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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 re...
Physical Quantity
A physical property that can be measured and
described by a number is called physical quantity.
Examples...
Types of physical quantities
1. Fundamental quantities:
The physical quantities which do not depend on any
other physical ...
Types of physical quantities
The physical quantities which depend on one or more
fundamental quantities for their measurem...
Units for measurement
The standard used for the measurement of
a physical quantity is called a unit.
Examples:
• metre, fo...
Characteristics of units
Well – defined
Suitable size
Reproducible
Invariable
Indestructible
Internationally acceptable
• This system was first introduced in France.
• It is also known as Gaussian system of units.
• It is based on centimeter,...
MKS system of units
• This system was also introduced in France.
• It is also known as French system of units.
• It is bas...
FPS system of units
• This system was introduced in Britain.
• It is also known as British system of units.
• It is based ...
International System of units (SI)
• In 1971, General Conference on Weight and Measures
held its meeting and decided a sys...
Seven fundamental units
FUNDAMENTAL QUANTITY SI UNIT SYMBOL
Length metre m
Mass kilogram kg
Time second s
Temperature kelv...
Definition of metre
The metre is the length of the
path travelled by light in a
vacuum during a time interval of
1/29,97,9...
Definition of kilogram
The kilogram is the mass of prototype
cylinder of platinum-iridium alloy
preserved at the Internati...
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 units
1. Radian: It is used to measure plane angle
θ = 1 radian
Two supplementary units
2. Steradian: It is used to measure solid angle
Ω = 1 steradian
Rules for writing SI units
1
Full name of unit always starts with small
letter even if named after a person.
• newton
• am...
Rules for writing SI units
2
Symbol for unit named after a scientist
should be in capital letter.
• N for newton
• K for k...
Rules for writing SI units
3
Symbols for all other units are written in
small letters.
• m for meter
• s for second
• kg f...
Rules for writing SI units
4
One space is left between the last digit of
numeral and the symbol of a unit.
• 10 kg
• 5 N
•...
Rules for writing SI units
5
The units do not have plural forms.
• 6 metre
• 14 kg
• 20 second
• 18 kelvin
not
• 6 metres
...
Rules for writing SI units
6
Full stop should not be used after the
units.
• 7 metre
• 12 N
• 25 kg
not
• 7 metre.
• 12 N....
Rules for writing SI units
7
No space is used between the symbols for
units.
• 4 Js
• 19 Nm
• 25 VA
not
• 4 J s
• 19 N m.
...
SI prefixes
Factor Name Symbol Factor Name Symbol
10
24 yotta Y 10
−1 deci d
10
21 zetta Z 10
−2 centi c
10
18 exa E 10
−3...
• 3 milliampere = 3 mA = 3 x 10
−3
A
• 5 microvolt = 5 μV = 5 x 10
−6
V
• 8 nanosecond = 8 ns = 8 x 10
−9
s
• 6 picometre ...
Some practical units for measuring length
1 micron = 10
−6
m
Bacterias Molecules
1 nanometer = 10
−9
m
Some practical units for measuring length
1 angstrom = 10
−10
m
Atoms Nucleus
1 fermi = 10
−15
m
Some practical units for measuring length
• Astronomical unit = It is defined as the mean distance of
the earth from the s...
Some practical units for measuring length
• Light year = It is the distance travelled by light in vacuum in
one year.
• 1 ...
Some practical units for measuring length
• Parsec = It is defined as the distance at which an arc of 1 AU
subtends an ang...
Some practical units for measuring area
• Acre = It is used to measure large areas in British system of
units.
1 acre = 20...
Some practical units for measuring mass
1 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 = 2.785 x 10
30
kg
• It is the biggest...
Some practical units for measuring mass
• 1 atomic mass unit =
1
12
x mass of single C atom
• 1 atomic mass unit = 1.66 x ...
Some practical units for measuring time
• 1 Solar day = 24 h
• 1 Sidereal day = 23 h & 56 min
• 1 Solar year = 365 solar d...
Seven dimensions of the world
Fundamental quantities
Length
Mass
Time
Temperature
Current
Amount of substance
Luminous int...
Dimensions of a physical quantity
The powers of fundamental quantities
in a derived quantity are called
dimensions of that...
=
Mass
length × breath × height
[Density] =
[M]
L × L × L
=
[M]
L3
= [ML−3
]
Dimensions of a physical quantity
Density =
M...
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 equation of displacement,
By writing the dimensi...
To convert units
Let us convert newton SI unit of force into dyne CGS unit of force .
The dimesions of force are = [LMT−2
...
To derive a formula
The time period ‘T’ of oscillation of a
simple pendulum depends on length ‘l’
and acceleration due to ...
Least count of instruments
The smallest value that can be
measured by the measuring instrument
is called its least count o...
LC of length measuring instruments
Ruler scale
Least count = 1 mm
Vernier Calliper
Least count = 0.1 mm
LC of length measuring instruments
Screw Gauge
Least count = 0.01 mm
Spherometer
Least count = 0.001 mm
LC of mass measuring instruments
Weighing scale
Least count = 1 kg
Electronic balance
Least count = 1 g
LC of time measuring instruments
Wrist watch
Least count = 1 s
Stopwatch
Least count = 0.01 s
Accuracy of measurement
It refers to the closeness of a measurement
to the true value of the physical quantity.
Example:
•...
Precision of measurement
It refers to the limit to which a physical
quantity is measured.
Example:
• Time measured by stud...
Significant figures
The total number of digits
(reliable digits + last uncertain digit)
which are directly obtained from a...
Significant figures
Mass = 6.11 g
3 significant figures
Speed = 67 km/h
2 significant figures
Significant figures
Time = 12.76 s
4 significant figures
Length = 1.8 cm
2 significant figures
Rules for counting significant figures
1
All non-zero digits are significant.
Number
16
35.6
6438
Significant figures
2
3
4
2
Zeros between non-zero digits are significant.
Rules for counting significant figures
Number
205
3008
60.005
Significant...
Rules for counting significant figures
3
Terminal zeros in a number without decimal are
not significant unless specified b...
Rules for counting significant figures
4
Terminal zeros that are also to the right of a
decimal point in a number are sign...
Rules for counting significant figures
5
If the number is less than 1, all zeroes before the
first non-zero digit are not ...
6
During conversion of units use powers of 10 to
avoid confusion.
Rules for counting significant figures
Number
2.700 m
2....
Exact numbers
• Exact numbers are either defined numbers or the
result of a count.
• They have infinite number of signific...
Rules for rounding off a measurement
1
If the digit to be dropped is less than 5, then the
preceding digit is left unchang...
2
If the digit to be dropped is more than 5, then the
preceding digit is raised by one.
Number
3.479
93.46
683.7
Round off...
3
If the digit to be dropped is 5 followed by digits other
than zero, then the preceding digit is raised by one.
Number
62...
4
If the digit to be dropped is 5 followed by zero or
nothing, the last remaining digit is increased by 1 if it is
odd, bu...
Significant figures in calculations
Addition & subtraction
The final result would round to the same decimal
place as the l...
Significant figures in calculations
Multiplication & division
The final result would round to the same number
of significa...
Errors in measurement
Difference between the actual value of
a quantity and the value obtained by a
measurement is called ...
Types of errors
Systematic errors
Gross errors
Random errors
1. Systematic errors
• These errors are arise due to flaws in
experimental system.
• The system involves observer, measuri...
Types of systematic errors
Personal errors
Instrumental errors
Environmental errors
a. Personal errors
These errors are arise due to faulty procedures
adopted by the person making measurements.
Parallax err...
b. Instrumental errors
These 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 et...
Advanced experimental setups
2. Gross errors
These errors are caused by mistake in using
instruments, recording data and calculating results.
Example:
...
3. Random errors
• These errors are due to unknown causes and
are sometimes termed as chance errors.
• Due to unknown caus...
Error analysis
For example, suppose you measure the oscillation period of
a pendulum with a stopwatch five times.
Trial no...
Mean value
The average of all the five readings gives the most probable
value for time period.
X =
1
n
Xi
X =
3.9 + 3.5 + ...
Absolute error
The magnitude of the difference between mean value and
each individual value is called absolute error.
∆Xi ...
Mean absolute error
The arithmetic mean of all the absolute errors is called
mean absolute error.
∆X =
1
n
∆Xi
∆X =
0.3 + ...
Reporting of result
• The most common way adopted by scientist and engineers
to report a result is:
Result = best estimate...
Relative error
The relative error is defined as the ratio of the
mean absolute error to the mean value.
relative error = ∆...
Percentage error
The relative error multiplied by 100 is called as
percentage error.
percentage error = relative error x 1...
Least count error
Least count error is the error associated with the
resolution of the instrument.
• The least count error...
Combination of errors
• Let ∆A be absolute error in measurement of A
• Let ∆B be absolute error in measurement of B
• Let ...
When X = A ± B
∆X
X
=
∆A+∆B
A ± B
∆X = ∆A + ∆B
When X = A × B or A / B
∆X
X
=
∆A
A
+
∆B
B
∆X =
∆A
A
+
∆B
B
X
When X = An
∆X
X
= n
∆A
A
∆X = n
∆A
A
X
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 of
something expressed in powers
of 10 is called order
of 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 ...
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Units and Measurement

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This presentation covers measurement of physical quantities, system of units, dimensional analysis & error analysis. I hope this PPT will be helpful for instructors as well as students.

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Units and Measurement

  1. 1. MEASUREMENT
  2. 2. Measurement in everyday life Measurement of mass Measurement of volume
  3. 3. Measurement in everyday life Measurement of length Measurement of temperature
  4. 4. 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.
  5. 5. Physical Quantity A 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
  6. 6. Types of physical quantities 1. 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
  7. 7. 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:
  8. 8. Units for measurement The 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
  9. 9. Characteristics of units Well – defined Suitable size Reproducible Invariable Indestructible Internationally acceptable
  10. 10. • 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
  11. 11. 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.
  12. 12. 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.
  13. 13. 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.
  14. 14. Seven fundamental units FUNDAMENTAL QUANTITY SI UNIT SYMBOL Length metre m Mass kilogram kg Time second s Temperature kelvin K Electric current ampere A Luminous intensity candela cd Amount of substance mole mol
  15. 15. Definition of metre The metre is the length of the path travelled by light in a vacuum during a time interval of 1/29,97,92,458 of a second.
  16. 16. 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.
  17. 17. Prototype cylinder of platinum-iridium alloy
  18. 18. 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.
  19. 19. Two supplementary units 1. Radian: It is used to measure plane angle θ = 1 radian
  20. 20. Two supplementary units 2. Steradian: It is used to measure solid angle Ω = 1 steradian
  21. 21. Rules for writing SI units 1 Full name of unit always starts with small letter even if named after a person. • newton • ampere • coulomb not • Newton • Ampere • Coulomb
  22. 22. Rules for writing SI units 2 Symbol for unit named after a scientist should be in capital letter. • N for newton • K for kelvin • A for ampere • C for coulomb
  23. 23. Rules for writing SI units 3 Symbols for all other units are written in small letters. • m for meter • s for second • kg for kilogram • cd for candela
  24. 24. Rules for writing SI units 4 One 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
  25. 25. Rules for writing SI units 5 The units do not have plural forms. • 6 metre • 14 kg • 20 second • 18 kelvin not • 6 metres • 14 kgs • 20 seconds • 18 kelvins
  26. 26. Rules for writing SI units 6 Full stop should not be used after the units. • 7 metre • 12 N • 25 kg not • 7 metre. • 12 N. • 25 kg.
  27. 27. Rules for writing SI units 7 No space is used between the symbols for units. • 4 Js • 19 Nm • 25 VA not • 4 J s • 19 N m. • 25 V A.
  28. 28. SI prefixes Factor Name Symbol Factor Name Symbol 10 24 yotta Y 10 −1 deci d 10 21 zetta Z 10 −2 centi c 10 18 exa E 10 −3 milli m 10 15 peta P 10 −6 micro μ 10 12 tera T 10 −9 nano n 10 9 giga G 10 −12 pico p 10 6 mega M 10 −15 femto f 10 3 kilo k 10 −18 atto a 10 2 hecto h 10 −21 zepto z 10 1 deka da 10 −24 yocto y
  29. 29. • 3 milliampere = 3 mA = 3 x 10 −3 A • 5 microvolt = 5 μV = 5 x 10 −6 V • 8 nanosecond = 8 ns = 8 x 10 −9 s • 6 picometre = 6 pm = 6 x 10 −12 m • 5 kilometre = 5 km = 5 x 10 3 m • 7 megawatt = 7 MW = 7 x 10 6 W Use of SI prefixes
  30. 30. Some practical units for measuring length 1 micron = 10 −6 m Bacterias Molecules 1 nanometer = 10 −9 m
  31. 31. Some practical units for measuring length 1 angstrom = 10 −10 m Atoms Nucleus 1 fermi = 10 −15 m
  32. 32. Some practical units for measuring length • Astronomical unit = It is defined as the mean distance of the earth from the sun. • 1 astronomical unit = 1.5 x 10 11 m Distance of planets
  33. 33. Some practical units for measuring length • Light year = It is the distance travelled by light in vacuum in one year. • 1 light year = 9.5 x 10 15 m Distance of stars
  34. 34. 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 = 3.1 x 10 16 m 1 AU 1”
  35. 35. 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 m2 • Hectare = It is used to measure large areas in French system of units. 1 hectare = 100 m x 100 m = 10000 m2 • Barn = It is used to measure very small areas, such as nuclear cross sections. 1 barn = 10 −28 m2
  36. 36. Some practical units for measuring mass 1 metric ton = 1000 kg Steel bars Grains 1 quintal = 100 kg
  37. 37. 1 pound = 0.454 kg Newborn babies Crops 1 slug = 14.59 kg Some practical units for measuring mass
  38. 38. Some practical units for measuring mass • 1 Chandrasekhar limit = 1.4 x mass of sun = 2.785 x 10 30 kg • It is the biggest practical unit for measuring mass. Massive black holes
  39. 39. Some practical units for measuring mass • 1 atomic mass unit = 1 12 x mass of single C atom • 1 atomic mass unit = 1.66 x 10 −27 kg • It is the smallest practical unit for measuring mass. • It is used to measure mass of single atoms, proton and neutron.
  40. 40. 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 = 10 −8 s
  41. 41. Seven dimensions of the world Fundamental quantities Length Mass Time Temperature Current Amount of substance Luminous intensity Dimensions [L] [M] [T] [K] [A] [N] [J]
  42. 42. Dimensions of a physical quantity The powers of fundamental quantities in a derived quantity are called dimensions of that quantity.
  43. 43. = Mass length × breath × height [Density] = [M] L × L × L = [M] L3 = [ML−3 ] Dimensions of a physical quantity Density = Mass Volume Example: Hence the dimensions of density are 1 in mass and − 3 in length.
  44. 44. Uses of Dimension To check the correctness of equation To convert units To derive a formula
  45. 45. To check the correctness of equation ∆x = displacement = [L] Consider the equation of displacement, By writing the dimensions we get, vit = velocity × time = length time × time = [L] at2 = acceleration × time2 = length time2 × time2 = [L] The dimensions of each term are same, hence the equation is dimensionally correct. ∆x = vit + 1 2 a t2
  46. 46. To convert units Let us convert newton SI unit of force into dyne CGS unit of force . The dimesions of force are = [LMT−2 ] So, 1 newton = (1 m)(1 kg)(1 s)−2 and, 1 dyne = (1 cm)(1 g)(1 s)−2 Thus, 1 newton 1 dyne = 1 m 1 cm 1 kg 1 g 1 s 1 s −2 = 100 cm 1 cm 1000 g 1 g 1 s 1 s −2 = 100 × 1000 = 105 Therefore, 1 newton = 105 dyne
  47. 47. To derive a formula The time period ‘T’ of oscillation of a simple pendulum depends on length ‘l’ and acceleration due to gravity ‘g’. Let us assume that, T ∝ 𝑙a 𝑔b or T = K 𝑙a 𝑔b K = constant which is dimensionless Dimensions of T = [L0M0T1] Dimensions of 𝑙 = [L1 M0 T0 ] Dimensions of g = [L1M0T−2] Thus, L0M0T1 = K [L1M0T0]a [L1M0T−2]b = K LaM0T0 LbM0T−2b L0M0T1 = K La+bM0T−2b a + b = 0 & − 2b = 1 ∴ b = − 1 2 & a = 1 2 T = K 𝑙1/2 𝑔−1/2 ∴ T = K 𝑙 𝑔
  48. 48. Least count of instruments The smallest value that can be measured by the measuring instrument is called its least count or resolution.
  49. 49. LC of length measuring instruments Ruler scale Least count = 1 mm Vernier Calliper Least count = 0.1 mm
  50. 50. LC of length measuring instruments Screw Gauge Least count = 0.01 mm Spherometer Least count = 0.001 mm
  51. 51. LC of mass measuring instruments Weighing scale Least count = 1 kg Electronic balance Least count = 1 g
  52. 52. LC of time measuring instruments Wrist watch Least count = 1 s Stopwatch Least count = 0.01 s
  53. 53. Accuracy of measurement It 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.
  54. 54. Precision of measurement It 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.
  55. 55. Significant figures The total number of digits (reliable digits + last uncertain digit) which are directly obtained from a particular measurement are called significant figures.
  56. 56. Significant figures Mass = 6.11 g 3 significant figures Speed = 67 km/h 2 significant figures
  57. 57. Significant figures Time = 12.76 s 4 significant figures Length = 1.8 cm 2 significant figures
  58. 58. Rules for counting significant figures 1 All non-zero digits are significant. Number 16 35.6 6438 Significant figures 2 3 4
  59. 59. 2 Zeros between non-zero digits are significant. Rules for counting significant figures Number 205 3008 60.005 Significant figures 3 4 5
  60. 60. Rules for counting significant figures 3 Terminal zeros in a number without decimal are not significant unless specified by a least count. Number 400 3050 (20 ± 1) s Significant figures 1 3 2
  61. 61. Rules for counting significant figures 4 Terminal zeros that are also to the right of a decimal point in a number are significant. Number 64.00 3.60 25.060 Significant figures 4 3 5
  62. 62. Rules for counting significant figures 5 If the number is less than 1, all zeroes before the first non-zero digit are not significant. Number 0.0064 0.0850 0.0002050 Significant figures 2 3 4
  63. 63. 6 During conversion of units use powers of 10 to avoid confusion. Rules for counting significant figures Number 2.700 m 2.700 x 10 2 cm 2.700 x 10 −3 km Significant figures 4 4 4
  64. 64. 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 definition 1 dozen = 12 objects 1 hour = 60 minute 1 inch = 2.54 cm By counting 45 students 5 apples 6 faces of cube
  65. 65. Rules for rounding off a measurement 1 If the digit to be dropped is less than 5, then the preceding digit is left unchanged. Number 64.62 3.651 546.3 Round off up to 3 digits 64.6 3.65 546
  66. 66. 2 If the digit to be dropped is more than 5, then the preceding digit is raised by one. Number 3.479 93.46 683.7 Round off up to 3 digits 3.48 93.5 684 Rules for rounding off a measurement
  67. 67. 3 If the digit to be dropped is 5 followed by digits other than zero, then the preceding digit is raised by one. Number 62.354 9.6552 589.51 Round off up to 3 digits 62.4 9.66 590 Rules for rounding off a measurement
  68. 68. 4 If 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. Number 53.350 9.455 782.5 Round off up to 3 digits 53.4 9.46 782 Rules for rounding off a measurement
  69. 69. Significant figures in calculations Addition & subtraction The 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
  70. 70. Significant figures in calculations Multiplication & division The final result would round to the same number of 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
  71. 71. Errors in measurement Difference between the actual value of a quantity and the value obtained by a measurement is called an error. Error = actual value – measured value
  72. 72. Types of errors Systematic errors Gross errors Random errors
  73. 73. 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.
  74. 74. Types of systematic errors Personal errors Instrumental errors Environmental errors
  75. 75. a. Personal errors These errors are arise due to faulty procedures adopted by the person making measurements. Parallax error
  76. 76. b. Instrumental errors These errors are arise due to faulty construction of instruments. Zero error
  77. 77. 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.
  78. 78. Advanced experimental setups
  79. 79. 2. Gross errors These 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.
  80. 80. 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.
  81. 81. Error analysis For example, suppose you measure the oscillation period of a pendulum with a stopwatch five times. Trial no ( i ) 1 2 3 4 5 Measured value ( Xi ) 3.9 3.5 3.6 3.7 3.5
  82. 82. Mean value The average of all the five readings gives the most probable value for time period. X = 1 n Xi X = 3.9 + 3.5 + 3.6 + 3.7 + 3.5 5 = 18.2 5 X = 3.64 = 3.6
  83. 83. Absolute error The magnitude of the difference between mean value and each individual value is called absolute error. ∆Xi = X − Xi Xi 3.9 3.5 3.6 3.7 3.5 ∆Xi 0.3 0.1 0 0.1 0.1 The absolute error in each individual reading:
  84. 84. Mean absolute error The arithmetic mean of all the absolute errors is called mean absolute error. ∆X = 1 n ∆Xi ∆X = 0.3 + 0.1 + 0 + 0.1 + 0.1 5 = 0.6 5 ∆X = 0.12 = 0.1
  85. 85. 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.
  86. 86. Relative error The relative error is defined as the ratio of the mean absolute error to the mean value. relative error = ∆X / X ∆X / X = 0.1 3.6 = 0.0277 ∆X / X = 0.028
  87. 87. Percentage error The 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 %
  88. 88. Least count error Least 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.
  89. 89. Combination of errors • Let ∆A be absolute error in measurement of A • Let ∆B be absolute error in measurement of B • Let ∆X be absolute error in measurement of X In different mathematical operations like addition, subtraction, multiplication and division the errors are combined according to some rules.
  90. 90. When X = A ± B ∆X X = ∆A+∆B A ± B ∆X = ∆A + ∆B
  91. 91. When X = A × B or A / B ∆X X = ∆A A + ∆B B ∆X = ∆A A + ∆B B X
  92. 92. When X = An ∆X X = n ∆A A ∆X = n ∆A A X
  93. 93. Estimation Estimation is a rough calculation to find an approximate value of something that is useful for some purpose.
  94. 94. Estimate the number of flats in Dubai city
  95. 95. Estimate the volume of water stored in a dam
  96. 96. Order of magnitude The approximate size of something expressed in powers of 10 is called order of magnitude.
  97. 97. 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 10 −31 kg ≈ 10 x 10 −31 kg ≈ 10 −30 kg • Mass of observable universe = 1.59 x 10 53 kg ≈ 1 x 10 53 kg ≈ 10 53 kg
  98. 98. Thank You
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This presentation covers measurement of physical quantities, system of units, dimensional analysis & error analysis. I hope this PPT will be helpful for instructors as well as students.

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