1 - measurement
TRANSCRIPT
MEASUREMENT
Measurement in everyday life
Measurement of mass Measurement of volume
Measurement in everyday life
Measurement of length Measurement of temperature
Need for measurement in physics
• To understand any phenomenon in physics we have to perform experiments.
• Experiments require measurements, and we measure several physical properties like length, mass, time, temperature, pressure etc.
• Experimental verification of laws & theories also needs measurement of physical properties.
Physical QuantityA physical property that can be measured and
described by a number is called physical quantity.Examples:• Mass of a person is 65 kg.• Length of a table is 3 m.• Area of a hall is 100 m2.• Temperature of a room is 300 K
Types of physical quantities1.Fundamental quantities:
The physical quantities which do not depend on any other physical quantities for their measurements are known as fundamental quantities.Examples:• Mass• Length
• Time• Temperature
Types of physical quantities
The physical quantities which depend on one or more fundamental quantities for their measurements are known as derived quantities.
Examples:• Area• Volume
• Speed• Force
2. Derived quantities:
Units for measurementThe standard used for the measurement of
a physical quantity is called a unit.Examples:• metre, foot, inch for length• kilogram, pound for mass• second, minute, hour for time• fahrenheit, kelvin for temperature
Characteristics of unitsWell – definedSuitable sizeReproducible
InvariableIndestructible
Internationally acceptable
• This system was first introduced in France.
• It is also known as Gaussian system of units.
• It is based on centimeter, gram and second as the fundamental units of length, mass and time.
CGS system of units
MKS system of units
• This system was also introduced in France.
• It is also known as French system of units.• It is based on meter, kilogram and
second as the fundamental units of length, mass and time.
FPS system of units
• This system was introduced in Britain.• It is also known as British system of units.• It is based on foot, pound and second as
the fundamental units of length, mass and time.
International System of units (SI)
• In 1971, General Conference on Weight and Measures held its meeting and decided a system of units for international usage.
• This system is called international system of units and abbreviated as SI from its French name.
• The SI unit consists of seven fundamental units and two supplementary units.
Seven fundamental unitsFUNDAMENTAL QUANTITY SI UNIT SYMBOL
Length metre mMass kilogram kgTime second s
Temperature kelvin KElectric current ampere A
Luminous intensity candela cdAmount of substance mole mol
Definition of metre
The metre is the length of the path travelled by light in a
vacuum during a time interval of1/29,97,92,458 of a second.
Definition of kilogram
The kilogram is the mass of prototype cylinder of platinum-iridium alloy
preserved at the International Bureau of Weights and Measures, at Sevres,
near Paris.
Prototype cylinder of platinum-iridium alloy
Definition of second
One second is the time taken by 9,19,26,31,770 oscillations of the
light emitted by a cesium–133 atom.
Two supplementary units1.Radian: It is used to measure plane
angle
= 1 radian
Two supplementary units2. Steradian: It is used to measure solid
angle
= 1 steradian
Rules for writing SI units
1Full name of unit always starts with small
letter even if named after a person.• newton• ampere• coulom
b
not• Newton• Ampere• Coulom
b
Rules for writing SI units
2Symbol for unit named after a scientist
should be in capital letter.
• N for newton• K for kelvin
• A for ampere• C for coulomb
Rules for writing SI units
3Symbols for all other units are written in
small letters.
• m for meter• s for second
• kg for kilogram• cd for candela
Rules for writing SI units
4One space is left between the last digit of
numeral and the symbol of a unit.• 10 kg• 5 N• 15 m
not• 10kg• 5N• 15m
Rules for writing SI units
5The units do not have plural forms.• 6 metre• 14 kg• 20
second• 18 kelvin
not• 6 metres• 14 kgs• 20 seconds• 18 kelvins
Rules for writing SI units
6Full stop should not be used after the
units.• 7 metre• 12 N• 25 kg
not• 7 metre.• 12 N.• 25 kg.
Rules for writing SI units
7No space is used between the symbols for
units.• 4 Js• 19 Nm• 25 VA
not• 4 J s• 19 N m.• 25 V A.
SI prefixesFactor Name Symbol Factor Name Symbol
yotta Y deci dzetta Z centi cexa E milli mpeta P micro μtera T nano ngiga G pico p
mega M femto fkilo k atto a
hecto h zepto zdeka da yocto y
• 3 milliampere = 3 mA = 3 x A• 5 microvolt = 5 μV = 5 x V• 8 nanosecond = 8 ns = 8 x s• 6 picometre = 6 pm = 6 x m• 5 kilometre = 5 km = 5 x m• 7 megawatt = 7 MW = 7 x W
Use of SI prefixes
Some practical units for measuring length1 micron = m
Bacterias Molecules
1 nanometer = m
Some practical units for measuring length1 angstrom = m
Atoms Nucleus
1 fermi = m
Some practical units for measuring length• Astronomical unit = It is defined as the mean distance
of the earth from the sun.• 1 astronomical unit = m
Distance of planets
Some practical units for measuring length• Light year = It is the distance travelled by light in vacuum
in one year.• 1 light year = m
Distance of stars
Some practical units for measuring length• Parsec = It is defined as the distance at which an arc of 1
AU subtends an angle of 1’’.• It is the largest practical unit of distance used in
astronomy.• 1 parsec = m
1 AU 1”
1 parsec
Some practical units for measuring area• Acre = It is used to measure large areas in British system
of units. 1 acre = 208’ 8.5” x 208’ 8.5” = 4046.8
• Hectare = It is used to measure large areas in French system of units.
1 hectare = 100 m x 100 m = 10000
• Barn = It is used to measure very small areas, such as nuclear cross sections.
1 barn =
Some practical units for measuring mass1 metric ton = 1000
kg
Steel bars Grains
1 quintal = 100 kg
1 pound = 0.454 kg
Newborn babies
Crops
1 slug = 14.59 kg
Some practical units for measuring mass
Some practical units for measuring mass• 1 Chandrasekhar limit = 1.4 x mass of sun = kg• It is the biggest practical unit for measuring mass.
Massive black holes
Some practical units for measuring mass
• 1 atomic mass unit = x mass of single C atom
• 1 atomic mass unit = 1.66 kg• It is the smallest practical unit for
measuring mass.• It is used to measure mass of single atoms,
proton and neutron.
Some practical units for measuring time
• 1 Solar day = 24 h
• 1 Sidereal day = 23 h & 56 min
• 1 Solar year = 365 solar day = 366 sidereal day
• 1 Lunar month = 27.3 Solar day
• 1 shake = s
Seven dimensions of the worldFundamental
quantitiesLength MassTime
TemperatureCurrent
Amount of substanceLuminous intensity
Dimensions[L] [M][T] [K][A][N][J]
Dimensions of a physical quantity
The powers of fundamental quantities in a derived quantity are called dimensions of that quantity.
¿Masslength×breath× height
[Density ]= [M ][ L ]× [ L ]× [ L ]
=[M ][L3 ]
=[M L−3]
Dimensions of a physical quantity
Density=MassVolume
Example :
H ence the dimensions of density are1∈mass∧−3∈length .
Uses of Dimension
To check the correctness of equation
To convert units
To derive a formula
To check the correctness of equation
∆ x=displacement=[L]
Consider the equationof displacement ,
B y writing the dimensionswe get ,
v i t=velocity × time=lengthtime ×time=[L]
a t2=acceleration×t ime2=l engthtime 2
×time2=[L]
The dimensions of each term are same, hence the equation is dimensionally correct.
∆ x=v i t+12 a t
2
To convert unitsLet us convert newton (SI unit of force )into dyne (CGS unit of force ) .
T hedimesions of force are=[LM T −2]
So ,1newton=(1m)(1kg)(1 s)− 2
a nd ,1dyne=(1cm)(1g)(1 s)−2
T hus , 1newton1dyne
=( 1m1cm )( 1kg1g )( 1 s1 s )− 2
=( 100 c m1cm )( 1000g1g )( 1 s1 s )− 2
Therefore,
To derive a formulaThe time period ‘T’ of oscillation of a simple pendulum depends on length ‘l’ and acceleration due to gravity ‘g’.
Let us assume that,
T or T
K = constant which is dimensionless
Dimensions of T
Dimensions of
Dimensions of g
Thus,
[L0M 0T1 ]=K [La+bM 0T −2b ]
a+b=0∧−2b=1
∴b=− 12∧a=12
T
T
Least count of instruments
The smallest value that can be measured by the measuring instrument
is called its least count or resolution.
LC of length measuring instrumentsRuler scale
Least count = 1 mm
Vernier Calliper
Least count = 0.1 mm
LC of length measuring instrumentsScrew Gauge
Least count = 0.01 mm
Spherometer
Least count = 0.001 mm
LC of mass measuring instrumentsWeighing scale
Least count = 1 kg
Electronic balance
Least count = 1 g
LC of time measuring instrumentsWrist watch
Least count = 1 s
Stopwatch
Least count = 0.01 s
Accuracy of measurementIt refers to the closeness of a measurement
to the true value of the physical quantity.Example:• True value of mass = 25.67 kg• Mass measured by student A = 25.61 kg• Mass measured by student B = 25.65 kg• The measurement made by student B is more
accurate.
Precision of measurementIt refers to the limit to which a physical
quantity is measured.Example:• Time measured by student A = 3.6 s• Time measured by student B = 3.69 s• Time measured by student C = 3.695 s• The measurement made by student C is most
precise.
Significant figures
The total number of digits(reliable digits + last uncertain digit)which are directly obtained from aparticular measurement are called
significant figures.
Significant figures
Mass = 6.11 g3 significant figures
Speed = 67 km/h2 significant figures
Significant figures
Time = 12.76 s4 significant figures
Length = 1.8 cm2 significant figures
Rules for counting significant figures
1All non-zero digits are significant.
Number16 35.66438
Significant figures2 34
2Zeros between non-zero digits are significant.
Rules for counting significant figures
Number205
3008 60.005
Significant figures3 45
Rules for counting significant figures
3Terminal zeros in a number without decimal are not significant unless specified by a least count.
Number400 3050
(20 1) s
Significant figures1 32
Rules for counting significant figures
4Terminal zeros that are also to the right of a
decimal point in a number are significant.Number64.00 3.60
25.060
Significant figures4 35
Rules for counting significant figures5
If the number is less than 1, all zeroes before thefirst non-zero digit are not significant.
Number0.0064 0.0850
0.0002050
Significant figures2 34
6During conversion of units use powers of 10 to
avoid confusion.
Rules for counting significant figures
Number2.700 m
2.700 x cm2.700 x km
Significant figures444
Exact numbers• Exact numbers are either defined numbers or the
result of a count.• They have infinite number of significant figures
because they are reliable.By definition1 dozen = 12
objects 1 hour = 60 minute1 inch = 2.54 cm
By counting45 students
5 apples6 faces of
cube
Rules for rounding off a measurement
1If the digit to be dropped is less than 5, then the
preceding digit is left unchanged.Number64.62 3.651546.3
Round off up to 3 digits64.6 3.65546
2If the digit to be dropped is more than 5, then the
preceding digit is raised by one.Number3.47993.46683.7
Round off up to 3 digits3.48 93.5684
Rules for rounding off a measurement
3If the digit to be dropped is 5 followed by digits otherthan zero, then the preceding digit is raised by one.
Number62.3549.6552589.51
Round off up to 3 digits62.4 9.66590
Rules for rounding off a measurement
4If the digit to be dropped is 5 followed by zero or
nothing, the last remaining digit is increased by 1 if it is odd, but left as it is if even.
Number53.3509.455782.5
Round off up to 3 digits53.4 9.46782
Rules for rounding off a measurement
Significant figures in calculations
Addition & subtractionThe final result would round to the same decimal
place as the least precise number.Example:• 13.2 + 34.654 + 59.53 = 107.384 = 107.4• 19 – 1.567 - 14.6 = 2.833 = 3
Significant figures in calculations
Multiplication & divisionThe final result would round to the same numberof significant digits as the least accurate number.
Example:• 1.5 x 3.67 x 2.986 = 16.4379 = 16• 6.579/4.56 = 1.508 = 1.51
Errors in measurement
Difference between the actual value ofa quantity and the value obtained by a
measurement is called an error.
Error =
Types of errors
Systematic errorsGross errors
Random errors
1. Systematic errors
• These errors are arise due to flaws in experimental system.
• The system involves observer, measuring instrument and the environment.
• These errors are eliminated by detecting the source of the error.
Types of systematic errors
Personal errorsInstrumental errors
Environmental errors
a. Personal errorsThese errors are arise due to faulty proceduresadopted by the person making measurements.
Parallax error
b. Instrumental errorsThese errors are arise due to faulty
construction of instruments.
Zero error
c. Environmental errors
These errors are caused by external conditions like
pressure, temperature, magnetic field, wind etc. Following are the steps that one must follow in order to eliminate the environmental errors:a. Try to maintain the temperature and humidity of the
laboratory constant by making some arrangements.b. Ensure that there should not be any external magnetic
or electric field around the instrument.
Advanced experimental setups
2. Gross errorsThese errors are caused by mistake in using instruments, recording data and calculating
results. Example:a. A person may read a pressure gauge indicating 1.01 Pa
as 1.10 Pa.b. By mistake a person make use of an ordinary electronic
scale having poor sensitivity to measure very low masses.Careful reading and recording of the data can reduce
the gross errors to a great extent.
3. Random errors
• These errors are due to unknown causes and are sometimes termed as chance errors.
• Due to unknown causes, they cannot be eliminated.
• They can only be reduced and the error can be estimated by using some statistical operations.
Error analysisFor example, suppose you measure the oscillation period of a pendulum with a stopwatch five times.
Trial no ( i ) 1 2 3 4 5Measured value ( ) 3.9 3.5 3.6 3.7 3.5
Mean valueThe average of all the five readings gives the most probable value for time period.
=
= =
= 3.64 = 3.6
Absolute errorThe magnitude of the difference between mean value and each individual value is called absolute error.
=
3.9 3.5 3.6 3.7 3.50.3 0.1 0 0.1 0.1
The absolute error in each individual reading:
Mean absolute errorThe arithmetic mean of all the absolute errors is called mean absolute error.
=
= =
= 0.12 = 0.1
Reporting of result• The most common way adopted by scientist and
engineers to report a result is:
Result = best estimate error
• It represent a range of values and from that we expect a true value fall within.
• Thus, the period of oscillation is likely to be within (3.6 0.1) s.
Relative errorThe relative error is defined as the ratio of the mean absolute error to the mean value.
=
/ = = 0.0277
/ = 0.028
Percentage errorThe relative error multiplied by 100 is called as percentage error.
percentage error = relative error x 100
percentage error = 0.028 x 100 percentage error = 2.8
Least count errorLeast count error is the error associated with the resolution of the instrument.
• The least count error of any instrument is equal to its resolution.
• Thus, the length of pen is likely to be within (4.7 0.1) cm.
Combination of errors
• Let be absolute error in measurement of • Let be absolute error in measurement of • Let be absolute error in measurement of
In different mathematical operations like addition, subtraction, multiplication and
division the errors are combined according to some rules.
=
∆ X=∆ A+∆ B
= +
=
W hen X=An
=
=
Estimation
Estimation is a rough calculation to find an approximate value of
something that is useful for some purpose.
Estimate the number of flats in Dubai city
Estimate the volume of water stored in a dam
Order of magnitude
The approximate size ofsomething expressed in powers
of 10 is called orderof magnitude.
To get an approximate idea of the number, one may round the coefficient a to 1 if it is less than or equal to 5 and to 10 if it is greater than 5.Examples:• Mass of electron = 9.1 x kg kg kg• Mass of observable universe = 1.59 x kg kg kg
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