CHAPTER 1
UNITS AND MEASUREMENT
INTRODUCTION
The comparison of any physical quantity with its standard unit is
called measurement.
Physical Quantities
All the quantities in terms of which laws of physics are described, and
whose measurement is necessary are called physical quantities.
Units
A definite amount of a physical quantity is taken as its standard unit.
The standard unit should be easily reproducible, internationally
accepted.
Shahil Sir
Physical Quantities
Quantitative versus qualitative
Most observation in physics are quantitative
• Descriptive observations (or qualitative) are usually imprecise
Quantitative Observations
Qualitative Observations
What can be measured with the instruments
How do you measure
on an aeroplane?
artistic beauty?
•
Shahil Sir
Physical Quantities
Are classified into two types:
•
•
Base quantities
Derived quantities
Base quantity
is like the brick – the
basic building block
of a house
Derived quantity is like
the house that was
build up from a collection
of bricks (basic quantity)
Shahil Sir
Fundamental quantities
• The quantities that are independent on other
quantities are called fundamental quantities. The
units that are used to measure these fundamental
quantities are called fundamental units.
Shahil Sir
Derived quantities.
• The quantities that are derived using the
fundamental quantities are called derived
quantities. The units that are used to measure
these derived quantities are called derived units.
Shahil Sir
Fundamental Units
Those physical quantities which are independent to each other are called fundamental
quantities and their units are called fundamental units.
Shahil Sir
System of international d’units
Physical quantity
Unit
Symbol
Length
Mass
Time
Electric current
Thermodynamic
temperature
Intensity of light
Quantity of substance
metre
kilogram
second
Ampere
Kelvin
m
kg
s
A
K
candela
mole
Cd
mol
Shahil Sir
Supplementary quantities
Plane angle radian rad
Solid angle steradian sr
Shahil Sir
Definitions of SI UNITS
• Metre: It is defined as the distance travelled by light in vacuum
during a time interval of 1/299, 792, 458 of a second
• Kilogram : The mass of a cylinder of platinum–iridium alloy kept
in the International Bureau of weights and measures preserved at
Serves near Paris is called one kilogram.
• Second : The duration of 9192631770 periods of the radiation
corresponding to the transition between the two hyperfine levels
of the ground state of caesium–133 atom is called one second.
Shahil Sir
• Ampere : The current which when flowing in each of two parallel
conductors of infinite length and negligible cross–section and
placed one metre apart in vacuum, causes each conductor to
experience a force of 2x10–7 newton per metre of length is known
as one ampere.
• Kelvin : The fraction of 1/273.16 of the thermodynamic
temperature of the triple point of water is called kelvin.
• Candela: The candela is the luminous intensity, in a given
direction, of a source that emits monochromatic radiation of
frequency 540 x1012 hertz and that has a radiant intensity in that
direction of 1/683 watt per steradian.
• Mole : The amount of a substance of a system which contains as
many elementary entities as there are atoms in 12x10 3 kg of
carbon–12 is known as one mole.
Shahil Sir
• Radian : The angle made by an arc of the circle
equivalent to its radius at the centre is known as
radian. 1 radian = 57017’45”.
• Steradian : The angle subtended at the centre by
one square metre area of the surface of a sphere
of radius one metre is known as steradian.
Shahil Sir
SOME MORE DEFINITIONS
• Angstrom is the unit of length used to measure the wavelength of light. 1 Å =
10-10 m.
• Fermi is the unit of length used to measure nuclear distances. 1 fermi = 1015metre.
• Light year is the unit of length for measuring astronomical distances.
• Light year = distance travelled by light in 1 year = 9.4605x1015m.
• Astronomical unit = Mean distance between the sun and earth = 1.5x1011 m.
• Parsec is the distance at which average radius of earth’s orbit subtends an
angle of 1 arc second
• Parsec = 3.26 light years = 3.084x1016 m
Shahil Sir
SI Units
Derived
Quantity
area
Relation with Base and
Derived Quantities
length × width
volume
density
length × width × height
mass ÷ volume
speed
acceleration
force
distance ÷ time
change in velocity ÷ time
pressure
work
power
mass × acceleration
force ÷ area
force × distance
work ÷ time
Shahil Sir
Unit
m2
m3
kgm-3
ms-1
ms-2
kgms-2
Nm-2
Nm
Js-1
Special
Name
newton (N)
pascal (Pa)
joule (J)
watt (W)
SI Units
►
SI Units – International System of Units
Base Quantities
Name of Unit
Symbol of Unit
length
metre
m
mass
kilogram
kg
time
second
s
electric current
ampere
A
temperature
kelvin
K
amount of substance
mole
mol
luminous intensity
candela
cd
Shahil Sir
SI Units
This Platinum Iridium
cylinder is the standard
kilogram.
Shahil Sir
SI Units
• Example of derived quantity: area
Defining equation:
area = length × width
In terms of units:
Units of area = m × m = m2
Defining equation:
volume = length × width × height
In terms of units:
Units of volume = m × m × m = m2
Defining equation:
density = mass ÷ volume
In terms of units:
Units of density = kg / m3 = kg m−3
Shahil Sir
SI Units
• Work out the derived quantities for:
Defining equation:
speed =
In terms of units:
Units of speed =
Defining equation:
acceleration =
In terms of units:
Units of acceleration =
Defining equation:
force = mass × acceleration
In terms of units:
Units of force =
Shahil Sir
SI Units
• Work out the derived quantities for:
Defining equation:
Pressure =
In terms of units:
Units of pressure =
Defining equation:
Work = Force × Displacement
In terms of units:
Units of work =
Defining equation:
Power =
In terms of units:
Units of power =
Shahil Sir
MEASUREMENT
OF
LENGTH AND TIME
Physical Quantities
• A physical quantity is one that can be measured
and consists of a magnitude and unit.
▲
70
4.5 m
km/h
▼
SI units are
common
today
Measuring length
Vehicles
Not
Exceeding
1500 kg In
Unladen
Weight
Shahil Sir
Measuring large Distances – Parallax Method
parallax is a displacement or difference in the apparent position of an object
viewed along two different lines of sight, and is measured by the angle or
semi-angle of inclination between those two lines. distance between the two
viewpoints is called basis.
measuring distance of a planet using parallax method
similarly, α = d/d
whereα = angular size of the planet (angle subtended by d at earth) and d is
the diameter of the planet.αis angle between the direction of the telescope
when two diametrically opposite points of the planet are viewed.
parallax method
The apparent displacement or the difference in apparent direction of an object
as seen from two different points not on a straight line with the object especially
: the angular difference in direction of a celestial body as measured from two
points on the earth's orbit.
Prefixes
► Prefixes
simplify the writing of very large or very
small quantities
Prefix
nano
micro
Abbreviation
n
μ
Power
10−9
10−6
milli
centi
deci
m
c
d
10−3
10−2
10−1
kilo
mega
k
M
103
106
giga
G
Shahil Sir
109
Prefixes
• Alternative writing method
• Using standard form
• N × 10n where 1 ≤ N < 10 and n is an integer
This galaxy is about 2.5 × 106
light years from the Earth.
The diameter of this atom is
about 1 × 10−10 m.
Shahil Sir
Measurement of Length
Length
• Measuring tape is used to measure relatively long
lengths
• For shorter length, a metre rule or a shorter rule
will be more accurate
Shahil Sir
Measurement of Length and Time
• Correct way to read the scale on a ruler
• Position eye perpendicularly at the mark on the
scale to avoids parallax errors
• Another reason for error: object not align or
arranged parallel to the scale
Shahil Sir
Measurement of Length
• Many instruments do not read exactly zero when
nothing is being measured
• Happen because they are out of adjustment or
some minor fault in the instrument
• Add or subtract the zero error from the reading
shown on the scale to obtain accurate readings
• Vernier calipers or micrometer screw gauge give
more accurate measurements
Shahil Sir
Measurement of Length
• Table shows the range and precision of some
measuring instruments
Instrument
Range of
measurement
Accuracy
Measuring tape
0−5m
0.1 cm
Metre rule
0−1m
0.1 cm
Vernier calipers
0 − 15 cm
0.01 cm
Micrometer screw gauge
0 − 2.5 cm
0.01 mm
Shahil Sir
Measurement of Length
Vernier Calipers
• Allows measurements up to 0.01 cm
• Consists of a 9 mm long scale divided into 10
divisions
Shahil Sir
Measurement of Length
Vernier Calipers
• The object being measured is between 2.4 cm
and 2.5 cm long.
• The second decimal number is the marking on the
vernier scale which coincides with a marking on
the main scale.
Shahil Sir
Measurement of Length
• Here the eighth marking on the vernier scale
coincides with the marking at C on the main scale
• Therefore the distance AB is 0.08 cm, i.e. the
length of the object is 2.48 cm
Shahil Sir
Measurement of Length
• The reading shown is 3.15 cm.
• The instrument also has inside jaws for measuring internal
diameters of tubes and containers.
• The rod at the end is used to measure depth of containers.
Shahil Sir
Measurement of Length
Micrometer Screw Gauge
• To measure diameter of fine wires, thickness of
paper and small lengths, a micrometer screw
gauge is used
• The micrometer has two scales:
• Main scale on the sleeve
• Circular scale on the thimble
• There are 50 divisions on the thimble
• One complete turn of the thimble moves the
spindle by 0.50 mm
Shahil Sir
Measurement of Length
Micrometer Screw Gauge
• Two scales: main scale and circular scale
• One complete turn moves the spindle by 0.50 mm.
• Each division on the circular scale = 0.01 mm
Shahil Sir
Measurement of Length
Precautions when using a micrometer
1. Never tighten thimble too much
– Modern micrometers have a ratchet to avoid this
2. Clean the ends of the anvil and spindle before making a
measurement
– Any dirt on either of surfaces could affect the reading
3. Check for zero error by closing the micrometer when there is
nothing between the anvil and spindle
– The reading should be zero, but it is common to find a small
zero error
– Correct zero error by adjusting the final measurement
Shahil Sir
Shahil Sir
Shahil Sir
Shahil Sir
Measurement of Time
Time
• Measured in years, months, days, hours, minutes
and seconds
• SI unit for time is the second (s).
• Clocks use a process which depends on a
regularly repeating motion termed oscillations.
Shahil Sir
Measurement of Time
Caesium atomic clock
► 1999 - NIST-F1 begins operation with an uncertainty of
1.7 × 10−15, or accuracy to about one second in 20
million years
Shahil Sir
Measurement of Time
Time
• The oscillation of a simple pendulum is an
example of a regularly repeating motion.
• The time for 1 complete oscillation is referred to
as the period of the oscillation.
Shahil Sir
Measurement of Time
Pendulum Clock
• Measures long intervals of time
• Hours, minutes and seconds
• Mass at the end of the chain attached
to the clock is allowed to fall
• Gravitational potential energy from
descending mass is used to keep the
pendulum swinging
• In clocks that are wound up, this
energy is stored in coiled springs as
elastic potential energy.
Shahil Sir
Measurement of Time
Watch
• also used to measure long intervals of time
• most depend on the vibration of quartz crystals
to keep accurate time
• energy from a battery keeps quartz crystals
vibrating
• some watches also make use of coiled springs to
supply the needed energy
Shahil Sir
Measurement of Time
Stopwatch
• Measure short intervals of time
• Two types: digital stopwatch, analogue stopwatch
• Digital stopwatch more accurate as it can measure
time in intervals of 0.01 seconds.
• Analogue stopwatch measures time in intervals of
0.1 seconds.
Shahil Sir
Measurement of Time
Errors occur in measuring time
• If digital stopwatch is used to time a race,
should not record time to the nearest 0.01 s.
• reaction time in starting and stopping the watch
will be more than a few hundredths of a second
• an analogue stopwatch would be just as useful
Shahil Sir
Significant figures: The significant figures are normally those digits in a measured quantity which are known reliably
plus one additional digit that is uncertain.
For counting of the significant figure rule are as:
(i) All non- zero digits are significant figure.
(ii) All zero between two non-zero digits are significant figure.
(iii) All zeros to the right of a non-zero digit but to the left of an understood decimal point are not
significant. But such zeros are significant if they come from a measurement.
(iv) All zeros to the right of a non-zero digit but to the left of a decimal point are significant.
(v) All zeros to the right of a decimal point are significant.
(vi) All zeros to the right of a decimal point but to the left of a non-zero digit are not significant.
Single zero conventionally placed to the left of the decimal point is not significant.
(vii) The number of significant figures does not depend on the system of units.
Addition or subtraction with significatn figure :In addition or subtraction , the result should be reported to the same number
of decimal places as that of the number with minimum number of decimal
places.
For ex: A= 334.5 kg; B= 23.45Kg then A + B =334.5 kg + 23.43 kg = 357.93 kg
The result with significant figures is 357.9 kg
Mutiplication and division in significant figure :In multiplication or division, the result should be reported to the same
number of significant figures as that of the number with minimum of
significant figures.
USES OF DIMENSIONAL EQUATION
1.
2.
3.
To check the correctness of physical equation.
To derive the relation between different physical phenomenon.
To change from one system of unit to another.
HOMOGENEITY OF DIMENSION
• If two equation have physically equal relation so, they also dimensionally
Equal.
KE = PE
½ mv2 = mgh
=
M(LT-1)2 = M x (LT-2-) x L
KE = PE
MLT-2 = ML2T-2
H
Some dimensional formula
Error
The lack in accuracy in the measurement due to the limit of accuracy
of the instrument or due to any other cause is called an error.
1. Absolute Error
The difference between the true value and the measured value of a
quantity is called absolute error.
If a1 , a2, a3 ,…, an are the measured values of any quantity a in an
experiment performed n times, then the arithmetic mean of these
values is called the true value (am) of the quantity.
The absolute error in measured values is given by
Δa1 = am – a1
Δa2 = am – a1
………….
Δam = Δam – Δan
2. Mean Absolute Error
The arithmetic mean of the magnitude of absolute errors in all the
measurement is called mean absolute error.
3. Relative Error The ratio of mean absolute error to
the true value is called relative
4. Percentage Error The relative error expressed in
percentage is called percentage error.
Problem. 2.16 Find the significant figure in the following :
(a) 0.007m2
(b) 2.64 x 1024kg
(c) 0.2370gcm-3
(d) 6.320 J
(e) 0.0006032m2
(f) 6.032 Nm-2
Propagation of Error
(i) Error in Addition or Subtraction Let x = a + b or x = a
–b
If the measured values of two quantities a and b are (a
± Δa and (b ± Δb), then maximum absolute error in
their addition or subtraction.
Δx = ±(Δa + Δb)
(ii) Error in Multiplication or Division Let x = a x b or x =
(a/b).
If the measured values of a and b are (a ± Δa) and (b ±
Δb), then maximum relative error
• Ans- (a) – 0.007 has one significant figure.
• Ans- (b) – 2.64 x 1024 has three significant figure.
• Ans–(c) – 0.2370 has four significant figure.
• Ans- (d) – 6.320 has four significant figure.
• Ans –(e)- 0.0006032 has four significant figure.
• Ans –(f) 6.032 has four significant figure.
Shahil Sir
• Problem 1.7 The length , breath and thickness of a metal sheet are 4.234m , 1.005m and 2.01cm respectively give
the area and volume of the sheet to correct significant figure
• Ans – L = 4.234m
•
B = 1.005m
• Thickness = 2.01cm = 2.01 x 10-2
• Area of metal sheet = 4.234 x 1.005
•
= 4.25517m2
• Since both length and breath have four significant figure , the area of the metal sheet after rounding off to four
significant figure given by –
• Area =4.255m2
• Volume of metal sheet = 4.234 x 1.005 x 2.01 x 10-2
• = 8.55289 x 10-2m-3
• After rounding off volume it gives three significant figure
• Volume = 8.55 x 10-2
Shahil Sir
• Q.1.29 A physical quantity P is related to four
observable a , b , c , d as follows
•
P = a3 b2/cd1/2
• The percentage error of measurements in a, b, c and
d are 1%,3%,4%and 2% respectively .what is the
percentage error . percentage error in quantity p? if
the value of p calculate using the above relation turns
out to be 3.763,to what value should
• You round off the result
Shahil Sir
The early systems of units :
• MKS : METER, KILOGRAM, SECOND
• CGS : CENTIMETER, GRAM, SECOND
• FPS : FOOT, POUND, SECOND
Shahil Sir
ERRORS
• The result of every measurement by any measuring instrument
contains some uncertainty . This uncertainty is called error.
• In general there are two types of errors
• Systematic Error
• Random Error
Shahil Sir
Systematic Error
⚫They are those errors tend to in one direction
either positive or negative
⚫Sources of systematic errors
⚫Instrumental errors-That arise due to imperfect
design of the measuring instrument for example
boiling point of water read as 104 degree Celsius
where 100 degree Celsius
⚫Imperfection in experimental technique or
procedure-the temperature of human body is under
armpit is lower than actual value
⚫Personal error-that arise due to carelessness
Shahil Sir
Random Error
⚫Which occurs irregularly
⚫ Random error occurs due to unpredictable
Shahil Sir
Shahil Sir
Shahil Sir
Shahil Sir
Combination of Errors
•
•
•
•
Error of a sum or a difference
± ΔZ = ± ΔA ± ΔB
The maximum value of the error ΔZ is ΔA + ΔB.
When two quantities are added or subtracted, the absolute error
in the final result is the sum of the absolute errors in the
individual quantities.
Shahil Sir
Error of a product or a quotient
• Suppose Z = AB OR Z = A/B and the measured values of A and B
are A ± ΔA and B ± ΔB. Then
• ΔZ / Z = (ΔA / A) + (ΔB / B)
• When two quantities are multiplied or divided, the fractional
error in the result is the sum of the fractional errors in the
multipliers.
Shahil Sir
Error due to the power of a measured quantity :
• Z = A2, then
ΔZ / Z = (ΔA / A) + (ΔA / A) = 2 (ΔA / A)
• If Z = Ap Bq / Cr, then
ΔZ / Z = p (ΔA / A) + q (ΔB / B) + r (ΔC / C)
• The fractional error in a physical quantity raised to the power is
the power times the fractional error in the individual quantity.
Shahil Sir
The error is communicated in different
mathematical operations as detailed below:
Shahil Sir
Shahil Sir
For counting of the significant figure rule
are as:
All non- zero digits are significant figure.
Ex –1.325 contains significant figures =4
All the zeros between two non-zero digits are significant figure
no matter where the decimal point is ,if at all
Ex–207.009 contains significant figures =6
All zeros to the right of a non-zero digit but to the left of
an understood decimal point are not significant. But such
zeros are significant if they come from a measurement.
Ex – 2400
significant figures = 2
2400 kg
significant figures = 4
Shahil Sir
For counting of the significant figure rule
are as:
All zeros to the right of a non-zero digit but to the left of
a decimal point are significant.
Ex – 300.24
significant figures = 5
All zeros to the right of a decimal point are significant.
Ex – 2.00
significant figures = 3
if the number is less than 1,the zeros to the right of a decimal
point but to the left of the first non-zero digit are not
significant. Single zero conventionally placed to the left of the
decimal point is not significant.
Ex – 0.00007
significant figures = 1
The number of significant figures does not depend on the
system of units.
Ex – 2.65 cm = 26.5 mm = 0.0265 m = 2.65 x 10-5 km
significant figures in each case Shahil
= 3Sir
Rounding off
• a) The result of computation with approximate numbers, which
contain more than one uncertain digit, should be rounded off.
• b) Preceding digit is raised by 1 if the insignificant digit to be
dropped is more than 5, and is left unchanged if the latter is less
than 5.
• Ex – 2.568 = 2.57,
3.642 = 3.64
• c) But what if the number is 2.745 in which the insignificant digit is
5. Here the convention is that if the preceding digit is even, the
insignificant digit is simply dropped and, if it is odd, the preceding
digit is raised by 1.
• Ex – 2.745 = 2.74,
5.635 = 5.64
Shahil Sir
Shahil Sir
ACCURACY
The accuracy of a measurement is a
measure of how close the measured
value is to the true of the quantity
PRECISION
Precision tells us to what resolution or
limit the quantity is measured.
Shahil Sir
Example:
Shahil Sir
Q1. State the no. of significant figures in the following
a) 0.007
Ans:- 1
b) 6.032
Ans:- 4
c) 2.64
Ans:- 3
d) 0.2370
Ans:- 4
Shahil Sir
Q2. Round off the following numbers to 2 places decimal
a) 2.038
Ans:- 2.04
b) 6.052
Ans:- 6.05
c) 7.625
Ans:- 7.62
d) 0.2356
Ans:- 0.24
Shahil Sir
❖ dimensional analysis
The derived unit of all the physical quantities can be suitably
Expressed in the term of fundamental unit ofMass – (M)
Length-(L)
Time -(T)
Ex. Area= L x B = m x m =
Ex . Velocity = d/t = {L/T} = LT-1
❖ Hence, the dimension of a physical quantities are the power
To which the fundamental unit of mass , length and time have
To be raised in order to obtain it unit .
It show the dependents on fundamental units.
Different type of variable and constant
* Dimensional variable- The quantities like area , volume Velocity force
posses dimension but not have constant value .+
• Non-dimensional variable- The quantities like angleSpecific gravity ,
strain etc. neither posses dimension Nor constant value.
*dimension constant- posses dimension also have a Constant value.
Ex- gravitational constant , Plank’s constant , Ryberg etc.
*Non-dimensional constant- Constant quantities having No dimension
like –Include pure no. 1,2,3,4, pi trignometrical function.
The dimensions of a physical quantity are the
powers to which the fundamental quantities are
raised to represent that physical quantity.
The equation which expresses a physical quantity
in terms of the fundamental units of mass, length
and time, is called dimensional equation.
According to the principle of homogeneity a
physical equation will be dimensionally correct if
the dimensions of all the terms occurring on both
sides of the equation are the same
Shahil Sir
Dimensions of fundamental quantities
Fundamental quantity
Length
Mass
Dimensional Formula
[L]
[M]
3
4
5
Time
Electric current
Thermodynamic Temp.
[T]
[A]
[K]
6
7
Luminous Intensity
Amount of Substance
[cd]
[mol]
Sl. No.
1
2
Shahil Sir
Dimensions of derived quantities
Shahil Sir
Main uses of the dimensional analysis
There are four main uses of the dimensional analysis(a) To convert a unit of given physical quantities from
one system of units to another system for which we
use n2 = n1 [M1/M2]a [L1/L2]b [T1/T2]c
(b) To check the correctness of a given physical relation.
(c) To derive a relationship between different physical
quantities.
(d) To derive dimensions of physical constants.
Shahil Sir
Example
Convert 7 joule into erg.
Dimension of work is [Ml2T-2]
SI Unit
• M1 = 1kg
• L1 = 1m
• T1 = 1s
• n1 = 7 J
cgs Unit
• M2 = 1g
• L2 = 1cm
• T2 = 1s
• n2 = ?
n2 = n1 [M1/M2]1 [L1/L2]2 [T1/T2]-2
= 7 [1kg/1g]1 [1m/1cm]2 [1s/1s]-2
= 7 [1000]1 [100]2 [1]-2
= 7 x 107 erg
Shahil Sir
Dimensional Analysis
Checking equations with dimensional analysis:
(L/T2)T2=L
L
(L/T)T=L
• Each term must have
same dimension
• Two variables can not be added if dimensions are different
• Multiplying variables is always fine
• Numbers (e.g. 1/2 or π) are dimensionless
Shahil Sir
Example
The frequency of vibration of a stretched string
depends on its length, mass per unit length and
tension. Derive a relation between them.
Shahil Sir
Derive the dimensions of a and b in Vanderwaal’s
equation (P + a/V2)(V – b) = RT
[P] = [ a]/[V2]
[a] = [p][V2]
= [ML-1T-2][L6]
= [ML5T-2]
[b] = [V]
= [L3]
Shahil Sir
Shahil Sir
Q1. State the dimensional formula of the following physical
quantities.
a) Force
Ans:- [MLT-2]
b) Pressure
Ans:- [ML-1T-2]
c) Surface tension
Ans:- [ML0T-2]
d) Torque
Ans:- [ML2T-2]
e) Angular momentum
Ans:- [ML2T-1]
Shahil Sir