area.area.

Area = LW = (233.3 mm)(9.3 Area = LW = (233.3 mm)(9.3 mm)mm)

Area = 2169.69 Area = 2169.69 mmmm22

But we are entitled But we are entitled to only to only twotwo significant digits. significant digits. Therefore, the Therefore, the answer becomes:answer becomes:

Area = 2200 mm2

Area = 2200 mm2

L = 233.3 mmL = 233.3 mm

W = 9.3 W = 9.3 mmmm

Lab Example (Cont.):Lab Example (Cont.): Find Find perimeterperimeter of sheet of metal measured of sheet of metal measured L =L = 233.3 mm233.3 mm and and W =W = 9.3 mm9.3 mm. . (Addition Rule)(Addition Rule)pp = 233.3 mm + 9.3 mm + 233.3 mm + = 233.3 mm + 9.3 mm + 233.3 mm +

9.3 mm9.3 mmpp = 485.2 mm= 485.2 mm

Note: The answer is Note: The answer is determined by the determined by the least preciseleast precise measure. (the measure. (the tenthtenth of a mm)of a mm)

Perimeter = 485.2 mm

Perimeter = 485.2 mm

L = 233.3 mmL = 233.3 mm

W = 9.3 W = 9.3 mmmm

Note: Note: The result The result has has moremore significant digits significant digits than the width than the width factor in this case.factor in this case.

Scientific NotationScientific Notation

0 000000001 10

0 000001 10

0 001 10

1 10

1000 10

1 000 000 10

1 000 000 000 10

9

6

3

0

3

6

9

.

.

.

, ,

, , ,

Scientific notationScientific notation provides a short-hand method for expressing provides a short-hand method for expressing very small and very large numbers.very small and very large numbers.

Examples:

93,000,000 mi = 9.30 x 107 mi

0.00457 m = 4.57 x 10-3 m

2

-3

876 m 8.76 x 10 m

0.00370 s 3.70 x 10 sv

53.24 x 10 m/sv 53.24 x 10 m/sv

Scientific Notation and Scientific Notation and Significant FiguresSignificant Figures

With With Scientific notationScientific notation one can easily keep track of one can easily keep track of significant significant digitsdigits by using only those digits that are by using only those digits that are necessary in the necessary in the mantissamantissa and letting the and letting the power of tenpower of ten locate the decimal.locate the decimal.

Mantissa x 10Mantissa x 10-4 -4

mm

Example.Example. Express the number Express the number 0.0006798 m0.0006798 m, accurate to three , accurate to three significant digits.significant digits.

6.80 x 10-4 m6.80 x 10-4 m

The “0” is significant—the last digit in The “0” is significant—the last digit in doubt.doubt.

Seven Fundamental UnitsSeven Fundamental Units

QuantityQuantity UnitUnit SymbolSymbol

LengthLength MeterMeter mm

MassMass KilogramKilogram kgkg

TimeTime SecondSecond SS

Electric CurrentElectric Current AmpereAmpere AA

TemperatureTemperature KelvinKelvin KK

Luminous IntensityLuminous Intensity CandelaCandela cdcd

Amount of Amount of SubstanceSubstance

MoleMole molmol

SUMMARYSUMMARY

Summary: Procedure for Summary: Procedure for Converting UnitsConverting Units

1. Write down quantity to be converted.

2. Define each unit in terms of desired unit.

3. For each definition, form two conversion factors, one the reciprocal of the other.

4. Multiply the quantity to be converted by those factors that will cancel all but the desired units.


Rule 2. When approximate numbers are added or subtracted, the number of significant digits should equal the smallest number of decimal places of any term in the sum or difference.

Summary – Significant Summary – Significant DigitsDigits

Rules for Rounding Rules for Rounding NumbersNumbers

Rule 1. If the remainder Rule 1. If the remainder beyond the last beyond the last digitdigit to be reported is less than 5, drop to be reported is less than 5, drop the last digitthe last digitRule 2. If the remainder is greater than Rule 2. If the remainder is greater than 5, increase the final digit by 1.5, increase the final digit by 1.

Rule 3. To prevent rounding bias, if the Rule 3. To prevent rounding bias, if the remainder is exactly 5, then round the remainder is exactly 5, then round the last digit to the last digit to the closest even numberclosest even number..

Classroom work and lab work should Classroom work and lab work should be treated differently unless told be treated differently unless told otherwise.otherwise.

Working with NumbersWorking with Numbers

In the classroom, In the classroom, we assume all we assume all given info is given info is accurate to 3 signi- accurate to 3 significant figures.ficant figures.

In lab, the number In lab, the number of significant figures of significant figures will depend on will depend on limitations of the limitations of the instruments.instruments.

Conclusion of Measurement Conclusion of Measurement Significant Digits ModuleSignificant Digits Module

Measurements

Documents

si units

unit of mass

uscu units

units of kms

older units

official units

base units

units of fts