Chapter 3A. Measurement and Significant Figures quantities. • Write the base units for mass, length, and time in SI and USCU units. • Convert one unit to another for the same quantity

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

A PowerPoint Presentation by

Paul E. Tippens, Professor of Physics

Southern Polytechnic State University

A PowerPoint Presentation byA PowerPoint Presentation by

Paul E. Tippens, Professor of PhysicsPaul E. Tippens, Professor of Physics

Southern Polytechnic State UniversitySouthern Polytechnic State University

© 2007

NASANASA

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

Objectives: After completing this Objectives: After completing this module, you should be able to:module, you should be 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 QuantitiesA A physical quantityphysical quantity is a quantifiable or is a quantifiable or assignable property ascribed to a assignable property ascribed to a partiparti-- cularcular phenomenon, body, or substance. phenomenon, body, or substance.

TimeTimeElectric Electric ChargeCharge

LengthLength

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

Units of MeasureUnits of Measure

Measuring Measuring diameter of disk.diameter of disk.

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

Based on definition, we say Based on definition, we say the diameter is the diameter is 0.12 m0.12 m or or 12 centimeters.12 centimeters.

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

1 m1 m1 second

299,792, 458t

SI Unit of Measure for MassSI Unit of Measure for Mass

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

SI Unit of Measure for TimeSI Unit of Measure for Time

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

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

Seven Fundamental UnitsSeven Fundamental Units

QuantityQuantity UnitUnit SymbolSymbol

LengthLength MeterMeter mmMassMass KilogramKilogram kgkgTimeTime SecondSecond SS

Electric CurrentElectric Current AmpereAmpere AATemperatureTemperature KelvinKelvin KK

Luminous IntensityLuminous Intensity CandelaCandela cdcdAmount of SubstanceAmount of Substance MoleMole molmol

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

Systems of UnitsSystems of UnitsSI System:SI System: The international system of units The international system of units established by the International Committee established by the International Committee on Weights and Measures. Such units are on Weights and Measures. Such units are based on strict definitions and are the only based on strict definitions and are the only officialofficial units for physical quantities.units for physical quantities.

US Customary Units (USCU):US Customary Units (USCU): Older units still Older units still in common use by the United States, but in common use by the United States, but definitions must be based on SI units.definitions must be based on SI units.

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

QuantityQuantity SI unitSI unit USCS unitUSCS unit

MassMass kilogram (kg)kilogram (kg) slug (slug)slug (slug)

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

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

ForceForce newtonnewton (N)(N) pound (lb)pound (lb)

Procedure for Converting UnitsProcedure for Converting 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 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 cm1 in. = 2.54 cm..

Step 1: Write down Step 1: Write down quantity to be converted.quantity to be converted. 12 in.12 in.

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

1 in. = 2.54 cm1 in. = 2.54 cm

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

1 in.2.54 cm2.54 cm

1 in

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

From Step 3. or1 in.

2.54 cm2.54 cm

1 in

2.54 cm12 in. 30.5 cm1 in.

21 in. in.12 in. 4.72 2.54 cm cm

Wrong Wrong Choice!Choice!

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 = 3600 s1 h = 3600 s..

Step 1: Write down Step 1: Write down quantity to be converted.quantity to be converted.

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

mi60h

Note: Note: Write units so that numerators and Write units so that numerators and denominators of fractions are clear.denominators of fractions are 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 = 3600 s1 h = 3600 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 s1 h = 3600 s

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 be Step 3, shown here for clarity, can really be done mentally and need not be written down.done mentally and need not be 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 = 3600 s1 h = 3600 s..

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

mi 5280 ft 1 h60 88.0 m/sh 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 MeasurementUncertainty of MeasurementAll measurements are assumed to be All measurements are assumed to be

approximate with the last digit estimated.approximate with the last digit 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 0.3 is estimated as 0.3 of the interval between 3 and 4.of the interval between 3 and 4.

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

0 1 2Length = 1.43 cmLength = 1.43 cm

The last digit is estimated, but is The last digit is estimated, but is significantsignificant. It . It tells us the actual length is between 1.40 cm tells us the actual length is between 1.40 cm and 1.50. It would not be possible to estimate and 1.50. It would not be possible to estimate yet another digit, such as 1.436.yet another digit, such as 1.436.

This measurement of length can be given in three significant digits—the last is estimated.

Significant Digits and NumbersSignificant Digits and Numbers

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

0.0062 cm 0.0062 cm 2 significant figures2 significant figures4.0500 cm 4.0500 cm 5 significant figures5 significant figures0.1061 cm 0.1061 cm 4 significant figures4 significant figures

50.0 cm 50.0 cm 3 significant figures3 significant figures

50,600 cm 50,600 cm 3 significant figures3 significant figures

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 Example:Example:

Least significant factor (45) has only Least significant factor (45) has only twotwo (2) (2) digits so only digits so only twotwo are justified in the answer.are justified in the answer.

The appropriate way The appropriate way to write the answer is:to write the 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.


Ex: Ex: 9.65 cm + 8.4 cm 9.65 cm + 8.4 cm –– 2.89 cm = 15.16 cm2.89 cm = 15.16 cm

Note that the Note that the least preciseleast precise measure is measure is 8.4 cm8.4 cm. . Thus, answer must be to nearest Thus, answer must be to nearest tenthtenth of cm of cm even though it requires 3 significant digits.even though it requires 3 significant digits.

The appropriate way The appropriate way to write the answer is:to write the answer is: 15.2 cm15.2 cm

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

A = LW = (8.71 cm)(3.2 cm) = 27.872 cmA = LW = (8.71 cm)(3.2 cm) = 27.872 cm22

Only 2 digits justified:Only 2 digits justified: A = 28 cm2A = 28 cm2

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

p = 8.71 cm + 3.2 cm + 8.71 cm + 3.2 cmp = 8.71 cm + 3.2 cm + 8.71 cm + 3.2 cm

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

Rounding NumbersRounding NumbersRemember that significant figures apply to Remember that significant figures apply to your your reported resultreported result. Rounding off your . Rounding off your numbers in the process can lead to errors.numbers in the 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 to just With calculators, it is usually easier to just keep all digits until you report the result.keep all digits until you report the result.

Rules for Rounding NumbersRules for Rounding Numbers

Rule 1.Rule 1. If the remainder If the remainder beyond the last digitbeyond the last digit toto be reportedbe reported is less than 5, drop the last digit.is less than 5, drop the last digit.

Rule 2.Rule 2. If the remainder is greater than 5, If the remainder is greater than 5, increase the final digit by 1.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 last remainder is exactly 5, then round the last digit to the digit to the closest even numberclosest even number..

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

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

4.994994.99499

0.094030.09403

95,63295,632

0.020320.02032

becomes becomes 4.994.99


becomes becomes 95,60095,600


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


ExamplesExamples

2.34522.3452

0.087570.08757

23,650.0123,650.01

4.995024.99502





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


ExamplesExamples

3.775003.77500

0.0244500.024450

96,650096,6500

5.095005.09500





Working with NumbersWorking with NumbersClassroom work and Classroom work and laboratory work should laboratory work should be treated differently. be treated 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 46 m/sm/s undergoes constant undergoes constant acceleration of acceleration of 2 m/s2 m/s22 for a time of for a time of 4.3 s4.3 s. . Find total displacement, given formula.Find total 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 is For class work, we assume all given info is accurate to 3 significant figures.accurate to 3 significant figures.

X = 217 mX = 217 m

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

Note that the precision of each measure Note that the precision of each measure is to the nearest tenth of a millimeter. is to the nearest tenth of a millimeter. However, the length has four significant However, the length has four significant digits and the width has only three.digits and the width 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 digits).Two (9.3 has least significant 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 mm233.3 mm and the width is and the width is 9.3 mm9.3 mm. Find area.. Find area.

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

Area = 2169.69 mmArea = 2169.69 mm22

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

Area = 2200 mm2Area = 2200 mm2

L = 233.3 mmL = 233.3 mm

W = 9.3 mmW = 9.3 mm

Lab Example (Cont.):Lab Example (Cont.): Find Find perimeterperimeter of of sheet of metal measured 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 + 9.3 mm= 233.3 mm + 9.3 mm + 233.3 mm + 9.3 mm

pp = 485.2 mm= 485.2 mm

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

Perimeter = 485.2 mmPerimeter = 485.2 mm

L = 233.3 mmL = 233.3 mm

W = 9.3 mmW = 9.3 mm

Note: Note: The result has The result has moremore significant significant digits than the width digits 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 shortprovides a short--hand method for expressing 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 m0.00370 s 3.70 x 10 s

v

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 significant digits.accurate to three significant digits.

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

The The ““00”” is significantis significant——the last digit in doubt.the last digit in doubt.

Seven Fundamental UnitsSeven Fundamental UnitsQuantityQuantity UnitUnit SymbolSymbol

LengthLength MeterMeter mmMassMass KilogramKilogram kgkgTimeTime SecondSecond SS

Electric CurrentElectric Current AmpereAmpere AATemperatureTemperature KelvinKelvin KK

Luminous IntensityLuminous Intensity CandelaCandela cdcdAmount of SubstanceAmount of Substance 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.



Summary Summary –– Significant DigitsSignificant Digits

Rules for Rounding NumbersRules for Rounding Numbers

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

Rule 2. If the remainder is greater than 5, Rule 2. If the remainder is greater than 5, increase the final digit by 1.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 last remainder is exactly 5, then round the last digit to the digit to the closest even numberclosest even number..

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

Working with NumbersWorking with Numbers

In the classroom, we In the classroom, we assume all given info assume all given info is accurate to 3 is accurate to 3 signisigni-- ficantficant figures.figures.

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

Conclusion of Measurement Conclusion of Measurement Significant Digits ModuleSignificant Digits Module

Documents

Chapter 3A. Measurement and Significant Figures quantities. • Write the base units for mass, length, and time in SI and USCU units. • Convert one unit to another for the same quantity