UNIT AND DIMENSION
Physics describes the laws of nature. This description is quantitative and involves
measurement and comparison of physical quantities. to measure a physical
quantity we need some standard unit of that quantity.
An elephant is heavier than a cow but exactly how many times? This question can
be easily answered if we have chosen a standard mass calling it a unit mass. If the
elephant is 150 times the unit mass and the cow is 15 times, we know that the
elephant is 10 times heavier than the cow. If I have the knowledge of the unit
length and some one says that Gandhi Maidan is 10 times the unit length from
here, I will have the idea whether I should walk down to Gandhi Maidan or I
should walk down to Gandhi Maidan or I should ride a rickshaw or I should go by
a bus. Thus, the physical quantities are quantitatively expressed in terms of a unit
of that quantity. The measurement of the quantity is mentioned in two parts, the
first part gives how many times of the standard unit, and the second part gives
the name of the unit. Thus; suppose I have to study for 3 hours the numeric part
3 says that it is 3 times of the unit of time and the second part hour says that the
unit chosen here is an hour.
Unit of a physical quantity is the reference standard used to measure it.
Who Decides the Units?
How is a standard unit chosen for a physical quantity? The first thing is that it
should have international acceptance. Otherwise, everyone will choose his or her
own unit for the quantity and it will be difficult to communicate freely among the
persons distributed over the world. A body named Conference General des Poids
et measures or CGPM also as General conference on weight and Measures in
English has been given the authority to decide the units by international
agreement. It holds its meetings and any changes in standard units are
communicated through the publications of the conference.
Physical quantities:
The quantities which can be measured by an instrument and by means of which
we can describe the lows of physics are called physical quantities.
e.g. length, velocity, acceleration, force, time, pressure, mass, density etc.
Physical quantity = Numerical value × unit
nu = const.
n
1
u
1 kg = 1000 gm
Physical quantities





Fundamental or
Derived
Supplementary
Basic
Quantities
Quantities
Quantities
Properties of Units:
(a)
The unit should be well-defined.
(b)
The unit should be of some suitable size.
(c)
The unit should be easily reproducible.
(d)
The unit should not change with time.
(e)
The unit should not change with physical condition like pressure,
temperature etc.
(f)
Unit should be of proper size.
There are a large number of physical quantities which are measured and every
quantity needs a definition of unit. However, not are the quantities are
independent to each other. As a simple example, if a unit of length is defined, a
unit of area is automatically obtained.
The fundamental quantities are those quantities which are independent to each
other and are other quantities may be expressed in terms of the fundamental
quantities.
Many different choices can be made for the fundamental quantities. For
example, one can take speed and time as fundamental quantities. Length is then
a derived quantity. One may also take length and time interval as the
fundamental quantities and then speed will be a derived quantity. Several
systems are in use over the world and in each system the fundamental quantities
are selected in a particular way.
The units defined for the fundamental quantities are called fundamental units
and those obtained for the derived quantities are called the derived units.
Fundamental quantities are also known as base quantities.
SI units:
In 1971 CGPM held its meeting and decided a system of units which is known as
the International System of Units. It is abbreviated as SI from the French name Le
system International d' Units, This system is widely used throughout the world.
In SI system, there are 7 fundamental quantities.
Table 1.1. Fundamental or Base quantities:
Quantity
Name of the unit
Symbol
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
Besides the seven fundamental units two supplementary units are defined. They
are the plane angle and solid angle. The unit for plane angle is radian with the
symbol rad and the unit for the solid angle is steradian with the symbol sr.
SI Prefixes:
The magnitudes of physical quantities vary over a wide range. The CGPM
recommended standard prefixes for certain powers of 10.
Table (1.2) Show these prefixes:
Table 1.2 : SI Prefixes
Power of 10
Prefix
Symbol
18
exa
E
15
Peta
P
12
tera
T
9
giga
G
6
mega
M
3
kilo
K
2
hecto
h
1
deka
da
–1
deci
d
–2
centi
c
–3
milli
m
–6
micro
µ
–9
nano
n
–12
pico
p
–15
femto
f
–18
atto
a
Dimension:
The dimension of a physical quantity are the powers raised to fundamental
quantities. The dimension of a physical quantity do not change with change of
system of units and a formula representing a physical quantity in term of
fundamental quantities is known as dimensional formula.
TYPES OF UNITS
FUNDAMENTAL UNITS:
The units defined for the fundamental quantities are called fundamental units.
1.
Unit of mass = Kilogram
(1 kilogram is defined as
the mass of a platinum –
iridium cylinder kept in
National Bureau of weights
and measurements, paris)
2.
Unit of length = Meter
(Travelled distance by light
in vacuum in 1/299, 792,
458 second or it is equal to
1650763.73 wave length
emitting from Kr86)
3.
Unit of Time = Sec.
(The time interval in which
Cesium-133 atom vibrates
9,192,631,770 times)
4.
Unit of Temperature = Kelvin
(It
is
defined
(1/273.16)
as
the
fraction
of
thermo
temperature
dynamic
of
triple
point of water.*)
5.
Unit of current = Ampere
(Amount of current which
produces a force of 2 × 10–
7
N on per unit length acts
between two parallel wires
of
infinite
negligible
area
length
and
cross-section
placed
at
1
m
distance in vacuum)
6.
Unit of luminious
(Amount of
Intensity = Candela
intensity on 1/60000 m 2
area at freezing point of
platinum
2042K
at
pressure of 101325 N/m 2.)
7.
Unit of quantity of
(It is the amount of
Substance = mole
a substance which has
same
number
of
elementry entities as in 12
gm of Carbon)
*Triple Point of Water is the temperature at which ice, water and water vapours
co-exist.
CLASSIFICATION OF UNITS SYSTEMS:
BASIC UNIT SYSTEMS:
Quantity
Name of system
C.G.S.
F.P.S.
M.K.S.
S.I.
Length
centimeter
foot
meter
meter (m)
Mass
gram
pounds
kilogram
kilogram (kg)
second (s)
Time
second
second
second
Kelvin (K)
Temperature
ampere (A)
Electric
candela (Cd)
Current
mole (mol)
Luminious
Intensity
Amount
of
Substance
In S.I. system there are two supplementary units.
Radian (rad) : Unit of plane angle.
Steradian (st) : Unit of solid angle.
S.I. PREFIXES
S.No.
Perfix
Symbol
Power of 10
1.
exa
E
18
2.
peta
P
15
3.
tera
T
12
4.
giga
G
9
5.
mega
M
6
6.
kilo
K
3
7.
hector
h
2
8.
deca
da
1
9.
deci
d
–1
10.
centi
c
–2
11.
milli
m
–3
12.
micro
µ
–6
13.
nano
n
–9
14.
pico
p
–12
15.
femto
f
–15
16.
atto
a
–18
Ex. :
1 micro volt = 1µV = 10–6 V
1 nano second = 1ns = 10–9 s
1 kilo-metre = 1km = 103 m
PRACTICAL UNITS OF LENGTH
1.
Light year = 9.46 × 1015 m
2.
Parsec = 3.084 × 1016 m
3.
Fermi = 10–15 m
4.
Angstrom (A°) = 10–10 m
5.
Micron/Micrometer = 10–6 m
6.
Nano meter = 10–9 m
7.
Picometer = 10–12 m
8.
Acto meter = 10–18 m
9.
Astro nomical unit (A.U.) = 1.496 × 1011 m
10.
Otto meter = 10–21 m
SOME IMPORTANT PRACTICAL UNITS
S.No.
1.
Quantity
Mass
Unit
Solar mass = 2 × 1030Dalton = 1.66 × 10–27 kg
Chander Shekhar = 1.4 times of
mass of sun
2.
Pressure
Pascal = 1 N/m2
Bar = 105 N/m2
3.
Area
barn = 10–28 m2
4.
Radio Activity
Becquerel
5.
Radiation doze for cancer
Roentgen
6.
Time
Shake = 10–8 sec
DIMENSIONS IN MECHANICS
Quantities
Dimensional eqn.
Distance
Displacement
M0L1T0
Length/depth/thickness
Wavelength
Mass,
Inertia,
Intertial mass,
M1L0T0
Gravitational mass
Speed,
Velocity,
M0L1T–1
Velocity of sound
Velocity of light
Acc. (a)
M0L1T–2
Acc. due to gravity (g)
Angular velocity,
Velocity gradient,
Decay constant ( )
M0L0T–1
linear frequency
Activeness
Wave Number
Propagation constant (K)
M0L–1T0
Rydberg constant
Gravitational constant (G)
M–1L3T–2
Force,
M1L1T–2
Weight
Tension
centripetal force
Work (W)
Energy (E)
M1L2T–2
Torque ()
Moment of couple
Heat (H)
Linear Momentum (P)
Impulse
M1L1T–1
Surface Tension (T)
M1L0T–2
Pressure, (P)
Coefficient of Elasticity
Young Modulus (Y)
M1L–1T–2
Bulk Modulus (K)
Stress
Plank Constant, (h)
Angular momentum (L)
M1L2T–1
Viscous coefficient ()
M1L–1T–1
DIMENSIONS IN HEAT
Quantities
Dimensional eqn.
Temperature
M0 L0 T0 1
Latent heat
M0 L2 T–2 0
Specific heat
M0 L2 T–2 –1
Coefficient of thermal expansion
M0 L0 T0 –1
Coefficient of thermal conductivity
M1 L1 T–3 –1
Mechanical equivalent (J)
M0 L0 T0
Stephen constant ()
M1 L0 T–3 k–4
Wien's constant (b)
M0 L1 T0 1
Gas constant (R)
M1 L2 T–2 –1 µ–1
Boltz mann constant (K)
M1 L2 T–2 –1
Solar Constant (S)
Intensity of Radiation
M1 L0 T–3
Energy flux
Pointing vector
EIMENSIONS IN ELECTRICTY
Quantities
Dimensional eqn.
Charge (Q)
A1 T1
Current (I)
A1
Potential gradient
Electric field (E)
Intensity of Electric field
Potential difference
M1 L1 T–3 A–1
M1 L2 T–3 A–1
Potential (V)
Potential energy
Electromotive force
Electrical capacitance (C)
M–1 L–2 T4 A2
Electric permittivity of free space ( 0)
M–1 L–3 T4 A2
Resistance (R)
M1 L2 T–3 A–2
Reactance (X)
Impedance (Z)
Electrical conductance
M–1 L-2 T3 A2
Admittance
Susceptance
Electrical flux ( )
M1 L3 T–3 A–1
Specific Resistance (K)
M1 L3 T–3 A–2
DIMENSION OF MAGNETIC QUANTITIES
Quantities
Dimensional eqn.
Magnetic field
Magnetic induction
M1 L0 T–2 A–1
Permeability of magnet (µ)
M1 L1 T–2 A–2
Momentum of magnet (M)
Bohr magneton (µB)
M0 L2 T0 A1
Self inductance (L)
Mutual inductance (M)
M1 L2 T–3 A–2
DIMENSION LESS QUANTITIES
S.No.
Quantities
1.
Efficiency ()
2.
Coefficient of amplification (µ)
3.
Q-factor
4.
Form-Factor
5.
Power coefficient
6.
Relative Electric Permitivity
7.
Refractive index (µ)
8.
Mec. coefficient of heat (J)
9.
Poison ratio
10.
Strain
11.
Angular displacement
12.
Angle / Solid angle
NOTE:
Dimension less quantity may have unit. But unitless quantities are
dimensionless.
Ex. angle – dimensionless but it has unit radian.
THE PRINCIPLE OF HOMOGENEITY OF DIMENSION
The dimension of physical quantity on the left hand side of dimensional equation
should equal to the net dimensions of all physical quantities on the right hand
side of it.
Ex. If in the form x = 3yz 2, x and z represent electrical capacitances and
magnetic induction the calculate dimensional equation of y.
Sol. By the principal of homogeneity of dimension
Dimension equation of x = Dimension equation of (3yz2)
M–1 L–2 T4 A2
= Dimension equation of (y) × (M1 L0 T–2 A–1)2
Dimension of (y)
= M–3 L–2 T8 A4
Force = mass × acceleration
= mass ×
velocity
time
length
= mass × time
time
= mass × length × (time)–2
Thus, the dimensions of force are 1 in mass, I in length and –2 in time. The
dimensions in all other base quantities Disc Zero.
Important views related to Dimensions:
(i)
Pure number and Pure ratio are dimensionless.
Ex.: 1, 2, , ex, log x, sin , cos  etc. and refractive index.
(ii)
Dimensionless quantity may have unit
Ex.: Angle and solid angle.
(iii)
The method of dimensions can not be applied to desire the formula if a
physical quantity depends on more than three physical quantities.
(iv)
For a given physical quantity there will be only one dimensional formula
of energy but for a given dimensional formula there can be several
physical quantities.
Work
ML2 T–2
Energy
Torque
Ex# Calculate the dimensional formula from the equation E =
E = mass × (velocity)2
1
is a number and has no dimensions.
2
L 
[E] = M   
T
2
= ML2 T–2
Uses of Dimension:
(1)
To check the accuracy of various formula or equation.
1
mv2
2
Principle of dimensional homogeneity:
The dimensions of each and every term on L.H.S. of the equation must be same
as that of each and every term on R.H.S.
NOTE:
If a formula is physically correct, it has to be dimensionally correct but, if a
formula is dimensionally correct, it is not necessary that it is also physically
correct.
Ex# Check the validity of the equation
x  ut 
1 2
at
2
where x is the distance travelled by a particle in time t which starts at a speed u
and has on acceleration a along the direction of motion.
Sol.
L.H.S.
Dimension of x = L
R.H.S.
Dimension of ut = LT–1 × T = L
Dimension of
1 2
at = LT–1 × T2 = L
2
 Dimensions of L.H.S. = Dimensions of R.H.S.
 Formula is dimensionally correct.
(2)
To convert the value of a physical quantity from one system of units to
another system of units.
Magnitude of a physical quantity always remains constant it will not
change if we express it in some other unit
So, Q = nu = constant
n
1
u
where,
n = number
u = unit
If n  then u 
and n  then u 
i.e. 1 kg = 1000 gm
 kg is a bigger unit than gram.
Let number and unit of a physical quantity in 1st and 2nd system of units are n 1,
u1 and n1, u2 respectively and Dimension of a physical quantity in mass, length
and time are a, b and c respectively.
 Q = Ma Lb Tc
II System
I System
n1 u1 = n2 u2
n1 M1a Lb1 T1c   n2 M2a Lb2 T2c 
a
b
M  L   T 
n2  n1  1   1   1 
 M2   L2   T2 
c
Ex# The value of circurtational constant in MKS system is 6.67 ×
Convert this value into CaS units.
I SystemII System
n1 = 6.67 × 10–11
n2 = ?
M1  kgM2  gm
L1  mL2  gm
T1  secT2  sec
G = M–1 L3 T–2
G = –1, b = 3, c = –2
1
3
 kg   m   sec 
n2 = n1 
 
 

 gm   cm   sec 
1
 gm   m 
= n1 
 

 kg   cm 
11
= 6.67  10 
2
3
1
 100  100  100
1000
10–
Nm2
1
.
kg2
= 6.67 × 10–8 CGS units
(3)
To derive a new formula.
Let a physical quantity depends on the another quantities P, Q and R
then
E  Pa Qb Rc
E = K Pa Qb Rc
…(i)
Now consider dimensional formula of each quantity on both sides—
a
b
x
y
z
x
y
z
x
y
z
Mx Ly Tz = M 1 L 1 T 1  M 2 L 1 T 2  M 3 L 3 T 3 
c
ax bx cx
ay by cy
az az az
=M 1 2 3 L 1 2 3 T 1 2 3
Now comparing the powers on both sides
ax1 + bx2 + cx3 = x
…(2)
ay1 + by2 + cy3 = y
…(3)
az1 + bz2 + cz3 = z
…(4)
Let after solving eqn. (2), (3) and (4), we get value of a, b and c are m, n and o
respectively then formula is
E = KPm Qn Ro
Ex# The time period of a simple pendulum depends on its length (l) and
acceleration due to gravity (g). Deduce the formula.
T  lx gy
T = K lx gy
M0 L0 T1 = KLx [LT–2]y
= KLx+y T–2y
x+y=0
and – 2y = 1
y= 
1
2
x 
1
2
1
2

T  K l g
TK
1
2
l
g
 n2  n1 
 , D = diffusion coefficient, n1 and n2 is
x

x
 2
1
Ex. In the formula; N =  D 
number of molecules in unit volume along x1 and x2. Which represents
distances where N is number of molecules passing through per unit area per
unit time calculate dimensional equation of D.
Sol. By Homogeneity theory of Dimension
Dimension of (N)
= Dimension of D ×
Dimension of n2  n1 
Dimension of  x2  x1 
1
L3
=
Dimension
of
D
×
L2 T
L
 Dimension of 'D' =
=
L
L3  L2 T
L2
 L2 T 1
T
USES OF DIMENSIONAL EQUATIONS
Following are the uses of dimensional equations.
(i)
Conversion of one system of units in to another.
(ii)
Checking the accuracy of various formula of equation.
(iii)
Derivation of Formula.
CONVERSION OF ONE SYSTEM OF UNITS INTO ANOTHER:
Let the numerical values are n1 and n2 of a given quantity Q in two unit system
and the units are—
U1 = M1a Lb1 T1c
and U2 = M2a Lb2 T2c
in two systems respectively)
Therefore, By the principle nu = constant
 n2 u2 = n1 u1
n2 M2a Lb2 T2c   n1 M1a Lb1 T1c 
 n2 
n1 M1a Lb1 T1c 
M2a Lb2 T2c 
a
b
c
M  L   T 
 n2   1   1   1  n1
 M2   L2   T2 
PRINCIPLE OF HOMOGENEITY
The dimensions of both sides in an equation are same.
Ex. s = ut 
1 2
gt
2
[L] = [LT–1 . T] + [LT–2 . T2]
[L] = [L] + [L]
DEFECTS OF DIMENSIONAL ANALYSIS
1.
While deriving a formula the proportionality constant cannot be found.
2.
The formula for a physical quantity depending on more than three other
physical quantities cannot be derived. It can be checked only.
3.
The equations of the type v = u + at cannot be derived. They can be
checked only.
4.
The equations containing trigonometrical functions (sin , cos , etc.),
logarithmic functions (log x, log x 3 etc.) and exponential functions (ex,
2
ex etc.) cannot be derived. They can be checked only.
ORDER OF MAGNITUDE
In physics, we come across quantities which vary over a wide range. To express
such widely varying numbers, one uses the powers of ten method.
In this method, each number is expressed as a × 10 b where 1 < a < 10 and b is a
positive or negative integer.
Thus, the diameter of the sun is expressed as 1.39 × 109 m. To get an
approximate idea of the number, one may round the number a to 1 if it is less
than or equal to 3.16 and to 10 if it is greater than 3.16. The number can then be
expressed approximately as 10b. We then get the order of magnitude of that
number. Thus, the diameter of the sun is of the order of 10 9 m. More precisely,
the exponent of 10 in such a representation is called the order of magnitude of
that quantity. The order of magnitude of 10 9 is 9.
The Structure of the World:
Man has always been interested to find how the world is structured. After
extensive experimental work people arrived at the conclusion that the world is
made up of just three types of ultimate particles, the proton, the neutron and the
electron. All objects which we have around us, are aggregation of atoms and
molecules. The molecules are composed of atoms and the atoms have at their
heart a nuclear containing protons and neutrons. Electrons move around this
nucleus in special arrangements. It is the number of protons, neutrons and
electrons in an atom that decides all the properties and behavior of a material.
Ch.-1. Introduction to Physics:
What is Physics?
Mathematics applied to Physics.
Ch.-2. Units and Dimensions, Dimensional Analysis Units:
Definition of Base Units
Dimensions, uses of dimensions
Order of magnitude
The structure of the world.
Ch.-3. Measurements
Least count, significant figures.
Methods of measurement and error analysis for Physical quantities
pertaining to the following experiments.
Experiments based on using Vernier Callipers and screw gauge
(micrometer).
Determination of g using simple pendulum Young modulus by scarles method
specific heat of a liquid using Celorimeter focal length of a Concave mirror and a
convex lens using uv method speed of sound using resonance column verification
of ohm's law using voltmeter and ammeter specific resistance of the material of
the wire using bridge and P.O. box.
Work and power:
Lifting m/c.
Significant figures:
Measurements made by any instrument are not absolutely correct. The degree of
accuracy or precision is shown by the significant figures upto which the
measurement has been recorded.
Let us say, the length of an object is 14.5 cm. It shows that the measurement has
been made to the nearest of
1
th of a centimetre which shows that figures 1
10
to 4 are absolutely correct and figure 5 is reasonably correct.
If the length recorded is 14.52 cm, then it shows that the measurement has been
made correctly upto
1
th of a centimeter. In this case, the figures 1, 4 and 5
100
are absolutely correct while the figure 2 is approximate.
Thus, significant figures are the number of digits upto which we are sure about
their accuracy. In other words, significant figures are those digits in a number
that are known with certainty plus one more digit that is uncertain.
For example, 14.5 cm has three significant digits and the measurement 14.52 cm
has four significant digits. Significant figures do not change if we measure a
physical quantity in different units.
For example, 14.5 cm = 0.145 m
= 14.5 × 10–2 m
Now 14.5 cm and 14.5 × 10–2 m both have three significant figures.
Rules for significant figures:
(1)
All non-zero digits are significant figures.
Example :
(2)
Number
Significant figures
17
2
178
3
1782
4
17825
5
All zeros occurring between non-zero digits are significant figures.
Example :
Number
Significant figures
401
3
4012
4
(3)
40056
5
400006
6
All zeros to the right of the last non-zero digit are not significant figures:
Example :
(4)
Number
Significant figures
20
1
210
2
2130
3
20350
4
All zeros to the right of a decimal point and to the left of a non-zero digit
are not significant figures:
Example :
Number
Significant figures
0.04
1
0.004
1
0.0045
2
0.0456
3
0.0004564
4
(5)
All zeros to the right of a decimal point and to the right of a non-zero
digit are significant figures:
Example :
Number
Significant figures
0.20
2
0.230
3
0.2370
4
Rounding off the measurements:
The following rules are applied in order to rounding off the measurements:
(i)
If the digit to be dropped in a number is less than 5, then the preceding
digit remains unchanged. For example, the number 8.64 is rounded off
to 8.6.
(ii)
If the digit to be dropped in a number is greater than 5, then the
preceding digit is raised by 1. For example, the number 8.66 is rounded
off to 8.7.
(iii)
If the digit to be dropped in a number is 5 or 5 followed by zeros, then
the preceding digit remains unchanged if it is even.
For example,
(i) the number 8.65 is rounded off to 8.6,
(ii) the number 8.650 is rounded off to 8.6.
(iv)
If the digit to be dropped in a number is 5 or 5 followed by zeros, then
the preceding digit is raised by I if it is odd.
For example,
(i) the number 8.75 is rounded off to 8.8,
(ii) the number 8.750 is rounded off to 8.8.
SIGNIFICIENT FIGURES
The numbers of figure required to specify a certain measurement perfectly are
called significant figure.
The last figure of a measurement is always doubtful, but is included in the
number of significant figure.
Example: If length of pencil measured by vernier callipers is 9.48 cm, the number
of significant fig. in the measurement is 3.
RULES FOR SIGNIFICANT FIGRUES
(i)
If a measurement contains no decimal point, the number of final zeros
are ambiguous and are not counted in significant fig. i.e. all non zero
digits are significant.
e.g. — In 3320 no. of significant figures = 3
(ii)
The power of 10 and the zeros on left hand side of a measurement are
not counted while counting the number of significant fig.
e.g. — 5 × 103 significant fig. 1
(iii)
the zeros after a decimal are counted as to significant fig.,
e.g. — 1.60 has three significant fig.
(iv)
The zeros appearing in between the non zero digits are counted as
significant figures,
e.g. — In 2.07, there are three significant figures.
(v)
The zeros appearing to the left of a non zero digit are not counted in
significant figures,
e.g. — 0.0702 has only three significant figure (702)
(vi)
When the position of decimal point changes, then the number of
significant figures does not change,
i.e. — 1.942, 194.2 all have four significant figures.
(vii)
The limit and accuracy of a measuring instrument is equal to the least
count of the instrument.
(viii)
In the sum and difference of measurements, the result contains the
minimum number of decimal places in the component measurements.
Ex. The length of string of simple pendulum is 101.4 cm and diameter of bob is
2.64 cm. What is th effective length of simple pendulum up to required
significant figures.
Sol.

0
r
Here
r=
0
 101.4 cm,
2.64
 1.32 cm
2
 101.4  1.32  102.72 cm
Since we take least number of decimal figures in a measurement which is 1 in
0
Hence Effective length = 102.7 cm.
(ix)
In the product and quotient of measurements, the result contains the
minimum
number
of
significant
figures
in
the
component
measurements.
Ex. The length, breadth and thickness of a block are given by
= 12 cm, b = 6
cm, t = 2.45 cm.
What is the volume of the block according to the idea of significant figures.
Sol. Volume = blt
= 6 × 12 × 2.45 = 176.4
= 1.764 × 104 cm3
The minimum number of significant figures is 1 in thickness.
Vernier Callipers and Screw Gauge:
The metre scale which commonly used in practice is the simplest instrument for
measuring length.
By metre scale we can measure upto 1 mm because the length of the smallest
division made on the scale is 1 mm. In order to measure still smaller lengths
 1
 1 
th or 
th of a millimeter, the instruments

 100 
10 
accurately upto 

commonly used in laboratory are:
1.
Vernier Callipers
2.
Screw Gauge
Vernier Callipers:
 1
th of millimeter. Vernier Callipers
10 
It is used to measure accurately upto 

comprises of two scales, Wz, main scale S and vernier scale V which is called
auxiliary scale. The main scale is fixed but the vernier scale is movable. The
divisions of vernier scale are usually a little smaller in size than the smallest
division on the main scale. It also has two jaws, one attached with the main scale
and the other with the vernier scale. The purpose of jaws are to grip the object
between them. Vernier has a strip, which slides along with vernier scale, over the
main scale. The strip is used to measure the depth of hollow object.
Vernier Constant (VC):
Suppose the size of one main scale division is S and that of one vernier scale
division is V units. Also suppose that length of n vernier division is equal to the
length of (n – 1) division of main scale. Thus, we have
(n – 1)S = nV
or nS – S = nV
or S – V =
S
n
The quantity (S – V) is called vernier constant (VC).
Least Count:
The smallest value of a physical quantity which can be measured accurately with
an instrument is called the least count (L.C.) of the instrument.
For vernier calipers, its least count is equal to its venier constant. Thus
Least count = S – V =
S
n
wherer, S = size of one main scale division
V = size of one vernier scale division
n = No. of division on vernier scale
=
Length of one division of main scale
No. of divisions on vernier scale
Length of the object = main scale reading + n (LC)
n = vernier division exactly coinciding with some main scale division.
Determination of zero error:
When jaws of the vernier are made touch other and the zero mark of the vernier
scale coincide with the zero mark of the main scale, there will no zero error in the
instrument. However, in practice it is never so. Due to wear and tear of the jaws
and due to some manufacturing defect, the zero mark of the main scale and
vernier scale may not coincide, it gives rise to an error, is called zero error. It may
be positive or negative zero error.
Positive and negative zero error:
When the zero mark of the vernier scale lies towards the right side of the zero of
the main scale when the jaws are in contact, the measured length will be greater
than the actual length. Because of this fact the zero error is called positive zero
error. On the other hand, when zero mark of the vernier scale lies towards the
left side of the zero of the main scale when jaws in contact with each other, the
length of the object measured by the instrument will be less than the actual
length of the object. Because of this reason is called negative zero error.
True reading = Observed reading – Zero error with proper sign.
Correction for positive zero error:
When its jaws are in contact with each other, suppose 3rd
vernier division coincides with the any of the divisions of main scale. They we
have
Zero error = + [0.00 cm + 3(L.C.)]
= + [0.00 + 3 × 0.01 cm]
= + 0.03 cm
Correct reading = Observed reading – (0.03 cm)
Figure for negative error:
Screw Gauge: It is used to measure small lengths like diameter of a wire or
thickness of sheet etc. It consists of a U' shaped metal frame as shown in fig.
A main scale which graduate in millimeter or half a millimeter. The main scale
also called pitch scale.
Pitch: It is defined as the linear distance moved by the screw forward or
backward when one complete rotation is given to the circular cap.
Least count (L.C.)
=
Pitch
 Total number of divisions on the circula r scale 
Ex. In Four complete revolution of the cap, the distance travelled on the pitch
scale is 2 mm. If there are 50 divisions on the circular scale, then calculate the
least count of the screw gauge.
Pitch =
L.C. =
2mm
 0.5 mm
4
0.05
mm  0.01 mm
50
Zero error:
When the studs P and Q of the screw gauge are brought in contact without apply
induce pressure and if the zero of the circular scale coincides with the reference
line, then there is no zero error, otherwise there will be zero error.
Positive zero error:
In this case, the zero of the circular scale lies below the reference line as the gap
between studs P and Q reduces to zero.
Suppose the zero line of the circular scale is 4 division below the reference line. In
other words, the 4th division of the head scale is in line with the line of
graduation.
Zero error = + 4 (L.C.)
= + 4 (0.01 cm)
= + 0.04 cm
Zero correction = – zero error
Negative zero error:
When zero of the circular scale lies above the reference line when the gap
between the studs P and Q become zero.
Zero error = – 3 × 0.01 mm
= – 0.03 mm
Zero correction = + 0.03 mm.