PHY115 – Sault College – Bazlur slide 4
Standards of Measure• When two people work together, they should both use
the same standards of measure.
PHY115 – Sault College – Bazlur slide 5
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.
PHY115 – Sault College – Bazlur slide 7
Derived SI Units• http://physics.nist.gov/cuu/Units/SIdiagram.html
PHY115 – Sault College – Bazlur slide 8
Prefixes for SI Units• http://en.wikipedia.org/wiki/SI• http://en.wikipedia.org/wiki/SI_prefix
PHY115 – Sault College – Bazlur slide 9
Prefixes for SI Units• http://en.wikipedia.org/wiki/SI_prefix
PHY115 – Sault College – Bazlur slide 10
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.
PHY115 – Sault College – Bazlur slide 11
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.
PHY115 – Sault College – Bazlur slide 12
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!
PHY115 – Sault College – Bazlur slide 13
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
PHY115 – Sault College – Bazlur slide 14
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
PHY115 – Sault College – Bazlur slide 15
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
PHY115 – Sault College – Bazlur slide 16
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.
PHY115 – Sault College – Bazlur slide 17
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.
^
^
^
PHY115 – Sault College – Bazlur slide 18
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.
PHY115 – Sault College – Bazlur slide 19
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
PHY115 – Sault College – Bazlur slide 20
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
PHY115 – Sault College – Bazlur slide 21
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
PHY115 – Sault College – Bazlur slide 22
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
PHY115 – Sault College – Bazlur slide 23
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
PHY115 – Sault College – Bazlur slide 24
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
PHY115 – Sault College – Bazlur slide 25
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.
PHY115 – Sault College – Bazlur slide 26
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
PHY115 – Sault College – Bazlur slide 27
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.
PHY115 – Sault College – Bazlur slide 28
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
PHY115 – Sault College – Bazlur slide 29
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) .
PHY115 – Sault College – Bazlur slide 30
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
PHY115 – Sault College – Bazlur slide 31
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
PHY115 – Sault College – Bazlur slide 32
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.
PHY115 – Sault College – Bazlur slide 33
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
PHY115 – Sault College – Bazlur slide 34
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
PHY115 – Sault College – Bazlur slide 35
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.
PHY115 – Sault College – Bazlur slide 36
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
PHY115 – Sault College – Bazlur slide 37
Error
%100x
valuetrue
valuetruevaluemeasured error percentage
%4
%100x5.2
1.0
%100x5.2
5.26.2
m
mm
mm error percentage
PHY115 – Sault College – Bazlur slide 38
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:
PHY115 – Sault College – Bazlur slide 39
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%
E.g., the true value of a bar is 2.50 m
Measured value is 2.55 m with 3 significant digits.
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
PHY115 – Sault College – Bazlur slide 40
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
PHY115 – Sault College – Bazlur slide 41
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
PHY115 – Sault College – Bazlur slide 42
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.
PHY115 – Sault College – Bazlur slide 43
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.)
PHY115 – Sault College – Bazlur slide 44
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.
PHY115 – Sault College – Bazlur slide 45
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%
E.g., the true value of a bar is 2.5 m
Measured value is 2.6 m with 2 significant digits.
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
PHY115 – Sault College – Bazlur slide 46
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.
PHY115 – Sault College – Bazlur slide 47
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.
PHY115 – Sault College – Bazlur slide 48
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
PHY115 – Sault College – Bazlur slide 49
Estimate to 1/10th of a cm
• What should be the estimated value?
1 cm
L = 6.7 cm
PHY115 – Sault College – Bazlur slide 50
Estimate to 1/10th of a mm
• What should be the estimated value?
1 cm
L = ?
PHY115 – Sault College – Bazlur slide 51
6 or 6.0 cm?
• What should be the estimated value?
1 cm
L = ?
PHY115 – Sault College – Bazlur slide 53
Estimate to 1/10th of the smallest unit
• What should be the estimated value?
1 cm
L = ?
PHY115 – Sault College – Bazlur slide 54
Estimate to 1/10th of the smallest unit
• 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
PHY115 – Sault College – Bazlur slide 55
Estimate to 1/10th of the smallest unit
• What should be the estimated value?
L = ?
1 cm
PHY115 – Sault College – Bazlur slide 56
Estimate to 1/10th of the smallest unit
• The estimated value is 0.7 x unit value
= 0.7 x 0.1 cm
= 0.07 cm
L = 6.7 cm + 0.07 cm
= 6.77 cm
1 cm
PHY115 – Sault College – Bazlur slide 57
Estimate to 1/10th of the smallest unit
• What should be the estimated value?
0 2 cm 4 6 8 10 12 14 16 18 20 22
L = ?
PHY115 – Sault College – Bazlur slide 58
Estimate to 1/10th of the smallest unit
• The estimated value is 0.7 x unit value
= 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
PHY115 – Sault College – Bazlur slide 59
Estimate to 1/10th of the smallest unit
• What should be the estimated value?
?
0 1 cm 2 3
L = ?
PHY115 – Sault College – Bazlur slide 60
Estimate to 1/10th of the smallest unit
• The estimated value is 0.7 x unit value
= 0.7 x 0.25 cm
= 0.175 cm
?
0 1 cm 2 3
L = 1.5 + 0.175 cm
= 1.675 cm
PHY115 – Sault College – Bazlur slide 61
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?
PHY115 – Sault College – Bazlur slide 62
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
PHY115 – Sault College – Bazlur slide 63
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.
PHY115 – Sault College – Bazlur slide 64
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
PHY115 – Sault College – Bazlur slide 65
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
PHY115 – Sault College – Bazlur slide 66
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
PHY115 – Sault College – Bazlur slide 67
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
PHY115 – Sault College – Bazlur slide 68
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.
PHY115 – Sault College – Bazlur slide 69
Examples of Rounding• http://www.astro.washington.edu/labs/clearingh
ouse/labs/Scimeth/mr-sigfg.html
PHY115 – Sault College – Bazlur slide 70
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?
PHY115 – Sault College – Bazlur slide 71
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
PHY115 – Sault College – Bazlur slide 72
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
PHY115 – Sault College – Bazlur slide 73
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?
PHY115 – Sault College – Bazlur slide 74
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.
PHY115 – Sault College – Bazlur slide 75
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
PHY115 – Sault College – Bazlur slide 76
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)!