phy115 – sault college – bazlurslide 1 measurement and the metric system

PHY115 – Sault College – Bazlur slide 1

Measurement

and

The Metric System


Standards of Measure• One cubit ?


Standards of Measure


Standards of Measure• When two people work together, they should both use

the same standards of measure.


Standards of Measure• http://news.bbc.co.uk/1/shared/spl/hi/sci_nat/

03/race_to_mars/timeline/html/1999.stm

• September 1999 Another Nasa space craft, Mars Climate Orbiter, is lost as it arrives at the Red Planet. A mix-up over units for a key space craft operation is blamed - one team used English units while the other used metric.


SI Units


Derived SI Units• http://physics.nist.gov/cuu/Units/SIdiagram.html


Prefixes for SI Units• http://en.wikipedia.org/wiki/SI• http://en.wikipedia.org/wiki/SI_prefix


Prefixes for SI Units• http://en.wikipedia.org/wiki/SI_prefix


Metric System• During the 1790s, a decimal system based on our

number system, the metric system, was being developed in France.– Easy to use– Easy to remember– Uses prefixes, that made the basic units larger or smaller by

multiples or fractions of 10

• For example:1km = 1000 m = 10,000 dm = 100,000 cm

1 mi = 1760 yd = 5280 ft = 63,360 in

• The only country left behind is the USA.


Imperial and U.S. customary systems of measurement

• http://en.wikipedia.org/wiki/Comparison_of_the_Imperial_and_US_customary_systems

• Both the Imperial (UK and Canada) and U.S. customary systems of measurement derive from earlier English systems.

• Comparison of Imperial and U.S. volume measures 1 liquid U.S. gallon = 3.785 411 784 litres ≈ 0.833 Imperial gallon

1 Imperial gallon = 4.546 09 litres ≈ 1.201 liquid U.S. gallons

On January 1, 1983, the metric systems and SI units were introduced in Canada.


Systems of Measurement

United States Customary System (USCS)• Formally called British System• Used in the US and Burma

– Length: foot– Weight/force: pound– Time: second

Systeme International (SI)• Also called the Metric or International System• Used everywhere else in the world!


Systeme International (SI)

Quantity Unit Symbol

Length meter m

Mass kilogram kg

Time second s

Force newton N

Energy joule J

Current ampere A

Temperature kelvin K


SI Conversions

• Major advantage – the decimal system – all digits are related to one another – multiples of 10!

1 kilometer = 1000 meters = 100,000 cm

1 meter = 100 cm = 0.001 kilometer


Scientific Notation• Scientists often use very large or very small

numbers that can not be conveniently written as fractions or decimal fractions.

• For example, the thickness of an oil film on water is about 0.0000001 m

• In scientific notation it is 1 x 10-7 m

0.1 = 1 x 10-1

0.001 = 1 x 10-3

10,000 = 1 x 104


Scientific Notation0.1 = 1 x 10-1

0.001 = 1 x 10-3

10,000 = 1 x 104

• Any number can be written as a product of a number between 1 and 10 and a power of 10.

• In general,

M x 10n;Where

M, is the a number between 1 and 10 and

n, is the exponent or power of 10.


Decimal to Scientific Notation578 = 5.78 x 102

0.025 = 002.5 x 10-2

3.5 = 3.5 x 100

• Place a decimal point after the first nonzero digit reading from left to right.

• Place a caret (^) at the position of the original decimal point.

• The exponent of 10 is the number of places from the caret to the decimal point.

• If the decimal point is to the right of the caret, the exponent of 10 is a negative number.

^

^

^


Scientific Notation to Decimal5.78 x 102 = 578

2.5 x 10-2 = 0.025

3.5 x 100 = 3.5

• Multiply the decimal part by the power of 10.– Move the decimal point to the right by the exponent

- If the exponent is a positive number– Move the decimal point to the left by the exponent

- If the exponent is a negative number

• Add zeros as needed.


Metric Length• The basic SI unit of length is the metre (m).

• Originally 1m = distance from the equator to either pole/10,000,000

• “The metre is the length of path traveled by light in a vacuum during a time interval of 1/299,792,458 s

– Km– m– cm


Conversion Factor• A conversion factor is an expression used to

change from one unit to another.

• Expressed as a fraction whose numerator and denominator are equal quantities in two different units.

• The information necessary for forming a conversion factor is usually found in their conversion table as follows:1 m = 100 cm

• So, the conversion factors are:1 m and 100 cm

100 cm 1 m


Conversion using Conversion Factor• So, convert 5m to cm:

5 m x 100 cm = 500 cm

1 m

Where the unit of the denominator should be the same as the original unit, so they cancels out.

• So, convert 7 cm to m:7 cm x 1 m = 0.07 m

100 cm


Conversion Factors as unit values• A conversion factor is an expression used to

change from one unit to another.

• 1 m = 100 cm

• So, the conversion factors are:1 m and 100 cm

100 cm 1 m

• These conversion factors can be read as:

per cm (or, 1 cm = m)

per m (or, 1 m = 100 cm)

1 m100

100 cm1

1 100


Conversion using units valueOr, it can be converted as follows:

5 m = 5 x 1 m = 5 x 100 cm = 500 cm

Similarly, 7 cm = 7 x 1 cm = 7 x 1 m = 0.07 m

100

1 m = 100 cm

100 cm = 1 mTherefore, 1 cm = (1/100) m


Metric-English ConversionTo change from an English unit to a metric unit or

from a metric unit to an English unit, we use a conversion factor, from the relation 1 in = 2.54 cm.

• So, the conversion factors are:1 in and 2.54 cm

2.54 cm 1 in


Area• The area of a plane surface

is the number of square units that it contains.

• To measure the surface area of an object, you must first decide on a standard unit of area.

• Standard units of area are based on the square of standard lengths, for example 1 square m.


Area• Find the area of a rectangle 5 m long and 3 m

wide.

• By simply counting the number of squares, we find the area of the rectangle is 15 m2.

• Or, by using the formula

A = l x w = 5 m x 3 m = (5 x 3) (m x m) = 15 m2


Volume• The volume of a figure is the number of cubic units

that it contains.• Standard units of volume are based the cube of

standard lengths, such as cubic meter, cubic cm, cubic in.


Volume• Find the volume of a rectangular prism 6 cm long, 4

cm wide, and 5 cm high.• To find the volume of the rectangular solid, count the

number of cubes in the bottom layer and then multiply by the number of layers.

• Or, V = l w h = 6 x 4 x 5 cm x cm x cm = 120 cm3


Mass• The mass of an object is the quantity of material

making up the object.• One unit of mass in the metric system is the gram (g).• The gram is defined as the mass of 1 cm3 of water at

its maximum density (at 4 C).• Since the gram is so small, kg is the basic unit of

mass in SI (Système international d'unités) .


Weight• The weight of an object is a measure of the gravitational force

or pull acting on an object.• The weight unit in the metric system is the newton (N).• An apple weighs about one newton (0.1kg x 10m/s2 =1kg.m/s2).• A newton is the amount of force required to accelerate a mass

of one kilogram by one meter per second squared.1 N = 1 kg·m/s²

• The pound (lb), a unit of force, is one of the basic English system units. It is defined as the pull of the earth on a cylinder of a platinum-iridium alloy that is stored in a vault at the U.S. Bureau of Standards.

• 1 N = 0.225 lb• 1 lb = 4.45 N


kg with weight • When the weight of an object is given in kilograms, the

property intended is almost always mass. • Occasionally the gravitational force on an object is

given in "kilograms", but the unit used is not a true kilogram: it is the deprecated kilogram-force (kgf), also known as the kilopond (kp).

• An object of mass 1 kg at the surface of the Earth will be subjected to a gravitational force of approximately 9.80665 newtons (the SI unit of force).

• http://en.wikipedia.org/wiki/Kilogram• http://en.wikipedia.org/wiki/Newton


Time• The basic unit of time is second (s) in both system.• It was defined as 1/86400 of a mean solar day.• Now the standard second is defined more precisely in

terms of frequency of radiation emitted by cesium atoms when they pass between two particular states; that is, the time required for 9,192,631,770 periods of this radiation.


Electrical Units• The ampere (A) is the basic unit and is

measure of the amount of electric current.

Derived units are:Columb (C) – is a measure of the amount of electrical

charge

Volt (V) – is a measure of electric potential

Watt (W) - is a measure of power


Accuracy vs. Precision• Accuracy: A measure of how close an

experimental result is to the true value.

• Precision: A measure of how exactly the result is determined. It is also a measure of how reproducible the result is.

– Absolute precision: indicates the uncertainty in the same units as the observation

– Relative precision: indicates the uncertainty in terms of a fraction of the value of the result


Accuracy• Physicists are interested in how closely a

measurement agrees with the true value.• This is an indication of the quality of the measuring

instrument.• Accuracy is a means of describing how closely a

measurement agrees with the actual size of a quantity being measured.


Error• The difference between an observed value and the

true value is called the error.• The size of the error is an indication of the accuracy.• Thus, the smaller the error, the greater the accuracy.

• The percentage error determined by subtracting the true value from the measured value, dividing this by the true value, and multiplying by 100.

%100x

valuetrue

valuetruevaluemeasured error percentage


Error

%100x

valuetrue

valuetruevaluemeasured error percentage

%4

%100x5.2

1.0

%100x5.2

5.26.2

m

mm

mm error percentage


Significant Digits• The accuracy of a measurement is indicated by

the number of significant digits.

• Significant digits are those digits in the numerical value of which we are reasonably sure.

• More significant digits in a measurement the accurate it is:


Significant Digits• More significant digits in a measurement the accurate

it is:

E.g., the true value of a bar is 2.50 m

Measured value is 2.6 m with 2 significant digits.

The percentage error is (2.6-2.50)*100/2.50 = 4%



The percentage error is (2.55-2.50)*100/2.50 = 0.2%

Which one is more accurate? The one which has more significant digits


Rules for Determining “Significant Digits”• All non zero digits are significant• All zeros between significant non zero digits are

significant. 450.09 5 significant digits• A zero in a number (> 1) which is specially tagged,

such as by a bar above it, is significant. 250,000 3 significant digits

• Zeros at the right in whole number. 5600 2 significant digits

• All zeros to the right of a significant digits and a decimal point. 5120.010 7 significant digits

• Zeros at the left in measurements less than 1 are not significant. 0.00672 3 significant digits


Determine the “Accuracy” and “Precision”

3463 m 4 S.D.s 1m

3005 km

36000 8800 V

1349000 km

0.00632 kg

0.0060 g

14.20 A

30.00 cm

100.060 g 6 SDs 0.001 g

0.00004 m

2.4765 m


Precision• Being precise means being sharply defined.

• The precision of a measuring instrument depends on its degree of fineness and the size of the unit being used.

• Using an instrument with a more finely divided scale allows us to take a more precise measurement.


Precision• The precision of a measuring refers to the smallest

unit with which a measurement is made, that is, the position of the last significant digit.

• In most cases it is the number of decimal places.

e.g.,• The precision of the measurement 385,000 km

is 1000 km. (the position of the last significant digit is in the thousands place.)

• The precision of the measurement 0.025m is 0.001m. (the position of the last significant digit is in the thousandths place.)


How precise do we need?• Physicists are interested in how closely a

measurement agrees with the true value.

• That is, to achieve a smaller error or more accuracy.

• For bigger quantities, we do not need to be precise to be accurate.


How precise do we need?• For bigger quantities, we do not need to be precise to be

accurate.

E.g., the true value of a bar is 25 m

Measured value is 26 m with 2 significant digits.

The percentage error is (26-25)*100/25 = 4%



The percentage error is (2.6-2.5)*100/2.5 = 4%

Which one is more precise? The one which has the precision of 0.1m

Which one is more accurate? Both are same accurate as both have 2 significant digits


Accuracy or Relative Precision• An accurate measurement is also known as a

relatively precise measurement.

• Accuracy or Relative Precision refers to the number of significant digits in a measurement.

• A measurement with higher number of significant digits closely agrees with the true value.


Estimate• Any measurement that falls between the

smallest divisions on the measuring instrument is an estimate.

• We should always try to read any instrument by estimating tenths of the smallest division.


Accuracy or Relative Precision• In any measurement, the number of significant figures are

critical. • The number of significant figures is the number of digits

believed to be correct by the person doing the measuring. • It includes one estimated digit. • A rule of thumb: read a measurement to 1/10 or 0.1 of the

smallest division. • This means that the error in reading (called the reading error) is

1/10 or 0.1 of the smallest division on the ruler or other instrument.

• If you are less sure of yourself, you can read to 1/5 or 0.2 of the smallest division.

• http://www.astro.washington.edu/labs/clearinghouse/labs/Scimeth/mr-sigfg.html


Estimate to 1/10th of a cm

• What should be the estimated value?

1 cm

L = 6.7 cm


Estimate to 1/10th of a mm


1 cm

L = ?


6 or 6.0 cm?


1 cm

L = ?


Estimate to 1/10th of the smallest unit




1 cm

L = ?



• The estimated value is 0.7 x unit value

= 0.7 x 1 cm

= 0.7 cm

1 cm

L = 6 + 0.7 cm

= 6.7 cm




L = ?

1 cm




= 0.7 x 0.1 cm

= 0.07 cm

L = 6.7 cm + 0.07 cm

= 6.77 cm

1 cm




0 2 cm 4 6 8 10 12 14 16 18 20 22

L = ?




= 0.7 x 2 cm

= 1.4 cm

0 2 cm 4 6 8 10 12 14 16 18 20 22

L = 12+1.4 cm

= 13.4 cm




?

0 1 cm 2 3

L = ?




= 0.7 x 0.25 cm

= 0.175 cm

?

0 1 cm 2 3

L = 1.5 + 0.175 cm

= 1.675 cm


MeasurementAn object measured with a ruler calibrated in

millimeters. One end of the object is at the zero mark of the ruler. The other end lines up exactly with the 5.2 cm mark.

• What reading should be recorded for the length of the object?

• Why?


Precision• Which of the following measured quantities is

most precise?

• Why?

126 cm

2.54 cm

12.65 cm

48.1 mm

0.081 mm


Exact vs. Approximate numbers• An exact number is a number that has been

determined as a result of counting or by some definition.

• E.g., 41 students are enrolled in this class• 1in = 2.54 cm

• Nearly all data of a technical nature involve approximate numbers.

• That is numbers determined as a result of some measurement process, as with a ruler.

• No measurement can be found exactly.


Calculations with Measurements• The sum or difference of measurements can be no

more precise than the least precise measurement.

42.28 mmUsing a micrometer

54 mmUsing a ruler,

Precision of the ruler is 1 mmBut actually it can be anywhere

between 53.50 to 54.50 mm

This means that the tenths and hundredths digits in the sum 96.28 mm are really meaningless,

the sum should be 96 mm with a precision of 1 mm


Calculations with Measurements• The sum or difference of measurements can be no

more precise than the least precise measurement.• Round the results to the same precision as the least

precise measurement.

42.28 mmUsing a micrometer

54 mmUsing a ruler,

Precision of the ruler is 1 mmBut actually it can be anywhere

between 53.50 to 54.50 mm

This means that the tenths and hundredths digits in the sum 96.28 mm are really meaningless,

the sum should be 96 mm with a precision of 1 mm


Calculations with Measurements• The product or quotient of measurements can be no

more accurate than the least accurate measurement.• Round the results to the same number of significant

digits as the measurement with the least number of significant digits.

• http://www.astro.washington.edu/labs/clearinghouse/labs/Scimeth/mr-sigfg.html

Length of a rectangle is 54.7 mWidth of a rectangle is 21.5 mArea is 1176.05 m2

Area should be rounded to 1180 m2

To express with same accuracy


Rounding Numbers• To round a number to a particular place value:

• If the digit in the next place to the right is less than 5, drop that digit and all other following digits. Replace any whole number places dropped with zeros.

• If the digit in the next place to the right is 5 or greater, add 1 to the digit in the place to which you are rounding. Drop all other following digits. Replace any whole number places dropped with zeros


Special case, Rounding Numbers• If the digit in the next place to the right is

exactly 5, add 1 to the digit in the place to which you are rounding if the previous digit is an odd number other wise just drop the digit. Replace any whole number places dropped with zeros.

• This rule means that if the digit to be dropped is 5 followed only by zeroes, the result is always rounded to the even digit.

• The rationale is to avoid bias in rounding: half of the time we round up, half the time we round down.


Examples of Rounding• http://www.astro.washington.edu/labs/clearingh

ouse/labs/Scimeth/mr-sigfg.html


Add the Measurements1250 cm, 1562 mm, 2.963 m, 9.71 m

• Convert all measurements to the same units.

• In this case m will be the best choice of units.

1250 cm = 12.5 m

1562 mm = 1.562 m12.5 m

1.562 m2.963 m9.71 m

26.735 mRound to ? Should we round before adding?


Calculations with Measurements• A rectangular has dimensions of 15.6 m by 11.4

m. What is the area of the rectangle?

A = L x W

= 15.6 m x 11.4 m

= 177.84 m2

= ? m2


Calculations with Measurements• A rectangular plot of land has an area of 78000

m2. one side has a length of 654 m. What is the length of the second side?

A = L x W W = A/L

= 78000 m2 / 654 m

= 119.266 m

= ? m


Calculations with MeasurementsSubtract the measurements: 2567 g – 1.60 kg

Express your answer in g.

• Convert all measurements to the same units.

1.60 kg = 1600 g 2567 g 1600 g

970 gRound to ? Should we round before subtracting?


Calculations with Measurements and Exact numbers

• To round the result of a calculation use the precesion and the accuracy of the measured number not the exact number.


Calculations with Measurements and Exact numbers

• 2 equal rectangular plots of land has an area of 75 m2. What is the area of one plot?

Area of one plot = Total Area / 2

= 75 m2 / 2

= 37.5 m2

= ? m2


So far…

• Accuracy and precision• Exact number and Approximate number• Estimate• Rounding• USCS (United States Customary System)• Systeme International (SI) or Metric system• Quantities, units and symbols of the SI system• Prefixes of SI system• Major advantage of the SI system (multiples of 10)!

phy115 – sault college – bazlurslide 1 measurement and the metric system

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