Crackiitjee.in.Phy.ch14

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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

in

elephant is 150 times the unit mass and the cow is 15 times, we know that the

e.

elephant is 10 times heavier than the cow. If I have the knowledge of the unit

iit je

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

cr ac k

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.

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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

in

English has been given the authority to decide the units by international

e.

agreement. It holds its meetings and any changes in standard units are

cr ac k

Physical quantities:

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communicated through the publications of the conference.

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


2


Physical quantities  







Fundamental or

Derived

Supplementary

Basic

Quantities

Quantities

in

Quantities

e.

Properties of Units: 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,

cr ac k

iit je

(a)

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


3


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

in

systems are in use over the world and in each system the fundamental quantities

e.

are selected in a particular way.

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The units defined for the fundamental quantities are called fundamental units and those obtained for the derived quantities are called the derived units.

SI units:

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Fundamental quantities are also known as base quantities.

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


4


Kilogram

kg

Time

Second

s

Electric current

Ampere

A

Temperature

Kelvin

K

Amount of substance

Mole

mol

Luminous intensity

Candela

cd

in

Mass

e.

Besides the seven fundamental units two supplementary units are defined. They

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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.

cr ac k

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


5

giga

G

6

mega

M

3

kilo

K

2

hecto

h

1

deka

da

–1

deci

d

–2

centi

c

–3

milli

–6

micro

µ

–9

nano

n

pico

p

e.

cr ac k

–12

Dimension:

in

9

iit je


m

–15

femto

f

–18

atto

a

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.


6


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

Unit of length = Meter

(Travelled distance by light

e.

2.

in

and measurements, paris)

3.

4.

cr ac k

iit je

in vacuum in 1/299, 792,

Unit of Time = Sec.

Unit of Temperature = Kelvin

458 second or it is equal to 1650763.73 wave length emitting from Kr86) (The time interval in which Cesium-133 atom vibrates 9,192,631,770 times) (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


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) Unit of luminious

(Amount of

Intensity = Candela

intensity on 1/60000 m 2

in

6.

e.

area at freezing point of

7.

Unit of quantity of

cr ac k

Substance = mole

iit je

platinum

2042K

at

pressure of 101325 N/m 2.) (It is the amount of 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.

8


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

Amount

in

Intensity

e.

of

cr ac k

iit je

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


9


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

iit je

e.

in

giga

cr ac k

Ex. :

4.

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


10


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

1.

Quantity

iit je

S.No.

e.

SOME IMPORTANT PRACTICAL UNITS

in

3.

Unit

Solar mass = 2 × 1030-

Mass

2.

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Dalton = 1.66 × 10–27 kg

Pressure

Chander Shekhar = 1.4 times of mass of sun 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


11


Quantities

Dimensional eqn.

Distance M0L1T0

Displacement Length/depth/thickness Wavelength Mass, Inertia,

M1L0T0

Intertial mass,

in

Gravitational mass

e.

Speed,

M0L1T–1

iit je

Velocity, Velocity of sound

Acc. (a)

cr ac k

Velocity of light

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


12


Force, M1L1T–2

Weight Tension centripetal force Work (W) Energy (E)

M1L2T–2

Torque () Moment of couple

in

Heat (H)

e.

Linear Momentum (P)

M1L1T–1

Surface Tension (T) Pressure, (P)

cr ac k

Coefficient of Elasticity

iit je

Impulse

Young Modulus (Y)

M1L0T–2

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.


13


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

in

Temperature

M1 L2 T–2 –1 µ–1

e.

Gas constant (R)

M1 L2 T–2 –1

Solar Constant (S) Intensity of Radiation

iit je

Boltz mann constant (K)

M1 L0 T–3

cr ac k

Energy flux Pointing vector

EIMENSIONS IN ELECTRICTY

Quantities

Dimensional eqn.

Charge (Q)

A1 T1

Current (I)

A1

Potential gradient M1 L1 T–3 A–1

Electric field (E) Intensity of Electric field Potential difference


14


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)

in

Impedance (Z) Electrical conductance

M–1 L-2 T3 A2

e.

Admittance

Electrical flux ( )

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Specific Resistance (K)

iit je

Susceptance

M1 L3 T–3 A–1 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


15


DIMENSION LESS QUANTITIES

S.No. 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

cr ac k

iit je

e.

in

1.

12.

NOTE:

Quantities

Angle / Solid angle

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


16


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)

in

M–1 L–2 T4 A2

e.

= Dimension equation of (y) × (M1 L0 T–2 A–1)2

= M–3 L–2 T8 A4

iit je

Dimension of (y)

cr ac k

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.


17


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

Work ML2 T–2

iit je

Energy

e.

physical quantities.

in

of energy but for a given dimensional formula there can be several

cr ac k

Torque

Ex# Calculate the dimensional formula from the equation E =

1 mv2 2

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.


18


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

in

correct.

1 2 at 2

iit je

x  ut 

e.

Ex# Check the validity of the equation

where x is the distance travelled by a particle in time t which starts at a speed u

Sol.

cr ac k

and has on acceleration a along the direction of motion. 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.


19


(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

in

where,

e.

n = number

If n  then u 

cr ac k

and n  then u 

iit je

u = unit

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


20


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 ×

Nm2 . 1 kg2

in

Convert this value into CaS units.

10–

e.

I SystemII System n1 = 6.67 × 10–11

iit je

n2 = ?

M1  kgM2  gm

cr ac k

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


21


= 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)

in

Now consider dimensional formula of each quantity on both sides— a

b

c

e.

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 

iit je

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

cr ac k

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


22


T = K lx gy M0 L0 T1 = KLx [LT–2]y = KLx+y T–2y x+y=0 and – 2y = 1

1 2 1 2

.in

x 

1



l g

ck i

TK

1 2

itj

 T  K l2 g

ee

y= 

cr a

 n2  n1   , D = diffusion coefficient, n1 and n2 is  x2  x1 

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)


23


= 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  L2 T 3

iit je

USES OF DIMENSIONAL EQUATIONS

e.

in

L2 =  L2 T 1 T

Following are the uses of dimensional equations.

Conversion of one system of units in to another.

(ii)

Checking the accuracy of various formula of equation.

(iii)

Derivation of Formula.

cr ac k

(i)

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)


24


Therefore, By the principle nu = constant  n2 u2 = n1 u1

n2 M2a Lb2 T2c   n1 M1a Lb1 T1c 

M2a Lb2 T2c  a

b

c

ee

M  L   T   n2   1   1   1  n1  M2   L2   T2 

.in

 n2 

n1 M1a Lb1 T1c 

PRINCIPLE OF HOMOGENEITY

1 2 gt 2

ck i

Ex. s = ut 

itj

The dimensions of both sides in an equation are same.

cr a

[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.


25


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

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.

iit je

positive or negative integer.

e.

In this method, each number is expressed as a × 10 b where 1 < a < 10 and b is a

Thus, the diameter of the sun is expressed as 1.39 × 109 m. To get an

cr ac k

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


26


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.

What is Physics?

iit je

Mathematics applied to Physics.

e.

in

Ch.-1. Introduction to Physics:

Ch.-2. Units and Dimensions, Dimensional Analysis Units:

cr ac k

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).


27


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:

in

Lifting m/c.

e.

Significant figures:

iit je

Measurements made by any instrument are not absolutely correct. The degree of accuracy or precision is shown by the significant figures upto which the

cr ac k

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


28


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

in

Now 14.5 cm and 14.5 × 10–2 m both have three significant figures.

All non-zero digits are significant figures. Example :

cr ac k

Number

iit je

(1)

e.

Rules for significant figures:

(2)

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


29


(3)

40056

5

400006

6

All zeros to the right of the last non-zero digit are not significant figures: Example :

Significant figures

20

1

210

2

.in

Number

3

ee

2130

ck i

All zeros to the right of a decimal point and to the left of a non-zero digit are not significant figures: Example :

cr a

(4)

4

itj

20350

Number

Significant figures

0.04

1

0.004

1

0.0045

2

0.0456

3

0.0004564

4


30


(5)

All zeros to the right of a decimal point and to the right of a non-zero digit are significant figures: Example :

Significant figures

0.20

2

0.230

3

0.2370

4

e.

Rounding off the measurements:

in

Number

(i)

iit je

The following rules are applied in order to rounding off the measurements: If the digit to be dropped in a number is less than 5, then the preceding

cr ac k

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.


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(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.

in

SIGNIFICIENT FIGURES

e.

The numbers of figure required to specify a certain measurement perfectly are

iit je

called significant figure.

The last figure of a measurement is always doubtful, but is included in the number of significant figure.

cr ac k

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.


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e.g. — 5 × 103 significant fig. 1 the zeros after a decimal are counted as to significant fig.,

(iii)

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

in

significant figures,

e.

e.g. — 0.0702 has only three significant figure (702)

When the position of decimal point changes, then the number of

iit je

(vi)

significant figures does not change,

cr ac k

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


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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.

minimum

number

of

significant

in

In the product and quotient of measurements, the result contains the figures

in

the

e.

(ix)

measurements.

cm, t = 2.45 cm.

iit je

Ex. The length, breadth and thickness of a block are given by

component

= 12 cm, b = 6

cr ac k

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.


34


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  100  th of a millimeter, the instruments 10 

accurately upto  

commonly used in laboratory are: Vernier Callipers

2.

Screw Gauge

e.

in

1.

Vernier Callipers:

iit je

 1 th of millimeter. Vernier Callipers 10 

It is used to measure accurately upto  

cr ac k

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):


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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

S n

in

or S – V =

e.

The quantity (S – V) is called vernier constant (VC). Least Count:

iit je

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.

cr ac k

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.


36


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

e.

Positive and negative zero error:

in

be positive or negative zero error.

iit je

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

cr ac k

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


37


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

in

thickness of sheet etc. It consists of a U' shaped metal frame as shown in fig.

e.

A main scale which graduate in millimeter or half a millimeter. The main scale

iit je

also called pitch scale.

Pitch: It is defined as the linear distance moved by the screw forward or

cr ac k

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


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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

in

between studs P and Q reduces to zero.

e.

Suppose the zero line of the circular scale is 4 division below the reference line. In

graduation. Zero error = + 4 (L.C.)

cr ac k

= + 4 (0.01 cm)

iit je

other words, the 4th division of the head scale is in line with the line of

= + 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.


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Crackiitjee.in.Phy.ch14

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