llainKiWeb.com I

MEASUREMENTS Accurate measurement is central to the development of any science. The

importance of measurement was apparent to ancient ci\ ilizations. Throughout history, the accuracy with which measurements could be made has been improved by the use of more and more sophisticated instruments. At almost every stage, improved measuring teclmiques have resulted in new concepts and ideas.

Q.J.l Descrioe the Importance of physics in daily life.

Ans:

Until about 1850, there are texts and courses in what was called natural or experimental philosophy. As a result and conclusion of natural philosophy accumulated, it became difficult for a single person to work in t11e whole fields and subdivisions appeared.

There was a huge increase in the volume of scientific k.Jlowledge up till the beginning of nineteenth centi~U"IIDIDi...necessa.!I._tO classify the study of nature into two branches, the biol&~I'M ~fe\tA...a.G~fOt living things and physical sciences which concern with non-living things. Phy~ics is an important and basic part of physical sciences besides its other disciplines such as chemistry, astronomy, geology etc. ·

Physics studies particularly simple systems, such as sing!: atoms. Scientific ~ methods are often expressed more clearly in these simple systems of ·physics than in ~1any other sciences. Because of this, physics is often regarded as a model for the

· ~tmtific Method". ~ . -.,"k Althcuf~h Physics is a fundamental science, its principles underliij.Ci, ~clr. of tech~gy, which is humankind's means of solving its probl_e~:~erthereforc, importat'Q".gr technologists and for engineers, who actually design rns to problems., to have a tlf'tO)aghP,ut understanding of the principles of physics'*~

(Chapter 01) MEASUREMENT 2

Q.l.2 (a) Define Physi~s. Describe the main frontiers of fundamental sciences.

(b) De...!g;.t>e some new branches of Physics anPcDcJJe. of Physics in ~oping technology. "'")'fj

Ans: • .,.'f>i.HYSICS: ~6 Ddh,JOn: ·c, ~'1>~.. 'rhe branch of science which deals with the study of mal/er and O~

energy and the relationship between them is called physics.) . ·

MAIN FRONTIERS OF FUNDAMENTAL SCIENCES:

1. The world of the extremely large, the universe itself, Radio telescopes now gather information from the far side of the universe and have recently detected, as radio waves. the "firelight" oj1t!l..e~" ft:bil'\started off the expanding universe nearly 20 billioUj:Mr~ '~~.1 ,.. :- _. •

2. The world of the extremely small, that of the particles such as, electrons, protons, neutrons, mesons and others.

3. / The world of complex matter and it is also the w~d of "middle-sized" things, from molecules at one extreme to the Earth at the other. This is all fundamental physics, which is the heart of science.

(b) BRANCHES OF PHYSICS:

By the end of 191h century, many physicists started believing that every thing

about physics has been discovered. However, about the beginning of the twentieth century, many new experimentai fdcts revealed that the laws formulated by the previous investigawrs need modificati0ns. Further researches gav~ birth to many new disciplines in physics which are given as under:

Nuclear Physics:

The branch physics which deals with atomic nuclei is called nuclear physic!> Particle Physics:

The branch of physics which is concerned with the ultimate particles of which the matter is composed of is called P3JttWnf(ciWeb.COm Relativistic Mechanics:

The branch of physics which deals with velocities approaching that of light is called relativistic mechanics.

Solid State Physics: ~

~c:L._ The br mch ot physics which is concerned with the structure and properties tfl solu:..-..,~;.ials ;s called solid state physics. . ~l>· . Role of~c~. in technology: ~

Physf6 pl. vs an important role in the development of ~ology and engineering. S~~~ and technology are a potent force for change ~ the outlook of

~

3 A Hand Book of Physics Part- I

mankind. The i9{J~ation media and fast means of communicat~s have brought all ?arts o~ the~~d in close contact with one another. Events in ofip~art of the wort~ Imm~~~'reverberate round the globe. . r'J~ ~~e are living in the age of information technology. The computef~works are

~aucts of chips, developed t1·om the basic ideas of physics. The chips a~ade of silicon. Silicon can be obtained from sand. It is upto us whether we make a sand~e or a computer out of it. ~

Q.1.3 What do you mean by physical quantities? What steps are involved to measure a base quantity?

Ans. PHYSICAL QUANTITIES:

Definition: , llamKiWeb.com I Any numbei' or set of numbers used for a quantitative description of

a pHysical phenomenon is called a physical quantity. }

Physical quantity is a tenn which is used to include lneasurable features of many different items. For example, area of a playing field, the mas.; of a bag of sugar and the speed of an aeroplane are all physical quantities.(In quoting any measurement of a physical quantity two things are to be stated:

(I) Numerical value of the quantity.

(2) Suitable unit.)

Types of Physical Quantities:

Physical q~antities are often divided into two catt!g,,r!t "·

( 1) Base quantities

(2) Derived quantities

1. Base Quantities:

( Base quantities are not defined in terms of other r·hysical quantities.

Distance (m)

10-20

10- 10

) Dtameter of a nucleu1

Diameter of an atom

) Typical examples of IJa mKtW•co..,and 100 =of a

time For other areas of physi-:c; other base quantities are; ~ Otameter ~ of the earth

temperature, electrical charge, luminous intensity and amount of 1o•o Distance to the

b wn su stance. Di•tance to the

nearest 1tar ~ Step~ to Measure Base Quantity: 1020 Diameter 01 the

'lh. ~The measurement of~. base quantity involves two step~ : Milky way .,...,"' ~ I Dtst~ the ~~ (i) The choice of a standard. 1():10 f '"f new-galaxy

. ~i) The establishment of a procedure for comparing ::t\e~ .. •c the quantity to be measured with the standard so -----.+..~

OJ,llat a number and a unit are determined as the 1Jl'-'!!'er ofma~n itude -~easure of that quantity. of ~ome di..,;tan..:L

[Chapter 01) MEASUREMENT 4

_2. • Derived ~ntities: ~

(Deriv~Qiuantities are those whose definitions are basef~ther physical qu~~~~r example; velocity, acceleration and force etc. are usually~~ as derived

qu~~:· . ~6 ~cipa Characteristics of an Ideal Standard: +c

N An ideal standard has two principal-characteristics: 0-'Q

(i) It is accessible

(ii) It is invariable

These two requirements are often incompatible and a compromise has to be made between them. 1

Thus physics is inherenllamKiwab~gmrd Kelvin (1824 - 1907), one of the pioneers in investigating energy relations in heat and thermal phenomena, stated this principle eloquently:

j:. .

I oft~n say that when you can measure what you are speaking about, and express it in numbers. you kno'vv something about it. But when yol! can not express it in numbers. Yours kn0" l~dg~ is of a meager and unsatisfactory kind; it may be the beginning of knowledgl!, but you have scarcely, in your thoughts, advanced to the stage of science, whatever the matter may be.

Q.l.4 (a) ·what is International system of units?

(b) What are base units? Define the base units of SI system.

(c) What are supplementary units and derived units? Give some examples.

Ans. (n) INTERNATIONAL SYSTEM OF UNITS:

( In 1960. an international committee agreed on a set of definitions and standard to descnbe the physical quantities. The system that was established is called the System International, abbrivated by SI ) The advantage of the SI system of units is that any quantitY. has only one unit in wlftch it can be measured. For example, in case of length the metre is the only unit or length useJ, together with multiple units such as the kilometre, anJ submultipll! unih 'iul·r 11~ thtlijrtd(tWeb.COm

The syskm intcrnati0nal (~ l) is built up from three kinds of units: i) Base Units ii) Supplementary Units ii~) Derievcd Units.

~ BASE UNITS· · 0~ D~on: . ~!J

.I~ The t! 1 .uled with bast! quantltif!l' like length, mass an.d~

a/~ led th~ Base Units. ~~ There •P&.1even base units for various physical quantities n~1ty, length, _mus,

temperature¥,.ectric current, luminous intensitv and amount nf' cuhct'Q .. ,.,.

5 A Hand Book of Physics Part-I

The names of base units given below in tabe~.

for these physical quantities along with symbols are

:oP .~...le ~, ... .

,,9-~

Table 1.1

Mass

Time

Electric current

Thermodynamics temp.

Luminous llaRJ Ki Amount of substance

Sl Unit

metre

kilogram

second

ampere

kelvin

e~eo mole

Standard definitions of base units are given as under:

Metre:

m

kg

s

A

K

cd

mol.

The unit of length is named as metre. In 1889, a metre bar of platinum-iridium alloy was choosen as the standard of length; this alloy ~s particularly chemically stable. However, the use of such a bar as a world standard is cumbersome; replicas must be made and compared with the world standard periodically. On October 14, 1960, the General Conference on Weights and Measures in France changed the standard of length to an atomic constant, namely, the wavelength of the orange-red light emitted by the individual atoms of Krypton- 86 in a tube filled with Krypton gas in which an electrical discharge is maintained. According to this; "One metre is a length equal to 1, 650, 763,73 times the wavelength in vacuum of the orange-red_ light emitted by Krypton 86-atom.

However, in 1983 the metre was redefined to be the distance traveled by light in vacuum during a time of 1/299, 792, 458 seconds. in fact, this latest definition establishes that the speed'oflight in vacuum is 299,792, 458 ms-1

•

Kilogram: 11 " The unit of mass 'i ClfillKf.Web(;)COJ"ft defined to be the mass of a

particular platinum (90 %) and iridium ( 1 0%) alloy cyrinder, 3.9 em in diameter and 3.9 em in height kept near Pairs, France.

~ Se<:ond: . ~ The unit of time is named as second. Until 1960, the standard of time was base< ~~n the mean solar day, the time interval between successive arrivals of the sun at it

~!lest point, average over a year. In 1967, an atomic standard was adopted. JN tw f~1_t energy states of the cesium (Cs133) atom · have slightly differef,~ergie: de~~ng on whether the spin of the outermost electron is parallel or Arallel to tt nucle in. Electromagnetic radiation (microwaves) of precise! roper frequenc causes t itions from one of these states to the other. ~~

1 •

(Chapter 01] MEASUREMENT 6

One second is redefined as the time required in which the outermost.electron of the cesium- 133 atom makes 9, 192, 631, 770 vibrations. ~

Ampere: 0~ ~*'>~t;. .. ~ T ~f'of electric current is named as ampere. It is that constant ~,.t which if

main · .%-two straight parallel conductors of infinite length, of negligible ctr~ cross-sect' d placed a metre apart in vacuum, would produce between these conductor~ ctsrce ~ to 2 x 1 0-7 newton per metre of length. This unit was established in 1971. ~

Kelvin:

The unit of thermodynamic temperature is kelvin. It is defined as "the fraction 11273.16 of the thermodynamic temperature of the triple-point of water.

It should be noted that the triple-point of a substance means the temperature at which solid, liquid and vapour At~~~eom triple-point of water is taken as 273.16 K. This standard l.f9!d~M i'b, 90/. • Mole:

The mole is the amount of substance of a system. It is defined as "the amount of a substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon -12.

It was adopted in 1971 . One mole of any substance contains 6.0225x 1023 entities.

Candela:

The unit of luminous intensity is candela. It is defined as the luminous intensity in the perpendicular direction of a surface of l/600000 square metre of a black body radiator at the solidification temperature of platinum under standard atmospheric pressure. This defu\ition was adopted by the 13th General Conference of Weights and Measures in 1967.

Difftrent System of Units:

The commonly used system of units are given as under:

1. F.P.S. system i.e., Foot-Pound-Second. This system is also called as British Engineering System.

In this system of units len~~...is..mt'Ulft.UetJ~ pounJs and time in second. l an I I' I • •

2. C.G.S. system i.e., Centimeter-Gram-Second.

In this system length is measured in centimetre, m_ass in gram and time in second.

3. M.K.S. system i.e., Metre-Kilogram-Second. ~

~ In this system length is measured in metre, mass in kilogram and time in second. (JO

(c) C,~PLEMENTARY UNITS: .:t~J>.~ !"l1V~eral Conference on Weights and Measures has not yet classi~ertain

units of the 1f6under either base units or derived units. These Sl urj,.~e called supplementary u~Q ~

/ ~

7 A Hand Book of Physics Part-1

Kind of Supplementary Units:

There are two kinds of supplementary units.

(i) Radian·

(ii) Steradian llamKiWeb.com

(i) Radian:

The radian is the plane angle between two radii of a circle which cut off on the circumference an arc, equal in length to the radius. It is shown in Fig. 1.1 (a).

(ii) Steradian:

The steradian is the solid angle (three­dimensional angle) subtended at the centre of a sphere by an area of its surface equal to the square of radius of the sphere. It is shown in Fig.l.l (b).

Derived Units: llamKIWeb. coR,

Fig. l.l(a)

\ .

Fig. l.t(b)

The Sl units derived from the base and supplementary units are called derived units. ) ·

~ In SI system, the seven pL:· :.teal quantities mentioned previously are regarded as ' ~ ase quantities and their units an.: called base units. Other physical quantities, which can

fined in terms of these base quantities such as velocity, acceleration, force a~rk etc. , therefore, called derived quantities and the units associated with th~ ~anti ties are c derived units. .~e Note: ~ are obtained by the combination of more than one bas~Y

~ ~11-

[Chapter 011 MEASUREMENT

Derived Unit

newton N

joule J

watt w coulomb · c

Electric potential difference volt v Pressure pascal Pa

Examples of Derived Units:

r;; speed: llamKiWeb.com It is defined as "distance traveled in unit time". In Sl units, distance i:: measured

in metre and time in second. So the unit of speed will be:

Speed Distance

= Time =

Unit of Speed = m = · ms-1 s

Thus,

The unit of speed is ms-1•

(ii) Acceleration:

Metre Second

l

It is defined as '·the rate of change of velocity". So the unit of acceleration will be;

Velocity = Distance/Time Acceleration = Time Time

Distance = (Time)2

Unit of acceleratiolla~'We b. com Thus,

The SI unit of acceleration is ms-2.

(iii) Force:

. ,,

~ It is defined as ''mass times acceleration". So the unit of force will be;

~h. . Force= Mass x Acceleration ;,-,~ k -2

"'1/J... · = g x ms

~6 Unit of force= kg ms-2

Thus, ·c Tht! SI un~t'dj force is newton (N) ( ·: · Kg ms-2 = N)

·-

9 A Hand Pook of Physics Part- 1

(iv) Work: 0(<\ A It _i!.\~'t! as .. the product of force and distance''. So the unit ~~ork wi ll be;

~...,.,, Work= Force x Distance ~.t. .

\\~~ = N x m ..-~6 Unit of work = Nm •(!!

(v)

be;

'1

l11us, ()~ The Sl unit ofwork is joule (J) .

JlJouw,rtum: ·

(': l Nm = 1 Joule)

It is detined as .. the pro~t of mass and velocity". So the unit of moml.!ntum wi ll · namK·Web.co,m rf

Momentum= Mass x Velocity

= kg x ms- 1

Unit of momentum= Kilogram metre per second (kg ms-1)

Thus,

The Sl unit of momentum is kg ms-1•

Q.l.S (a) What is scientific notation and explain the use of prefix?

(b) Write a note on conversion of unit.

Ans. (a) SCIENTIFIC NOTATION:

Numbers are expressed in standard form called scientific notation, which employs powers of ten. The 'internationally accepted practice is that there should be only one non zero digit kft of decimal. Thus, the! number 134.7 should be -wTitten as 1.347 x 102 and 0.0023 should be expressed as 2.3 x 10-3

.

Use of Prefix:

In physics, sometimes_ we use v;iW~e numbers and sometimes. very small numbers. Prefixes are used toll&mK b!eo-trfmbers as mu!uplcs of ten.· For example;

One light year = 9460000000000000

= 946 x 10 13 m

~ Similarly, = 9.46 x 1015 m '

~L Radius of proton= 0.000.000,000,000,000,001 ,2 m .A'\. ~~ ~· ~ = 12x I0- 16 m ~ :0~

· 8~ = 1.2 x w - ts m ~~0

Another if!G~ Prefix: ~?).~ co --~•: ......... "'"" •t<:P nrP.fix to exoress appropriate units. For example;

[Chilpter 01] MEASUREMENT 10

Centi means I I 1 00, therefore centimetre means 1/1 00 metre, or; ~ 1 ~ vO 100 m = 1 centimetre 4?">).

:o· . ~· .~e I m = 102 centimetres ;~ ~ . . _I_ _ fl)6 '!$" Sm1llarly, 1000 m - I mm •c

~~ oh.. or 1m = 103 mm · ~,

1 and 1 OOO km = I metre

or I km = I03 metre Hence centi, kilo, milli arc called prefix.

Table 1.4

Conventions for Indicating ulla K·Web.com Use of SI units requires special care, more

particularly in writing prefixes. Some Prefaxcs for Following points should be kept in mind while

using units.

(i) Full name of the unit does not begin with a capital letter even if named after a scientist e.g; newton.

(ii) The symbol of unit named after a ~cientist has initial capital letter such as N for .newton.

(iii) The prefix should be written before the unit without any space, such as I X 1 o-J m is written as l mm. Standard prefixes are given in table 1.4.

(iv) A combination of base units is written each with o.ne s~ace ap3ii .£or Clji!W••llew.toi: metre 1s wrttten as "tlla J\.IVV ~ D •

(v) Compound prefixes are not allowed. For exam pie, 11-l.llF may be written as 1 pF.

Powers ofTen

Factor Prefix Symbol 10-18 atto a 10-15 femto f to-•2 pi co p to-9 nano n I 0-<i micro ll 10-3 milli m · 10-2 centi c to-• deci d to' dec a da

4~~rrf kilo k

106 mega M 109 gig a G 1012 tera T

•

1015 peta p

l<t' 1018 ex a E ,.~ ~ . ~-fl>il A numb~r such as 5.0 x 104 em may be expressed in scientific.~ as ~S.O x 10m. ~

(vii) ~en a multiple of a base unit is raised to a power, the ~applies to the ;f?~ultiple and not the base unit alone. Thus, 1 km2

= I ~i = 1 x I 06 m2•

11 A Hand Book of Physics Part-1

(viii) Measur_:KEnt in practical work should be recorded immediately in the most conv~t. unit, e.g. micrometer screw gauge measure~ in mm, and the

_ .l.~ of calorimeter in grams (g). But before calculation ~the result, all -~""~easurements must be converted to the appropriate SI base u?;tf;..J..

. ~~~xplain the terms crro-:- and uncertainty. ,.~6 ~s. ERROR: ·~ All physical measurements are uncertain or imprecise to some extent. It is vg,

difficult to eliminate all possible errors or uncertainties in a measurement. The error in a measurement may occur due to:

(1) negligence or inexperience of a person.

(2) the faulty apparatus.lla K"'W · b , (3) inappropriate methoa or t~ique. e -.. com

Types of Errors:

There are two major types of errors whidb ~ given below:

(i) Random error (ii) Systematic erorr

(i) Random Error:

Random error is said to occur when repeated measurements of the quantity, give different values under the same conditions. It is due to mme unknown causes. Reduction of Random Error:

Repeating the measurement several tim'!s and taking an average can reduce the effect of random errors. (ii) Systematic Error:

Systematic error refers to an effect that influences all measurements of a particular quantity equally. It produces a consistent difference in readings. Occurre11ce of Systematic £rror:

It occurs to some definite rule. It may occur due to zero error of instruments, poor calibration of instruments or ir.rJtrcct ma;i.iPiiitc.b Reduction of Systematic Error: 1ami\IVV~ .CO

Systematic error can be n: -i:R;..d by comparing the instruments with another which is known to be more accurate. Thus for systematic error a correction factor can be applied.

~CERTAINTY: . ,

~ L The uncertainty is also usually described as an error in a measurement. It ~ occ~Jti}o ':()~

-(1$~ inadequacy <>r limitation of an instrument. · .~e . v ~'t;

(2) •crei!ral variations of the object being measured. ~-

(3) nat:OJ imperfections of a person's senses. · ~'If.

jCh;'lptcr Oil ivtEASUREMENT 12

Q. t. 7 Explain the significant figures and degree of accuracy. Give examples.

Ans. SIG~I~NT FIGURES AND DEGREE OF ACCU~Y: \~R,l! make some sort of measurement. the measured val~~ some error or

u~~~·~ This error or uncertainty may be due to the faulty instrume' due to the ~essness. lack of experience or training of the observer. The instrument us~~y also

~~c responsible for this error. For example, if a certain thickness is measured with ~tre rod, a sere\\ gauge or a vernier callipers. The error or uncertainty in the thickness isTflast as measured with the screw gauge. To make this point clear, one may real ize that every instrument is calibrated to a certain smallest division, which puts a limit to the degree of accuracy which may be achieved on making measurement with it. A reading which may

be with in the two marked divJi~ffil(fV'tiW!etJ.rtfan therefore be considered as correct.

Suppose we want to measure the length of a straight line with the help of a metre rod calibrated in millimeters. Let the end pojnt of the line lies between 10.3 and 10.4 em marks. By convention if the end of ~he line does not touch or cross the midpoint of the smallest division, the reaJ ing is confined to the previous division. In case the end of the "line seems to be touching or have crossed the midpoint, the reading is extended to the next division.

Approximate Values of Some Times lntervals<O

Table 15 Interval s

Age of the universe S.x 1017

A e of the Earth 1.4 X 1017

One year J .2 X 107

One day 8.6 X 10~

Period of typical radio waves 1 x 10-6

Period of vibration of an atom in a solid 1 x 1 o-13

Period of visible light Wa\'eS 2 X I o-•s

By applying the above rule the position of the edge of a line recorded as 12.7 em with the help of a metre rod calibrated in millimeters may lie between 12.65 em and

12.75 em. Thus in this .examlilf the~~ ~erta~~ ± 0.05. em. lt is, i~ ~act , equivalent to an uncertamty o~l4m~~,~~Me~Ycftln\ of the mstrument d1v1ded into two parts, half above and half below the recorded reading.

The uncertainty or accuracy in the value of a measured quantity can be indicated conveniently by using significant figures. The recorded value of the length of the straight ~ne i.e. 12.7 em contains three digits (1 , 2, 7) out of which two digits (1 and 2),

f.b. ... "'tely known while the third digit i.e, 7 is a doubtful one. 'V'\..~ ~~~ . ~ Dcfin~: ~-~:_..p~

~,(n any measurement, the accurately known digits ~-.me first doub~~gil are called as significant digils. N

13

· In other .;.: a significant figure is one ''btl h is kno ~be reasonably reliable. If the abo . e menf easurement is taken by a better measun·1g

which is exact up to a hundredth o l a timetre. it wpuld have been recorded as

12.70 em. ln this case, the number of significant figUJ es is four. Thus, we can say that as we tmprove the quantrty of our measuring instrument and techniquP.s, we extt nd the measured result to more and more significant figu ··es· and correspondingiy improve the experimental accur .. c)

of the result. · llamKiWeb~c General Rules:

While · calculating a · result from t lie measurements, it .is. important· to give due attention to significant figures and we must know the followi ng rules in deciding how· many significant figures are to be retained in the final result.-

(i) All digits I, 2, 3,' 4, 5, 6, 7, 8, 9 are significa11t. However, zeros may or may not be significant. In case of zeros, the following rules may be

adopted.

(a) A zero between two significant figures is itself significant.

(b) Zeros to the left of significant figures are not significant. For example, rione of th~.: zeros in 0.00467 or 02.59 is significant.

r\ H.and Book ot l'hys1cs l';ut - 1

\la~nit~ of some masses

"~ Mau(kg) ~~

1()-30 • El

0 Prot (S 10-2& UraniiMTI ~

1G-20 I DNA motecu1e ~ 10- 15

m 1()-10

1G-&

100

100

1010

10111 .

1020

' een

Ou tanker

MountE~

(c) Zeros to the right of a significant figun: may or may not be stgni ticant. In dttcimal fractiol)~iliJ'iUllt. g&llt ~!a sign11icant ! r g ur~.: Jrl.! stgrll ti -=,mt. Fo1 example, all ttll~ ltt.IW8D.'00fnrc sianiticant. However. in integers such as 8,000 kg, the number of signiticant zeros is determined by the accuracy of the measuring instrument. If the measuring scale has a least count of .. 1 kg then there are four significant figures written in scientific

~ notation as 8.000 x 103 kg. If the leost count of the scale is 10 kg, then the . ~~ number of significant figyres will ' 'e 3 written in scientific notation as

l • • 8.00 x 10 kg and so on. 0~

• · !;1'\ld) When a measurement is recorded in scientifk notation or stand:\il'rm, the ~ figures other thart the powers of ten are significant fig~rlt!lf* example, a

• +oQ measurement recorded as 8. 70 >< 104 kg has three signrfig~res. till In m~iplying or dividing number!£, keep a number of\)1fniflcant figures. in the

·'- - · ---·-:"..,. In the leut ar .. .uate factor i.e.,

[Chapter 01] MEASUREMENT 14

the factor containing the least number of significant fig~s. For example, the comput~ofthe follo\ving using a calculator gives; ici, »~ 5.348 X 10-l X 3.64 X } 0

4 = 1 4576898

3 ~~ _ ..\e 1 ., ... 6 . 2 x 1 o ·~, . .>J ~

~~As the factor 3.64 x 10·', the least a~curate in the above <:alculation~e, three ~~ significant figures. the answer sh 1uld be written to three significant tigureAr..rv.

. The other figures are insignificant and should be deleted. While deleti~g,...-~e ligures, the last signiticant figure to be retained is rounded off for which the fol_}owing rule are followed.

(a) If the first digit dropped is less than 51 the last digit retained should remain unchanged. •

(b) If the first digit d~~W§ti?,•MeQ~gftlto be retained is increased by one.

(c) If the digit to be dropped is 5, the previous digit which is to be retained is increased by one if it is odd and retained as such if it is even. For example, the following numbers are rounded off to three significant figures as follows:

43 .75

56.8546

73.650

64.350

is rounded off as 43.8

is rounded of: as 56.9

is rounded off ac:: T' .. r;

is rounded off as M 4

Following this rule, the corre :t ansv .. er ol the computation given in section (ii) is 1.46 X 103

•

(iii) In adding or subtractin!J. m:r, ,h.._r, . rhr rll'mhl·r of decimal places retained in the answer should equal the 511\atksL llllllllll.:r of decimal places in any of the quantities bl.!ing added or ~L,:'\racted. In this case, the number of significant tlgures is not important. IL is the position of decimal that matters. For example, suppose -w~ \\ish to add tti~'K4w.!~ Of.M1¥ in metres. , (a) T.:.. l _..tbf 2.7543

3 42 4.10

0.003 1.273

~ 75.523 . . 8.1273 . 0~ C1&,~answer: 75.5 m 8.13 m · :0~

~sc (a) the number 72.1 has the smallest number of decimal~! us the answer i/'8! ded off to the same position which is then 75 .5 m. In case e number 4.10 has the~~lkst number of decimal places and hence, the answ~i): ounded off the same decimal p~ns which is then 8.13 m. l

15 A Hand Book of Physics Part- 1

Determination of ~ificant Digits:

Let u find the number of si

5142 .- ...\' All digits are significant. _, rJJ;..~""

so~ ... ~00 0.2029

All digits are significant.

Digit 5 is significant, zeros may or may not be significant.

All digits after the decimal point are significant, but decimal point is not significant.

All digits after the decimal point are significant.

1 cases:

0.1000

0.0010 Digit 1 and 0 on the right are significant. Zeros on the left of I are not significant. This i;J.iue to ~.(a~t.th~~ the number is a fraction and may be written as 10 X lcJiaml\JtW"eo,.com

1.00 X 1 o-3 Digits 1 and 0, 0 before 1 o-3 are significant.

1.0020

Q.1.8 Explain the terms precision and :.ccuracy.

Ans. 'PRECISION:

Defir6tion:

A precise measurement is the one which has less precision or absolute uncertainty.

ACCURACY:

Definition:

An accurate measurement is the one ll'hich has lc!ss fractuma! or percentage uncertainty or error.

Note: A precise reading will be taken -to large number of significant figures, but be careful to use instruments of appropriate precision.

In measurements made in physics, the terms precision and accuracy are frequently used. They should be uisti ttt.Atnlfli.Weftcl'il'iMn of a measurement is determined by the instrument or Clevie~ ~~~g us~·ancMM'e' Accuracy of a measurement depends on the fractional or percentage uncertainty in that measurement.

For example when the length of an object is recorded as 25.5 em by using n metre rod having smallest division in millimetre, it is the difference of two readings of the ~itial and final positions. The uncertainty in the single reading as discussed before is ~Las ± 0.05 em which is now doubled and is called absolute uncertainty e~'ti,!Jt ± OJy~. Absolute uncertainty, ih fact, is equal to the !Past count of the ~· ing instru"'lf~ ~t means that .:t\e

Pr~ion or absolute uncertainty (least count) = ± 0.1 em ~:f•'\ O.q ~~

(Ch.lPter 01] MEASUREMENT 16

d F .

1 . 0. 1 em

an .J\. ract10na uncertamty = 25 - = 0.004 ~ :\... . ::"1 em ;~

vo t9~ . _\~~ . 0.1 em 100 0.~· ·~'liyo.IOre, Percentage uncertamty = 25.5 em x 100 = lOU ~

~~ Another measurement.taken by vernier callipers with least count a~l em is ,,~orded as 0.45 em In this case . •("

0 Precision or absolute uncertainty (least count)=± 0.01 em ~

d F .

1 . 0.01 em

0 an racuona uncertamty = 0 45 = .OJ . em

~ . 0.01 em 100 2.0 There,ore, Percentage uncertamtr ~ O.~cm x fiJ ~ 100 ~ 2.0%

Thus thl! reading 25.5 ~lqa{'tt~I)Yrf!~e·fiS ough less preci::. ·but is more accurate having less percentage uncertainty or error whereas the reading 0.45 em taken by vernier callipers is more precise but is less accurate. In fact, it is the relative measurement wluch is important. The smaller a physical quantity, the more prt!cise instrument should be used. Here the measurement 0.45 em demands that a more precise instrument such as micrometer screw gauge with least count 0.001 em should have been used.

Q.1.9 How total uncertainty in the final result is assessed?

Ans. ASSESSMENT OF TOTAL UNCERTAINTY IN THE FINAL RESULT:

To assess the total uncertainty or error, it is necessary to evalurte the likely uncertainties in all the fa~tors involved in that calculation. The maximum possible uncertainty or error in the firal result c~n be found as follows:

1. For Addition and Subtraction:

Absolute uncertainties are added. For example, the distance x determined by the differen~:e between two separate posttion measurements,

x1 = 10.5 ± 0.1 em and x2 = 26.8 ± 0.1 em is recorded as

2. For Multi~li;nt~n'7 ~~~~ ~\tiWe b.CO Percentage uncertainties are added. For example the maximum possible uncertainty

in the vulue of resistance ]{ of a conductor determined from the measwements of potential difterence V and current l ~Y using R = V /1 is found as follows:

~ v = 5.2 ± 0.1 v I • = 0.84 ± 0.05 A ~

~~ ~ ~e %age uncertainty for V is ~

The·~,uncertainty f~r I is ,...

0.1 V .JOO = 5.2Vx100

O.OSA 100 = 0.84 A X 100

/ .

17 A Hand Book of Physics Part-1

Hence total ~~inty in the value of resistance R when V is divided by I is 8%. The result is thus ~Cl as; ~

»• 52V ~~i · ·:;_t\lli = o.84 A = 6.19 VA -• = 6.19 ohms with a o/oage unc..,-j~ of 8%.

\\1>-~.e., R = 6.2 ± 0.5 ohms ~ The result is rounded off to two significant digits because both V and R hav~o

significant figures and uncertainty being an estimate only, is recorded by one signific~ figure. ·

I 3. For Power Factor:

Multiply the percentage uncertainty by that power. Fo_r example, in the calculation ~eE ~~ .

4 V = -n~

3

. %age uncertainty in V = 3 x %age uncertainty in

radius r.

As uncertainty is multiplied by power factor, it increases the precision demand of measurement. If the radius of a small sphere is measured as 2.25 em by a vernier callipers with least count 0.01 em, then the radius r is recorded as;

r = 2.25 ± 0.01 em.

Absolute uncertainty = Least count= ± 0.0 I em

%age uncertainty in r = O.Ol em x 100 = 0 4%

2.25.cm 100 ·

Total percentage uncertainty in V = 3 X 0.4%

Some Specific Temperatures

............... *.

akri'lium melta w -••

. lla K eb'9.--....... 4

Thus volume V = 3n~

4 = 3 x 3.14 x (2.25 cm)3

~ · ~~. = 47.689 cm3 with 1.2% uncertainty

4'/il!us the result should be recorded as 86 V = 47.7 ± 0.6 cm3

4. Fo;~rlainty in the Average Value of many Measure~ngs' m -~~~ th,. Av~r~ae value of measured values.

-----~------~~------~

[Chapter 01] MEASUREMENT 18

(ii) Find deviation of each measured value from the average value.

{iii) Th~an deviation is the uncertainty in the average ~e. (iv..)_ .~example the six reading of the ·micrometer screw ~to measure the

.~eV diameter of a wire in mm are c-.. '1/~ ~' 1.20, 1.22, 1.23, 1.19, 1.22, 1.21. ~6

..._,.,.~ 1.20 + 1.22 + t.23 + t.19 + 1.22 + 1.21 ·c.0 '"" Then Average = 6 ~ = 1.21 mm

The deviation of the readings, which are the difference without regards to the sign, between each reading and average value are O.OC 0.01, 0.02, 0.02, 0.01 , 0.

M fd . . - 0.01 + 0.01 + 0.02 + 0.02 + 0.01 + 0

ean 0 evlatlOnS- II. K·Web .. com = ~.TJ,mm

Thus likely uncertainty in the mean diameter 1.21 mm is 0.01 mm recorded as 1.21 ± 0.01 mm. 5. For the Uncertainty in a Timing Experiment:

The uocertainty in the time period of a vibrating T r r h rave time o 121 t

body is found by dividing the least count of timing device by the number of vibrations. For example, the time of 30 vibrations of a simple pendulum recorded by a stopwatch accurate up to one tenth of a second is 54.6 s, the period

Moon to Earth

Sun to Earth

Pluto to Earth

T 54.6 s 1 82 "th . 03.0ls = 0.003 s = ---w- = . s w1 uncertamty

Thus period T is quoted as T = 1.82 + 0.003 s.

I min 20 s

8 min 20 s

s h 20 s

Hence, it is advisable to count large number of swings to reduce timing uncertainty.

Example 1.1

The length, breadth att~We:heCQM33. m, 2.105 m and 1.05 em respectively. Calculate th~ ft!ddt~ "b'r"~"tKew'"sli'eet correct up to the appropriate significant digits.

Solution:

Given data:

~ Length of sheet = f. = 3.233 m

~~readth of sheet = b = 2.1 OS m

/~kness Of Sheet = h = 1.05 Cm = 1.05 X 1 o-l m

To deter~~ Volum~e sheet = V = ?

19 A Hand Book of Physics Part-1 •

Calculations: _'((\ As we knoMJtat; .-.l~~me = Length x Breadth x Thickness

.. t.~, . ~TJt V = ( X b X h

~~ Substituting the values, we have; V = 3.233 m x 2.105 m x 1.05 x 10-2m ..

or V = 7.14573825 x 10-2 m3

As the factor 1.05 em has minimum number of significant figures equal to three, therefore, volume is recorded up to three significant figures. Hence,

V = 7.15 x 10-2 m3 Ans.

Example 1.2 llamKiWeh · · · The mass of a metal box measured by a 'revS9aDJce is 2.2 kg. Two silver ·

coins of masses 10.01 g and 10.02 g measured by a beam balance are added to it. What is now the total mass of the box correct up to the appropriate precision. · Solution: Given data:

Mass ·or a metal box = m1 = 2.2 kg Mass of 151 silver coin= m2= IO.~lg = 0.01001 kg Massof2ndsilvercoin = m3= 10.02 g = 0.01002 kg

To determine: Total mass of the box= M = ?

Calculations: Total mass when silver coins are added to box

M = 2.2 kg+ 0.01001 kg+ 0.01002 kg or M = 2.22003 kg Since least precise is 2.2 kg, having one decimal place, hence total mass should be

to one decimal place which is ~~l.P~ate precision. Thus, Total mass = M 1\IWeb.com

Example 1.3 The diameter and length of a metal cylinder measured with the hrlp of

vernier callipers Df least count 0.01 em are 1.22 em and 5.35 em. Cah:ul.ah tht> ~volume V of the cylinder ud uncertainty in it.. ·

~~tion: . · . Ga~ata: .

~tcountofvemiercallipers = 0.01 em

oi~ of m~tal cylinder • - L --~ --• ..... 1 ,.._,1t",4,_p

= d = 1.22 em

= l = S.3Scm

[Chapter m) MEASUREMENT

To determine:

Volume ~e ·cylinder = V = ?

u~q;inthevolwne = ?

Calc~s: ~~Absolute uncertainty in length=

~" %age unce~ty in length =

Similarly,

0.01 em

0.01 em 100 _ 0 20 5.53 em x 100 - · Vo

Absolute uncertainty in diameter = 0.01 em

0 • • d' 0.01 em 100 0 Vo age uncertamty m ·antmKi'Wib:tiMn .s%

As we know that;

xa2l v = 4 ......... (i)

Hence

Total. uncertainty in V = 2 (% age uncertainty in diameter

+% age uncertainty in length)

= 2 X (0.8 + 0.2) = }.8o/o

Now, substituting the values in equation (i), we have;

V = 3.14 x (1.22 cmi x 5.35 em 4

or V = 6.2509079 cm3

Where

Uncertainty in volume= 1.8%

Thus, V = (6.2 ± O.~cm3 KlWjbo~(ijfi29 = 0.1) where 6.2 cm3 is calculatMiDl~ ~!~1 em ~s the uncertainty in it.

20

Q.l.lO What is meant by the dimensions of physical quantities? Explain with exaamples.

Ans. DIMENSIONS OF PHYSICAL QUANTITIES:

It is sometimes convenie~t to express a unit in terms of base units only. BecalJ!e~ multitude of names of units, and because the same units can be expre~n

ays, a comparison of units can best be made by ~ing this form. .~e

Definitio ~- ':: ~..P.· "1'14f-,s'~~ers of the base units in terms of whi'h a phys~uantity

can be reprpynted are known as dimensions.

21 A Hand Book of Physics Part-I . The dime~ of a physical quantity can be obtained by exJZssing that physical

quantity in t'-<f~Y ~bois of base units. . ~ • , -fymbols of length, mass and time denoted by L, M and ~ctively are

~nd~eate the dimensions of a physical quantity. Square brackets [t'ftE used to ... ote the dimensions of a physical quantit)'. $6

·~ . ·Dimensions of Some Physical Quantities: OA (i) Velocity: .,,

The dimensions of velocity are:

As we know that>

Velocity= .,KiWeb.com So, [V] = [ ¥ J or [V] = [L r']

(ii) Acceleration:

The dimensions of acceleration are:

As we know that; Velo~ity

Acceleration = Time =

Acceleration

Dimensions of acceleration

Force:

Distance = (Timei

=[~]

Distanceffime Time

(iii)

It can be written as: llamKiWeb.coni Force= Mass x Acceleration

M Velocity

= ass x Time

~ • M Distanceffime v~ = ass x Time

~L.. . Dsitance ~ Force = Mas!; x (Time)2 6

~C'0Dimensions offorcc ~ [~~ J ~-~~~~~~~- r F 1 =· [MLT2]

(Chapter 01) MEASUREMENT 22

(iv) Work:

· .~~ Work= Force x Distance ~~

= Mass x Acceleration x Distance ~ Velocity . 6.L

Weknow~t:

vo _fll.l>.

~~,

~'IJ.~ = Mass x T' x Distance · 4U

Ime ·c 0~ Distanceffime .

= Mass x T ' . x Distance I me

Distance ' . = Mass x Time2 x Distance

M (Distance/

l~a~iW'eb.com Dimension of worlC = L r J

[W] = [ML 2r 2] •

(v) Power: We know that

~· .

Power =

=

Work Force x Distance ---Time Time

Mass x Acceleration x Distance Time

M D. Acceleration

= ass x Istance x T' I me

M D. Velocityffime

= ass x Istance x T' I me

. Distanceffime 1 = Mass x Distance x T' · x -T. Ime 1me

. Distance 1 = Mass x D1stance x T' 2 x T---:--Jme . 1me

Dimensionofpower=l~~lWeb.CO~

= [~] [ p ] = [ML2rJ]

(v~ Area: C$)'1/s

~· Area = Length x Length

" ens ions of area= [L x L]

.,.. [A] = [L2] ... 0

~

\

23

(vi) Volume: 0~ As e"Q~

A Hand Book of Physics Part-1

Volume = Area x Length ~ ~"~ IF Length x Length x Length

Dimensionsofvolume = (L x L x L)

[ v] = [L3]

Applications of Dimensions:

Using the method of dimensions called the dimensional analysis, we can check the correctness of a given formula or an equation and can also derive it.

(i) Checking the HomogemflatftKiWett.'eom Jn order to check the correctness of an equation, we are able to show that the

dimensions of the quantities on both sides of the equation are the same, irrespective of the form of the formula. This is called the principle of homogeneity of dimensions.

(ii) Deriving a Possible Formula:

The success of this method for deriving a relation for a physical quantity depends on the correct guessing of various factors on which the physical quantity depends.

Example 1.4

Check the correctness of :he relation v = - /E! where v is the speed of . -\Jm transverse wave on a stretched string of tension F, length land mass m.

Solution:

Given equation:

v = ~ In order to check the CO'ai.~~ i. ~laii_on, we compare dimensions on both

sides of an equation. II Cl Jl II'\ I YV 9 0 CO . Dimensions ofleft hand side of the equation = [v]= [L 1 1

] ......... (i) I

Dimensionofrighthandsideoftheequation = ([F] x [/] x [m-1 ])~

~~ - ([ML12] x (L] X (M"1])l

~· I 0~ ~~ . = [L2r2J2 . e~~

;e~ , • I .~...,\1 ( " ) v = [L!] ~;:t'-:': .... u

Fr;~Qwations (i) and (ii), ·we-come to know that dimens~~both sides of the ""nn~tinn ~re ilif.bme. therefore, equation is dimensionally correct. ·

" (Chapter 01] MEASUREMENT

Example l.S

Derive ~ relation for the time peri~d of (Fig. 1.2) us~imensional analysis. The various possible tAf'tnrr•~ perioM~Y depend are: ~~ength of the pendulum(/)

~'b~ Mass of the bob (m)

Angle 9 which the thread makes with the vertical Acceleration due to gravity (g) ·

Solution:

To determine:

Relation for the time peritleff¥tKiWe8:CGm Calcuhltions:

The relation for the time period T will be of the

. . . . . . e .

. . . .

24

. T oc m• X r X 9c: X gd

e fonn:

m

or T = Constant m• f 9c: gd .... ... .. (i) ..---' . where, we are to find the values of powers a, b, c and d. Fig. 1.2

Writing the dimensions of both sides of the equation (i), we have; [T] = Constant x [Mt [L]b {LL-1t [Lr2

]d

Comparing the Q.imensions on both sides, we have; [T] = [Tr2d

[Mt = [Mt [L]o = [L]b+d+c:-c:

Equating powers on both the sides, we have:

-2d 1

= or d = -- · LLamtQY'J,&.com a 0

.

. I 9 = [LL -It = [L 0t = or b = -d =- and 2 1

Substituting the values of a, b, 9 and d in equation (i), we have: I . I

~A . T = Constant ~x m; x ii x I x g-:; :0~0~ "~ T = Constant - · · .~.10

g . ..L~' i9RJ11erical value of the constant cannot be determined ~~mensional

analysis, ho~tv% can be found by experiments. · ~'II-

25 A Hand Book of Physics Part-1

Example 1.6 O~ q. dimensions and hence the SI units of coefficien~~iscosity in the

relati toke's law for the drag force F for a spherical object o · s r moving

2 ~ocity v given as F = 6n11 rv. . ~ t . ~~ IOn: ·~

Given data: 0~ Drag force for spherical object = F

Radius of the spherical object = r

Velocity of the object = v

To determine: . I Ia K"W h com Dimensions of coefficient of viscosity = ? " SI units of coefficient of viscosity = ?

Calculations:

In the given equation i.e. F = 6n11rv, where 6n is a number having no dimensions. It is not accounted in dimensional analysis. Therefore,

[F]

or

= [11rv]

[F) = [r][v]

Substituting the dimensions ofF, rand v in R.H.S, we have; ·

_ [ML12]

[11] - [L][L11]

[11] = [ML-11 1]

SUM.M.ARY ~> Physics is the study of entire physical world.

~ ~ The most basic quantities that can be used to describe the physical world~e ~~· mass, length and time. All other quantities, called derived quantities, fl1. be

escribed in terms of some combinations of the base quantities. e"' · +

.. > ternationally adopted system of units used by all the s · ts anq almost a ~o5euntries of the World is International System (S~'J.t:!. It consists of seven o~e units. two suoolementary units and a number of derived units.

[Chapter 01) MEASUREMENT 26

,.._> Errors due Jt\_ incorrect design or calibratio~s of the measu~g device are called system~p~rrors. Random errors arc due to unknown cause~ fluctuations in t~i~tity being measured. ~·

~> ·~1 accuracy of a measurement is the extent to which systematic e~&'akes ·a ~'b~ m.easured value differ from its true value +c-

0 i)-> The accuracy of a me_asurement can be indicated by the number of signift~

figures, or by a stated uncertainty.

The significant figures or digits in a measured or calcu.Jated quantity are those digits that are know11: to be reasonably reliable.

The result of multiplicatt~Mifi\VW~tlo e0'11'fnificant figures than any factor in the input data. m>~JJ ~rf your caT'c"ulator result to correct number of digits.

In case of addition or subtraction the precision of the result can be only as great as the least precise term added or subtracted.

Each basic measurable physical property represented by a specific symbol written • with in square brackets is called a dimension. All other physical quantities can be derived as combinations of the basic dimensions.

Equations must be dimensionally consistent. Two terms can be added only when they have the same dimensions.

SHORT QUESTIONS & ANSVVERS 1.1 Name several repetitive phenomena occurring in nature, which could serve

as reasonable time standards.

Ans. Any phenomenon that ree~ats itself after r~~~lar time intervals can be used as a time standard. The rotaticfla KiW~C:Gm determines the length of the day, has been used as a time standard from the earliest times. Some other repetitive. phenomena, which can be adopted to define a time standard, are (i) Heart beat (ii) Human pulse rate (iii) Oscillations of a simple pendulum (iv) Revolution of the moon around the ·Earth (v) Characteristic vibrations of

.l: crystals such as quartz crystal (vi) Radioactive decay rate of some substances i.e~ ~ Half life of a radioactive substance. (JO

1.2~"t:&e the drawbacks to use the period of a pendulum as a time stan~c!}O· Ans. ~;ow that the period of a pendulum is . ~

o - n ·. 11-~ +oo T = 2n -\j i N ~ . g~----~----~~~---