Units and 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 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

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

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 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

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, 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 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

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.

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 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
• ampere
• coulomb
not
• Newton
• Ampere
• Coulomb

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

3
Symbols for all other units are written in
small letters.
• m for meter
• s for second
• kg for kilogram
• cd for candela

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

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

6
Full stop should not be used after the
units.
• 7 metre
• 12 N
• 25 kg
not
• 7 metre.
• 12 N.
• 25 kg.

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.

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

• 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

Some practical units for measuring length
1 micron = 10
−6
m
Bacterias Molecules
1 nanometer = 10
−9
m

1 angstrom = 10
−10
m
Atoms Nucleus
1 fermi = 10
−15
m

• 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

• 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

• 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”

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

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

• 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

• 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.

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

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]

Dimensions of a physical quantity
The powers of fundamental quantities
in a derived quantity are called
dimensions of that quantity.

=
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.

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 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

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

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

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 instruments
Ruler scale
Least count = 1 mm
Vernier Calliper
Least count = 0.1 mm

LC of length measuring instruments
Screw Gauge
Spherometer

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:
• 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 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.

Significant figures
The total number of digits
(reliable digits + last uncertain digit)
which are directly obtained from a
particular measurement are called
significant figures.

Significant figures
Mass = 6.11 g
3 significant figures
Speed = 67 km/h

Significant figures
Time = 12.76 s
Length = 1.8 cm

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.
Number
205
3008
60.005
Significant figures
3
4
5

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

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

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

6
During conversion of units use powers of 10 to
avoid confusion.
Number
2.700 m
2.700 x 10
2
cm
2.700 x 10
−3
km
Significant figures
4
4
4

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

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

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
3.48
93.5
684

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
62.4
9.66
590

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
53.4
9.46
782

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

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

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

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, measuring
instrument and the environment.
• These errors are eliminated by detecting
the source of the error.

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 error

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 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.

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.

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 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

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

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:

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

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 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

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 %

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.

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.

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 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

Units and Measurement

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