Units and Measurement: Class 11 Physics Chapter 1 Hinglish mein

Physics Part 1 · Hinglish notes, simulations aur practice · Official NCERT chapter PDF ↗

Published 18 Oct 2026 · Updated 18 Oct 2026

80 min
1 simulation
25 practice questions
25 MCQs

Is chapter mein aap kya seekhenge

Samjhana ki unit kya hoti hai, aur base quantities, derived quantities aur systems of units mein fark karna
Saat SI base quantities aur unki units ke naam batana, aur plane aur solid angles ko radians aur steradians mein naapna
Significant figures ginna, values ko scientific notation mein likhna aur order of magnitude ka estimate lagaana
Products, quotients, sums aur differences ke liye significant-figure rules lagaana, aur sahi tarah round off karna
Measured values se calculate kiye gaye result ki uncertainty ka estimate lagaana
Physical quantities ke dimensional formulae aur dimensional equations likhna
Dimensional analysis se equations check karna, units convert karna aur relations nikaalna, aur iski limits batana

NCERT Physics Class 11, Chapter 1, pp. 1–12 par based original companion text. Yeh NCERT se juda nahi hai aur NCERT ne ise endorse nahi kiya hai.

1.1 Introduction

Kolkata ka ek tailor note karta hai ki sleeve 58 cm lambi honi chahiye. Pune ka ek pharmacist kisi powder ka 250 mg taulta hai. Dono ek hi kaam kar rahe hain: ek quantity ko ek agreed standard se compare karna aur ginna ki woh standard usmein kitni baar fit hota hai. Wahi standard unit (मात्रक) hai. Poori physics aise hi comparisons par tiki hai, isliye hum units se shuru karte hain, aur is baat se ki kisi measured number par kitna bharosa kiya ja sakta hai.

Isliye ek measured result ke do parts hote hain: ek pure number, yaani uska numerical measure, aur ek unit. “Corridor 40 hai” se kuch pata nahi chalta; “corridor 40 m hai” ek measurement hai. Agar tum chhoti unit par switch karte ho, to number usi proportion mein badh jaata hai, taaki dono ka product same rahe (Fig. 1.1):

Unit badalna

n_{1} u_{1} = n_{2} u_{2}

(1.1)

jahan

$n_{1}, n_{2}$: dono numerical measures
$u_{1}, u_{2}$: dono units

Jaise, $2.5 m$ hi $250 c m$ bhi hai: unit 100 guna chhoti hui aur number 100 guna bada.

3/3Unit B aadhi lambi hai aur 6 baar fit hoti hai: $3 u_{A} = 6 u_{B}$ .

Fig. 1.1Ek hi length do units mein naapi gayi. Unit ko aadha karne se number double ho jaata hai.

Figure ke upar ek plank rakha hai. Pehle step mein woh akela hai, saath mein sawaal hai ki woh kitni units lamba hai. Doosre step mein A likhe teen barabar blocks uske neeche ek ke baad ek rakhe hain aur theek uski length ke barabar hain, isliye n barabar 3. Teesre step mein B likhe chhe chhote blocks, har ek A se aadha lamba, bhi plank ke barabar aate hain, isliye n barabar 6, aur label 3 u A barabar 6 u B dikhata hai ki number guna unit nahi badalta.

Speed distance divided by time hai. Density mass divided by volume hai. Force mass times acceleration hai. Kyunki physical quantities aise judi hoti hain, isliye sabke liye kuch hi independent standards kaafi hain. Hum kuch base quantities chunte hain (inhe fundamental quantities bhi kehte hain), jaise length, mass aur time, aur har ek ke liye ek base unit tay karte hain. Baaki har unit base units ko multiply aur divide karke banti hai; yeh derived units (व्युत्पन्न मात्रक) hain, jaise metre per second. Base units ka ek chuna hua set, unse banne wali saari derived units ke saath, system of units kehlata hai (Fig. 1.2).

4/4Mass × acceleration se force milta hai: $1 k g m s^{- 2}$ ko newton (N) kehte hain.

Fig. 1.2Derived units base units ko multiply aur divide karke banti hain.

Left mein teen boxes mein base quantities hain: length metres mein, time seconds mein aur mass kilograms mein. Doosre step mein length se ek arrow area ke box tak jaata hai, square metres mein. Teesre step mein length aur time se arrows speed tak jaate hain, metres per second mein, aur divide by time likha ek aur arrow acceleration tak jaata hai, metres per second squared mein. Chauthe step mein mass aur acceleration se arrows force tak jaate hain, kilogram metre per second squared mein, jiska naam newton hai.

Definition

Unit

Kisi physical quantity ki ek fixed amount, jis par poori duniya sehmat hai, aur jo us quantity ka paimaana hai. Measure karne ka matlab hai yeh pata karna ki quantity mein aisi kitni units hain.

Example $1.5 k g$ chawal mein 1.5 numerical measure hai aur kilogram unit hai.

Try karke dekho

Apni desk ko do units mein naapo

Tumhe chahiye

30 cm ka ruler ya measuring tape
tumhara apna haath

Pehle guess karo

Tum ek desk ko pehle hand-spans mein aur phir centimetres mein naapte ho. Kaun sa number bada hoga?

hand-spans ki sankhyacentimetres ki sankhyadono numbers barabar honge

Apna haath phaila kar desk ki length hand-spans mein naapo, nearest aadhe span tak.
Ruler se apne ek hand-span ki length naapo.
Ab desk ko seedhe ruler ya tape se naapo.
Spans ki sankhya ko ek span ki length se multiply karo aur result ko step 3 se compare karo.

Measurement	Tumhari value
Desk (hand-spans mein)
Ek span (cm)
Ruler se desk (cm)
Spans × span ki length (cm)

Tumhari entries apne aap save hoti hain.

1.2 International system of units

Beesvi sadi ke beech tak alag-alag desh, aur science ki alag-alag branches bhi, alag systems mein naapti thin. Teen systems khaas taur par common the. Unmein length, mass aur time ke liye chuni gayi base units alag thin (Table 1.1).

Table 1.1Chaar systems mein length, mass aur time ki base units. SI MKS ki teen units rakhta hai aur chaar aur base units jodta hai.

System	Length	Mass	Time
CGS	centimetre	gram	second
FPS (British)	foot	pound	second
MKS	metre	kilogram	second
SI	metre	kilogram	second

Aaj lagbhag poori duniya ek hi system use karti hai, Système International d'Unités (French mein International System of Units), yaani SI. Ise 1960 mein General Conference on Weights and Measures (CGPM) ki ek meeting mein apnaya gaya; CGPM ek intergovernmental body hai jiske saath Paris ke paas BIPM, yaani International Bureau of Weights and Measures, kaam karta hai. SI ko apni maujooda saat base units 1971 mein mili, jab mole joda gaya. November 2018 mein CGPM ne saaton units ko constants of nature ki fixed values ke terms mein dobara define kiya, aur nayi definitions 20 May 2019 se laagu hui. SI multiples powers of ten mein chalte hain, isliye millimetres, metres aur kilometres ke beech convert karne mein sirf decimal point khiskaana padta hai. Ab se is book mein har quantity SI units mein hai, jab tak hum kuch aur na kahein.

Table 1.2Saat SI base quantities aur units. 2019 se har unit kisi constant of nature ki ek exact value declare karke fix ki jaati hai. In values ko yaad karne ki zaroorat nahi hai.

Base quantity	Unit	Symbol	Define karne wala constant (exact value)
Length	metre	m	vacuum mein light ki speed, $c = 299 792 458 m s^{- 1}$
Mass	kilogram	kg	Planck constant, $h = 6.626 070 15 \times 10^{- 34} J s$
Time	second	s	caesium-133 ki hyperfine frequency, $Δ ν_{C s} = 9 192 631 770 H z$
Electric current	ampere	A	elementary charge, $e = 1.602 176 634 \times 10^{- 19} C$
Thermodynamic temperature	kelvin	K	Boltzmann constant, $k = 1.380 649 \times 10^{- 23} J K^{- 1}$
Amount of substance	mole	mol	Avogadro constant, $N_{A} = 6.022 140 76 \times 10^{23} m o l^{- 1}$
Luminous intensity	candela	cd	$540 \times 10^{12} H z$ light ki luminous efficacy, $K_{c d} = 683 l m W^{- 1}$

Ek constant kisi unit ko kaise define kar sakta hai? Metre ko lo. Vacuum mein light hamesha same speed se chalti hai, aur SI bas declare kar deta hai ki woh speed exactly 299 792 458 metres per second hai. Tab ek metre woh distance hai jo light ek second ke $1 / 299 792 458$ hisse mein tay karti hai. Koi bhi achhi equipped laboratory ise bana sakti hai, kisi master rod se milaane ki zaroorat nahi. Baaki units bhi isi tarah fix ki jaati hain: second caesium atom ki ek frequency se, kilogram Planck constant ke through, aur isi tarah aage.

Mole ke baare mein ek savdhaani: hamesha batao ki tum kaun si entities gin rahe ho. Hydrogen atoms ke ek mole aur hydrogen molecules ke ek mole, dono mein lagbhag $6.022 \times 10^{23}$ particles hote hain, lekin doosre ka mass pehle ka double hai.

History se

Ek kilogram jise tum haath mein pakad sakte the

1889 se 2019 tak kilogram ek khaas metal cylinder ka mass tha, jo platinum aur iridium se bana tha aur Paris ke paas BIPM ki ek vault mein rakha tha. Deshon ke paas numbered copies thin; India ki national copy, No. 57, New Delhi ki CSIR-National Physical Laboratory mein rakhi hai, jo Indian Standard Time ko bhi maintain karti hai. Ek sadi mein, dhyaan se kiye gaye comparisons se pata chala ki copies aur original ke beech dasiyon micrograms ka fark aa gaya tha, aur koi nahi bata sakta tha ki kaun sa badla. 2019 se kilogram Planck constant ke through fix hai aur ise Kibble balance naam ke instrument se banaya ja sakta hai, isliye ab yeh kisi ek object par depend nahi karta jis par scratch aa sakta tha, jise saaf karna padta tha ya jo kho sakta tha.

Baaki har unit in saat se derive hoti hai. Kai derived units ke khaas naam hain, jaise newton (force), joule (energy), watt (power), pascal (pressure) aur hertz (frequency). SI ke bahar ki kuch units itni convenient hain ki woh iske saath use hoti rehti hain (Table 1.3): kheti ki zameen hectare mein bikti hai, anaaj ke daam per quintal bataye jaate hain, aur diamonds carats mein taule jaate hain.

Table 1.3SI ke saath use hone wali kuch units.

Unit	Symbol	SI units mein value
minute	min	$60 s$
hour	h	$3600 s$
day	d	$86 400 s$
degree (angle)	°	$(π / 180) r a d$
litre	L	$10^{- 3} m^{3}$
tonne	t	$10^{3} k g$
quintal	q	$100 k g$
hectare	ha	$10^{4} m^{2}$
carat (gems)	ct	$2 \times 10^{- 4} k g$
bar	bar	$10^{5} P a$
standard atmosphere	atm	$101 325 P a$
astronomical unit	au	$1.496 \times 10^{11} m$
electronvolt	eV	$1.602 \times 10^{- 19} J$

Prefixes kisi unit ko powers of ten se scale karte hain (Table 1.4). Symbols kuch fixed rules follow karte hain, jinse likhi hui value duniya mein har jagah bina confusion ke padhi jaati hai:

Table 1.4SI prefixes jo tum sabse zyada use karoge.

Prefix	Symbol	Factor
tera	T	$10^{12}$
giga	G	$10^{9}$
mega	M	$10^{6}$
kilo	k	$10^{3}$
centi	c	$10^{- 2}$
milli	m	$10^{- 3}$
micro	µ	$10^{- 6}$
nano	n	$10^{- 9}$
pico	p	$10^{- 12}$
femto	f	$10^{- 15}$

Unit symbols seedhe (upright) print hote hain aur unmein kabhi plural s ya full stop nahi lagta: 5 kg, 5 kgs nahi.
Number aur symbol ke beech space chhodo, jaise 3 m ya 25 °C mein; sirf angle ke signs exception hain, jaise 30°.
Logon ke naam par rakhi gayi units ke symbols capital hote hain (N, Pa, W), lekin naam khud lower case mein likhe jaate hain (newton, pascal, watt).
Ek baar mein ek hi prefix use karo: 1 ng likho, 1 mµg nahi; prefix aur uski unit milkar ek hi symbol banate hain, isliye km² ka matlab (km)² hai.

Plane angle aur solid angle

Do aur SI units angles naapti hain. Plane angle do lines ke beech ka khulaav hai jo ek point par milti hain. Us point ke around radius $r$ ka ek circle banao; lines length $s$ ka ek arc kaat deti hain. Angle ko is ratio se define kiya jaata hai

Plane angle

θ = \frac{s}{r}

(1.2)

jahan

$s$: arc ki length
$r$: circle ka radius

Iski unit radian (rad) hai. Ek radian woh angle hai jiska arc radius ke barabar ho. Poore chakkar ka arc $2 π r$ hota hai, isliye poora chakkar $2 π r a d$ hai, aur $1 r a d = 180^{\circ} / π \approx {57.3}^{\circ}$ . Kyunki $θ$ ek ratio hai, isliye isse fark nahi padta ki tum kitna bada circle banate ho (Fig. 1.3).

3/3Jab arc radius ke barabar ho, to angle $1 r a d \approx {57.3}^{\circ}$ hota hai.

Fig. 1.3Angle arc aur radius ka ratio hai, isliye yeh circle ke size par depend nahi karta.NCERT Fig. 1.1 · p. 2

Left mein ek point se do rays angle theta par nikalti hain. Pehle step mein radius r ka ek blue arc mark hai, jiski arc length s hai. Doosre step mein unhi rays ke beech radius 2r ka ek orange arc dikhta hai; uski length 2s hai, isliye arc aur radius ka ratio nahi badalta. Teesre step mein right mein ek alag drawing aisa angle dikhati hai jiski arc length uske radius r ke barabar hai; label kehta hai ki yeh angle 1 radian hai, lagbhag 57.3 degrees.

SimulationArc, radius aur angle

Ek circle jis par do points A aur B hain aur centre O se radii bane hain. Tum B ko circle par ek-ek degree ke steps mein drag kar sakte ho aur radius badalne ke liye ek handle drag kar sakte ho. Readouts angle AOB degrees mein, arc length AB aur chord AB centimetres mein, aur radius r dikhate hain, saath mein arc length ka formula bhi.

Fig. 1.4Angle badalne ke liye point B ko drag karo, aur circle ka size badalne ke liye radius handle ko drag karo. Arc AB ko radius

r

se compare karo.

Try karo

B ko tab tak drag karo jab tak arc AB radius ke barabar na ho jaaye. Angle kya dikhata hai? Ise 180° divided by π se compare karo.
Angle ko 90° par set karo, phir radius handle ko drag karo. Har baar arc AB ko r se divide karo: kya ratio badalta hai?
AB ko diameter bana do. Arc kitne radii lamba hai, aur angle kitne radians ka hai?

Solid angle iska three-dimensional version hai: ek cone ka khulaav, jaise torch ki beam. Cone ki nok ko radius $r$ ke sphere ke centre par rakho; cone sphere ki surface se area $d A$ ka ek chhota patch kaat deta hai. Solid angle hai

Solid angle

d Ω = \frac{d A}{r^{2}}

(1.3)

jahan

$d A$: sphere par patch ka area
$r$: sphere ka radius

Iski unit steradian (sr) hai. Double size ke sphere par usi cone mein patch chaar guna bada hota hai, isliye ratio fixed rehta hai (Fig. 1.5). Poore sphere ka area $4 π r^{2}$ hai, isliye ek point ke around saari directions milkar $4 π s r \approx 12.6 s r$ banati hain. Radian aur steradian do lengths aur do areas ke ratios hain, isliye dono dimensionless hain: unka naam hai lekin koi dimension nahi.

2/3Radius $2 r$ par patch ka area $4 d A$ hai, aur $(2 r)^{2} = 4 r^{2}$ : wahi solid angle.

Fig. 1.5Solid angle area divided by radius squared hai, isliye yeh sphere ke size par depend nahi karta.NCERT Fig. 1.1 · p. 2

Ek sphere bana hai jiska centre O hai. Pehle step mein O se do lines ek patla cone banati hain jo radius r ke sphere par area dA ka ek chhota blue patch kaatta hai. Doosre step mein radius 2r ka ek sphere use gher leta hai, aur wahi cone chaar guna area ka ek orange patch kaatta hai, isliye solid angle nahi badalta. Teesre step mein poora chhota sphere shade hai aur label dikhata hai ki poora sphere 4 pi steradians subtend karta hai.

Try karke dekho

Dhaage ke ek tukde se ek radian dhoondho

Tumhe chahiye

ek gol steel ki plate ya dhakkan
dhaage ka ek tukda
ek marker
ek protractor

Pehle guess karo

Plate ke kinaare par radius jitne lambe dhaage ke kitne tukde fit honge?

exactly 36 se thoda zyadaexactly 4lagbhag 10

Dhaage ko plate ke centre se uske kinaare tak pakdo aur utni length par kaat lo: yeh ek radius lamba hai.
Ek nishan se shuru karke dhaage ko kinaare ke saath bichhao, aur jahan woh khatam ho wahan doosra nishan lagao.
Dono nishanon ko centre se jodo. Lines ke beech ka angle ek radian hai; ise protractor se naapo.
Dhaage ko kinaare ke saath aage badhaate raho aur gino ki ek poore chakkar mein kitni lengths fit hoti hain.

Abhi nahi kar sakte? Simulation try karo · Fig. 1.4

Quick check

4 mein se 1

Inmein se kaun si SI base unit nahi hai?

newtonkelvincandelamole

Answer dekho

Answer (A) newton

Newton ek derived unit hai,

1 N = 1 k g m s^{- 2}

. Candela aur mole anjaan lag sakte hain, lekin dono base units hain.

1.3 Significant figures

Ek hi pencil ko do rulers se naapo to tum alag-alag sankhya mein digits likhoge (Fig. 1.6). Sirf poore centimetres wale scale par pencil ka sira 7 aur 8 ke beech hai, aadhe se thoda pehle, isliye tum $7.4 c m$ record karte ho: 7 nishan se padha gaya aur 4 aankh se andaaza lagaya gaya. Millimetre marks wale scale par sira 7.4 aur 7.5 cm ke beech hai, aur tum ek aur digit ka andaaza laga sakte ho: $7.46 c m$ . Har reading mein aakhri digit ko chhod kar har digit seedhe scale se aata hai, aur aakhri ek dhyaan se lagaya gaya guess hai.

Kisi bhi measured value ko likhne ka yahi rule hai: saare digits jin par tumhe pakka bharosa hai, phir ek estimated digit, aur uske baad kuch nahi. Yeh sab milkar measurement ke significant figures (सार्थक अंक) hain. Tum imaandaari se kitne claim kar sakte ho, yeh instrument ke sabse chhote division, yaani uske least count (अल्पतमांक), par depend karta hai. Pencil ke liye $7.4613 c m$ likhna aisi precision ka claim hoga jo koi bhi ruler nahi de sakta.

3/3Significant figures = saare pakke digits + pehla estimated digit.

Fig. 1.6Scale jitna barik, utne zyada digits tum padh sakte ho. Tum jo aakhri digit likhte ho woh hamesha estimate hota hai.

Upar ek peeli pencil zero se shuru hokar rakhi hai, aur uski nok se ek dashed line neeche jaati hai. Pehle step mein sirf poore centimetres wala ruler nok ko 7 aur 8 ke beech dikhata hai, aur reading 7.4 centimetres dikhti hai jismein 4 estimated digit ke rang mein hai. Doosre step mein neeche millimetre marks wala ruler aata hai; nok 7.4 aur 7.5 ke beech hai, aur reading 7.46 centimetres ho jaati hai, ab 6 estimated digit hai. Teesre step mein 7.4 ke neeche ek bracket par sure likha hai aur 6 ke around ek box par estimated likha hai.

Kyunki yeh ginti measurement se tay hoti hai, isliye yeh us unit par depend nahi kar sakti jo tum likhte ho. Ek sikke ka mass $3.79 g$ hi $0.00379 k g$ bhi hai: extra zeros sirf decimal point khiskaate hain, aur dono forms mein teen significant figures hain. Yeh dhyaan mein rakhte hue, likhe hue number mein significant figures ginne ke rules yeh hain:

Har non-zero digit significant hota hai. $3.47$ mein teen significant figures hain.
Non-zero digits ke beech ke zeros significant hote hain, decimal point chahe kahin bhi ho. $40.08$ aur $4008$ dono mein chaar hain.
1 se chhote number mein, pehle non-zero digit se pehle wale zeros nahi gine jaate, aur na hi woh zero jo aam taur par decimal point se pehle likha jaata hai. Yeh sirf decimal point ki jagah batate hain: $0.0036$ mein do significant figures hain.
Bina decimal point wale whole number ke aakhri zeros significant nahi hote. Is convention se $5600 m$ mein do significant figures hain.
Decimal point ke baad aakhri zeros significant hote hain. $2.50$ mein teen significant figures hain aur $0.07100$ mein chaar.

4/5Aakhir ka zero, decimal point ke baad, naapa gaya tha: yeh gina jaata hai. Total: 4.

Fig. 1.7Shuru ke zeros kabhi nahi gine jaate. Beech ke zeros hamesha gine jaate hain. Aakhri zeros sirf decimal point ke baad gine jaate hain.

Number 0.005070 bade digits mein likha hai, har box mein ek. Doosre step mein pehle teen zeros fade hokar cross ho jaate hain aur unpar place-holders label hai, kyunki woh sirf decimal point ki jagah batate hain. Teesre step mein 5, 0 aur 7 wale boxes green ho jaate hain aur label kehta hai ki yeh gine jaate hain, kyunki zero non-zero digits ke beech hai. Chauthe step mein aakhri zero bhi green ho jaata hai, kyunki woh decimal point ke baad aata hai, aur total 4 significant figures padhta hai. Paanchve step mein whole number 5070 metres sirf 3 figures ke saath dikhaya gaya hai, bagal mein 5.070 times 10 cubed metres 4 figures ke saath.

Rule 4 asli information chhupa sakta hai. Ek groundsman boundary rope ko tape se 6500 cm naapta hai, nearest centimetre tak sahi, yaani chaaron digits naape gaye the. Phir bhi $6500 c m$ likhne par value mein sirf do significant figures lagte hain, kyunki uske zeros whole number ke aakhir mein hain. Unit ko metres mein badlo to wahi length $65.00 m$ padhti hai, jahan rule 5 se chaaron digits gine jaate hain. Zeros par khatam hone wala whole number yeh bata hi nahi sakta ki woh zeros naape gaye the ya nahi.

Scientific notation yeh shaq door karta hai. Value ko $a \times 10^{b}$ ki tarah likho, jahan $1 \leq a < 10$ aur $b$ ek positive ya negative whole number hai. $a$ ka har digit significant hai, aur power of ten sirf scale tay karta hai. Rope $6.500 \times 10^{3} c m$ hai, ya $6.500 \times 10^{- 2} k m$ : kisi bhi unit mein chaar significant figures. Agar tape sirf nearest metre tak padha gaya hota, to tum $6.5 \times 10^{3} c m$ likhte. Scientific notation na ho to rules 4 aur 5 par chalo.

Formulas ke kuch numbers measurements hote hi nahi. Sphere ke surface area $4 π r^{2}$ mein 4, $\frac{1}{2} m v^{2}$ mein $\frac{1}{2}$ , aur $1 m i n = 60 s$ mein 60 exact hain. Aise numbers ko unlimited significant figures wala samjho (4 = 4.000…), isliye yeh kabhi result ki precision ko limit nahi karte. $π = 3.14159 \dots$ jaise constants ko utne figures tak le sakte ho jitne data ko chahiye; aam taur par 3.14 ya 3.142 kaafi hai.

Scientific notation se size ka jaldi andaaza bhi milta hai. Agar $a \leq 5$ ho to $a$ ko 1 par round karo, aur agar $a > 5$ ho to 10 par. Tab value lagbhag ek single power of ten, $10^{n}$ , hoti hai, aur hum kehte hain ki woh $10^{n}$ ke order ki hai; exponent $n$ uska order of magnitude (परिमाण की कोटि) hai.Kuch books 5 ki jagah $\sqrt{10} \approx 3.16$ par round karti hain. Hum NCERT ka rule follow karte hain; dono tab tak same answer dete hain jab tak $a$ 3.16 aur 5 ke beech na ho. Mount Everest $8.849 \times 10^{3} m$ ooncha hai; kyunki $8.849 > 5$ , yeh $10^{4} m$ ke order ka hai. Moon $3.84 \times 10^{8} m$ door hai, $10^{8} m$ ke order ka. Isliye lagbhag $10^{8} / 10^{4} = 10^{4}$ Everests, ek ke upar ek rakhe jaayein, to Moon tak pahunch jaayenge. Fig. 1.8 kuch aur lengths ko powers of ten ki isi seedhi par dikhata hai.

3/3Earth ( $10^{7} m$ ) aur Earth–Sun distance ( $10^{11} m$ ): baal se 15 orders of magnitude aage.

Fig. 1.8Kuch lengths ke orders of magnitude. Har tick pichhle wale ka das guna hai.

Metres mein sizes ka ek horizontal axis left mein ten to the minus ten se right mein ten to the eleven tak jaata hai, har power of ten par ek tick. Pehle step mein ek atom, ek virus aur ek insaani baal ten to the minus ten, minus seven aur minus four par mark hain. Doosre step mein ek insaan aur ek cricket pitch ten to the zero aur ten to the one par dikhte hain. Teesre step mein Earth ka diameter aur Earth–Sun distance ten to the seven aur ten to the eleven par dikhte hain, aur baal se Earth–Sun distance tak ek bracket par ten to the fifteen times likha hai.

Chalo explore karein

Tumhara dil kitni baar dhadkega?

Tumhe chahiye

seconds dikhane wali ghadi ya phone

Pehle guess karo

70 saal ki zindagi mein heartbeats ki sankhya ke order of magnitude ka guess lagao.

10^{5}

10^{7}

10^{9}

10^{12}

30 seconds tak apni pulse gino aur use double karke beats per minute nikaalo.
Ek ghante ke minutes (60), ek din ke ghante (24) aur ek saal ke din (365) se multiply karo.
70 saal se multiply karo aur answer ko $a \times 10^{b}$ ki tarah likho.
Iska order of magnitude nikaalo, aur ek classmate ke answer se compare karo.

1.3.1 Significant figures ke saath arithmetic operations ke rules

Tum ek toy car ko $1.25 m$ par time karte ho aur $3.8 s$ aata hai. Calculator speed $0.328947368 m / s$ batata hai. Sirf pehle do digits ka koi matlab hai: time do figures tak pata tha, isliye speed $0.33 m / s$ hai. Calculate kiya gaya result kabhi bhi un measurements se zyada precise nahi ho sakta jinse woh bana. Do rules pakka karte hain ki use isi tarah report kiya jaaye.

PropertyMultiply aur divide karna

Result mein utne significant figures rehte hain jitne sabse kam significant figures wali measured value mein hain.

Ek rectangular plot $12.5 m$ by $8.2 m$ hai. Product $102.5 m^{2}$ hai, lekin $8.2 m$ mein sirf do significant figures hain, isliye area $1.0 \times 10^{2} m^{2}$ hai. Yahan scientific notation zaroori hai: “100” likhne se doosra significant figure chhup jaata. Isi tarah, sunlight $1.496 \times 10^{11} m$ ko $3.00 \times 10^{8} m / s$ par paar karne mein calculator se $498.67 s$ leti hai, jise teen figures tak $499 s$ report kiya jaata hai.

PropertyAdd aur subtract karna

Result mein utne decimal places rehte hain jitne sabse kam decimal places wali measured value mein hain.

Ek dukaandaar $12.6 k g$ chawal ki bori, jo $0.1 k g$ tak padhne wale platform scale par tauli gayi, ek carton mein $0.845 k g$ daal ke packet ke saath rakhta hai, jo $1 g$ tak padhne wale kitchen scale par taula gaya. Calculator $13.445 k g$ deta hai. Lekin bori ka mass kilogram ke hundredths ya thousandths ke baare mein kuch nahi batata, isliye sum ke woh columns bhi unknown hain (Fig. 1.9). Total $13.4 k g$ hai.

3/3Ek decimal place rakho: $13.4 k g$ .

Fig. 1.9Sum mein, jis column mein koi unknown digit ho, answer ka wahi column bhi unknown ho jaata hai.

Do masses, 12.6 aur 0.845 kilograms, ek ke neeche ek likhe hain aur unke decimal points ek line mein hain, aur unka sum 13.445 ek line ke neeche likha hai. Doosre step mein 12.6 ki khaali hundredths aur thousandths jagahon mein question marks aa jaate hain, aur un do columns ko shade karke unreliable label diya gaya hai. Teesre step mein sum ke aakhri do digits fade ho jaate hain aur answer 13.4 kilograms dikhta hai.

Yeh rule sach mein decimal places ka hai, significant figures ka nahi. Ek balance par ek bottle $152.37 g$ ki aati hai aur ek tablet $0.42 g$ ki. Dono gram ke hundredth tak pata hain, isliye total $152.79 g$ apne aakhri digit tak reliable hai, bhale hi $0.42 g$ mein sirf do significant figures hain; sum ko do figures tak round karna teen achhe digits ko phenk dena hoga. Subtraction ulta bhi kar sakta hai. Do clinical thermometers ${36.8}^{\circ} C$ aur ${36.6}^{\circ} C$ padhte hain, har ek mein teen significant figures, phir bhi unka difference, ${0.2}^{\circ} C$ , mein sirf ek hai. Lagbhag barabar do measurements ka chhota difference hamesha, relatively, kisi bhi ek measurement se kahin kam certain hota hai.

1.3.2 Uncertain digits ko round off karna

Calculation aam taur par utne digits se zyada deta hai jitne tum rakh sakte ho, isliye result ko round karna padta hai. Jab chhoda jaane wala hissa aakhri rakhi jagah ki aadhi unit se saaf taur par zyada ya kam ho, to rule wahi jaana-pehchaana hai: teen significant figures tak, $4.837$ ban jaata hai $4.84$ aur $4.832$ ban jaata hai $4.83$ .

Dilchasp case tab hai jab chhoda jaane wala digit exactly 5 ho, jo theek beech mein hai. Aisi values ko hamesha upar round karne se results average mein upar khisak jaayenge. Convention yeh hai: agar 5 se pehle wala digit even hai, to use waise hi chhod do; agar odd hai, to use ek badha do. Isliye $6.125$ round hokar $6.12$ banta hai (2 even hai) aur $6.135$ round hokar $6.14$ (3 odd hai). Bahut saari roundings mein aadhe ties neeche jaate hain aur aadhe upar. Tie rule sirf tab lagta hai jab chhoda gaya hissa exactly 5 ho; $6.1251$ aadhe se zyada hai aur upar round hokar $6.13$ banta hai (Fig. 1.10).

3/3 $6.125 \to 6.12$ (2 even hai) lekin $6.135 \to 6.14$ (3 odd hai).

Fig. 1.10Teen significant figures tak rounding. Ties even padosi ki taraf jaate hain.

Ek number line 6.12 se 6.14 tak jaati hai, har thousandth par chhote ticks aur 6.125 aur 6.135 par dashed halfway marks ke saath. Pehle step mein value 6.123 left mein 6.12 par kood jaati hai, kyunki aadhe se kam chhoda gaya. Doosre step mein 6.128 right mein 6.13 par kood jaati hai, kyunki aadhe se zyada chhoda gaya. Teesre step mein do halfway values dikhti hain: 6.125 neeche 6.12 par jaati hai kyunki 2 even hai, aur 6.135 upar 6.14 par jaati hai kyunki 3 odd hai.

Do aadatein picture poori karti hain. Kai steps wali calculation mein intermediate results mein ek extra digit rakho aur sirf final answer ko round karo; Section 1.3.3 dikhata hai ki warna kya galat hota hai. Aur constants ko sensible figures tak use karo: Avogadro constant $6.02214076 \times 10^{23} m o l^{- 1}$ ko aam taur par $6.02 \times 10^{23} m o l^{- 1}$ liya jaata hai, aur $π$ ko 3.14 ya 3.142, taaki data ki precision se match kare. Neeche wala inspector tumhari type ki hui kisi bhi value ke significant figures ginta hai aur use round karta hai, theek inhi rules se.

InteractiveSignificant-figure inspector

Measured value ke liye ek text box, 0.005070, 3200 aur 6.125 jaise example buttons ke saath. Har digit apne box mein dikhta hai: gine jaane wale digits ki solid outline hai aur counts likha hai, shuru ke zeros dashed hain aur unpar place label hai, aur shaq wale aakhri zeros dotted hain aur unpar question mark hai. Cards significant figures ki sankhya, scientific notation mein value aur uska order of magnitude dikhate hain, saath mein use kiye gaye rules samjhane wala note. Ek menu value ko kam significant figures tak round karta hai aur rounding samjhata hai, exact 5 ke liye even-digit rule bhi.

Fig. 1.11Ek measured value type karo ya koi example chuno. Inspector gine jaane wale digits ko mark karta hai, value ko scientific notation mein likhta hai, uska order of magnitude nikaalta hai aur use round karta hai.

Example 1.1NCERT Example 1.1 · p. 6

Ball bearing ka surface area aur volume

Ek steel ball bearing ka radius $0.6150 c m$ hai, jo screw gauge se naapa gaya. Iska surface area aur volume sahi significant figures tak nikaalo.

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Example 1.2NCERT Example 1.2 · p. 6

Marathon runner ki average speed

Ek runner marathon, $42.195 k m$ , $2.1 h$ mein poori karti hai (sirf ghante ke nearest tenth tak time kiya gaya). Uski average speed sahi significant figures ke saath likho.

Pehle khud try karo. Phir solution ek-ek step karke kholo.

1.3.3 Arithmetic calculations ke results mein uncertainty nikaalne ke rules

Significant figures precision ka ek rough guide hain. Jab hume har measurement ki uncertainty pata ho, to hum isse behtar kar sakte hain aur calculate kiye gaye result ki uncertainty ka estimate laga sakte hain. Teen observations mein woh sab aa jaata hai jo tumhe chahiye.

(1) Product mein relative errors add hote hain. Mass $m = 2.50 \pm 0.01 k g$ ki ek trolley $v = 1.20 \pm 0.02 m / s$ par chalti hai. Har value ka relative error (आपेक्षिक त्रुटि) uski uncertainty divided by value hai: mass ke liye $0.01 / 2.50 = 0.4 %$ aur speed ke liye $0.02 / 1.20 = 1.7 %$ . Tab momentum $p = m v = 3.00 k g m / s$ mein lagbhag $0.4 % + 1.7 % = 2.1 %$ ka relative error hota hai, jo $0.06 k g m / s$ hai. Isliye hum $p = 3.00 \pm 0.06 k g m / s$ report karte hain. Fig. 1.12 dikhata hai ki yeh rule kahan se aata hai, sabse aasaan product ki picture se, yaani rectangle ka area.

3/3Corner $Δ l Δ b$ negligible hai, isliye $Δ A / A = Δ l / l + Δ b / b$ .

Fig. 1.12Extra area lagbhag poora do strips se aata hai; corner do chhoti quantities ka product hai.

Ek green rectangle jiski length l neeche aur breadth b side mein hai, uska area A barabar l b hai. Doosre step mein right side ke saath width delta l ki ek patli strip judti hai, jiska area b delta l hai, aur upar height delta b ki ek patli strip, jiska area l delta b hai. Teesre step mein area delta l times delta b wala chhota corner square negligible ke roop mein highlight hota hai, aur result delta A over A barabar delta l over l plus delta b over b dikhta hai.

DerivationProduct ka relative error

Maan lo asli length aur breadth $l \pm Δ l$ aur $b \pm Δ b$ ke andar hain. Area zyada se zyada itna ho sakta hai

\begin{aligned} (l + Δ l) (b + Δ b) \\ = l b + b Δ l \\ + l Δ b + Δ l Δ b \end{aligned}

Aakhri term do chhoti quantities ka product hai. $0.1 c m$ ke errors ke liye yeh $0.01 c m^{2}$ hai, baaki terms ke saamne negligible. Isliye area mein sabse bada error hai

Δ A = b Δ l + l Δ b

(1.4)

$A = l b$ se divide karne par rule apne sabse kaam ke form mein milta hai:

Relative errors add hote hain

\frac{Δ A}{A} = \frac{Δ l}{l} + \frac{Δ b}{b}

(1.5)

Yahi result quotient $l / b$ ke liye bhi sahi hai, kyunki worst case mein ek value zyada badi aur doosri zyada chhoti hoti hai. Square ke liye, $l^{2} = l \times l$ , isliye relative error double ho jaata hai; cube ke liye teen guna.

(2) Sums aur differences mein absolute errors add hote hain. Length $25.4 \pm 0.1 c m$ ki ek rod se $3.86 \pm 0.01 c m$ kaata jaata hai. Bacha tukda zyada se zyada $25.5 - 3.85 = 21.65 c m$ lamba ho sakta hai ya kam se kam $25.3 - 3.87 = 21.43 c m$ , isliye yeh $21.54 \pm 0.11 c m$ hai: sirf $21.5 c m$ likhne layak hai. Badi uncertainty haavi rehti hai, isiliye sums aur differences ko sabse kam decimal places tak round kiya jaata hai, aur isiliye difference mein kisi bhi value se kam significant figures ho sakte hain.

(3) Barabar significant figures ka matlab barabar precision nahi hai. Ek stopwatch ek sprint start ke liye $2.05 s$ aur ek chhoti race ke liye $8.75 s$ padhti hai, dono $\pm 0.01 s$ tak aur dono teen significant figures ke saath. Pehle ka relative error $0.01 / 2.05 \approx 0.5 %$ hai, doosre ka sirf $0.01 / 8.75 \approx 0.1 %$ . Significant figures ginna ek rough guide hai; relative error precision ka exact measure hai.

Aakhir mein, yeh rahi wajah ki tum intermediate steps mein ek extra digit kyun rakhte ho. Edge $1.27 m$ wale cube ka volume do steps mein nikaalo. Ek face ka area $1.6129 m^{2}$ hai; ise $1.61 m^{2}$ par round karo, edge se multiply karo, aur tumhe $2.0447 \approx 2.04 m^{3}$ milta hai. Lekin asli volume, ${1.27}^{3} = 2.048 m^{3}$ , round hokar $2.05 m^{3}$ banta hai. Beech wale step mein chaar figures rakho, $1.613 m^{2}$ , aur product $2.0485$ sahi tarah $2.05 m^{3}$ par round hota hai.

Quick check

4 mein se 1

3.6 c m \times 1.452 c m

ko sahi significant figures tak likho.

5.2 c m^{2}

5.2272 c m^{2}

5.23 c m^{2}

5 c m^{2}

Answer dekho

Answer (A) $5.2 c m^{2}$

3.6

mein do significant figures hain, sabse kam, isliye product

5.2272

5.2 c m^{2}

report kiya jaata hai. Teen figures (5.23) rakhna kam precise factor ki jagah zyada precise factor ko follow karna hai.

1.4 Physical quantities ke dimensions

Do quantities size aur units mein bahut alag ho sakti hain aur phir bhi ek hi kism ki cheez ho sakti hain. Green light ki wavelength aur Mount Everest ki height, dono lengths hain. Snail ki speed aur light ki speed, dono length divided by time hain. Dimensions ka idea isi kism ko pakadta hai, aur size ko ignore karta hai.

Har physical quantity saat SI base quantities se banai ja sakti hai, isliye hum in saat ko basic dimensions maante hain aur har ek ko square brackets mein likhte hain: mass $[M]$ , length $[L]$ , time $[T]$ , amount of substance $[m o l]$ , electric current $[A]$ , temperature $[K]$ aur luminous intensity $[c d]$ . Kisi quantity ko base quantities ke product ki tarah likho; har ek par jo exponent hai woh us base quantity mein uska dimension (विमा) hai, aur yeh saare exponents milkar quantity ke dimensions hain. Kisi symbol ke around square brackets, jaise $[v]$ , ko “ $v$ ke dimensions” padha jaata hai.

Mechanics mein inmein se teen kaafi hain: $[M]$ , $[L]$ aur $[T]$ . Kisi mechanical quantity ke dimensions nikaalne ke liye uski definition ko peeche mass, length aur time tak le jaao. Momentum mass times velocity hai, aur velocity length per unit time hai, isliye

Momentum ke dimensions

[p] = [M] \frac{[L]}{[T]} = [M L T^{- 1}]

Exponents seedhe brackets se padh lo: momentum mass aur length mein first power ka hai, aur time mein power $- 1$ ka. Force, yaani mass times acceleration, $[M L T^{- 2}]$ aata hai. Area, length times length, $[L^{2}]$ hai; yeh zor dene ke liye ki mass aur time ka koi role nahi, hum ise $[M^{0} L^{2} T^{0}]$ likh sakte hain. Do cheezein jaan-boojh kar chhodi gayi hain. Size nahi aata: ek din mein ek metre khisakne wala glacier aur rifle ki goli, dono dimensions $[L T^{- 1}]$ ke saath chalte hain. Na hi quantity ka khaas matlab aata hai, jaise tum Section 1.6.2 mein dekhoge. Fig. 1.13 chaar common quantities ke exponents ki picture dikhata hai.

3/4Mass se multiply karo: $[M L T^{- 2}]$ .

Fig. 1.13Dimensions powers hain. Line ke upar ke bars positive powers hain, neeche ke negative.

M, L aur T label wale teen bars ek zero line par khade hain, plus aur minus one aur two par grid lines ke saath. Pehle step mein, speed ke liye, M 0 hai, L plus 1 aur T minus 1. Doosre step mein, acceleration, T bar neeche minus 2 tak badh jaata hai. Teesre step mein, force, M bar plus 1 tak uthta hai jabki L plus 1 aur T minus 2 rehte hain. Chauthe step mein, pressure, L bar palat kar minus 1 par aa jaata hai, yaani M plus 1, L minus 1 aur T minus 2. Har step mein upar quantity ka naam aur dimensional formula dikhta hai.

1.5 Dimensional formulae aur dimensional equations

Har derived quantity ek relation se define hoti hai, aur woh relation uske dimensions tay karta hai. Power ko lo, work karne ki rate: $P = W / t$ , jahan work $W$ force times distance hai. Har symbol ko uske dimensions se replace karo aur har base quantity ke exponents jodo (Fig. 1.14). Result, $[M L^{2} T^{- 3}]$ , power ka dimensional formula (विमीय सूत्र) hai: shaamil base quantities ki ek chhoti list, har ek apne exponent ke saath. Equation ki tarah likhne par, $[P] = [M L^{2} T^{- 3}]$ , yeh ek dimensional equation (विमीय समीकरण) ban jaata hai. Teen aur yeh rahe:

Dimensional equations

\begin{aligned} [a] & = [M^{0} L T^{- 2}] \\ [p] & = [M L T^{- 1}] \\ [ρ] & = [M L^{- 3} T^{0}] \end{aligned}

(1.6)

Table 1.5 un quantities ke dimensional formulae ikattha karta hai jo tum is book mein sabse zyada use karoge. Ise yaad karne ki zaroorat nahi: har row apne defining relation se ek-do line mein nikal aati hai.

3/3M, L aur T ki powers jodo: $[P] = [M L^{2} T^{- 3}]$ .

Fig. 1.14Dimensional equation likhna: relation, substitute, powers jodo.

Teen boxes ek ke upar ek rakhe hain aur unke beech arrows hain. Pehle step mein upar wala box relation P barabar W over t barabar F d over t dikhata hai. Doosre step mein beech wala box har symbol ko uske dimensions se replace karta hai: M L T to the minus 2 times L, divided by T. Teesre step mein neeche wala box result M L squared T to the minus 3 deta hai, aur ek side panel powers jodta hai: M 1 hai, L 1 plus 1 barabar 2 hai, aur T minus 2 minus 1 barabar minus 3 hai.

Table 1.5Kuch common quantities ke dimensional formulae. Ek jaisi quantities ke ratios dimensionless hote hain.

Quantity	Defining relation	SI unit	Dimensional formula
Area	length × breadth	$m^{2}$	$[L^{2}]$
Volume	length × breadth × height	$m^{3}$	$[L^{3}]$
Density	mass ÷ volume	$k g m^{- 3}$	$[M L^{- 3}]$
Velocity, speed	displacement ÷ time	$m s^{- 1}$	$[L T^{- 1}]$
Acceleration	velocity ÷ time	$m s^{- 2}$	$[L T^{- 2}]$
Momentum, impulse	mass × velocity	$k g m s^{- 1}$	$[M L T^{- 1}]$
Force	mass × acceleration	N	$[M L T^{- 2}]$
Work, energy, torque	force × distance	J (torque ke liye N m)	$[M L^{2} T^{- 2}]$
Power	work ÷ time	W	$[M L^{2} T^{- 3}]$
Pressure, stress	force ÷ area	Pa	$[M L^{- 1} T^{- 2}]$
Frequency	1 ÷ period	Hz	$[T^{- 1}]$
Angle, strain, refractive index	ek jaisi quantities ka ratio	rad, ya koi nahi	$[M^{0} L^{0} T^{0}]$
Spring constant	force ÷ extension	$N m^{- 1}$	$[M T^{- 2}]$
Gravitational constant	$F r^{2} / (m_{1} m_{2})$	$N m^{2} k g^{- 2}$	$[M^{- 1} L^{3} T^{- 2}]$
Planck constant, angular momentum	energy ÷ frequency	J s	$[M L^{2} T^{- 1}]$
Electric charge	current × time	C	$[A T]$

Example

Defining relation se dimensional formula

Shahad ko hilaana mushkil hai kyunki uski layers ek doosre par slide karne ka virodh karti hain. Zaroori force $F = η A \frac{Δ v}{Δ x}$ hai, jahan $A$ ek layer ka area hai, $Δ v$ distance $Δ x$ door do layers ki speed ka difference hai, aur $η$ liquid ki ek property hai jise viscosity kehte hain. $η$ ka dimensional formula aur SI unit nikaalo.

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Quick check

3 mein se 1

Har quantity ko uske dimensional formula se match karo.

Left mein koi item tap karo, phir right mein uska match.

Answer dekho

Answer

power : $[M L^{2} T^{- 3}]$
pressure : $[M L^{- 1} T^{- 2}]$
momentum : $[M L T^{- 1}]$
frequency : $[T^{- 1}]$

Power energy per time hai, pressure force per area hai, momentum mass × velocity hai aur frequency ek time ka reciprocal hai.

1.6 Dimensional analysis aur uske applications

Dimensions sirf labels nahi hain. Sirf ek hi kism ki quantities ko add, subtract ya compare kiya ja sakta hai, isliye dimensions is par sakht limits lagaate hain ki ek sahi equation kaisi dikh sakti hai. In limits ko use karna dimensional analysis (विमीय विश्लेषण) kehlata hai. Isse hum equations check kar sakte hain, units convert kar sakte hain, aur kabhi-kabhi poori theory hone se pehle hi kisi relation ka form predict kar sakte hain.

Kisi bhi calculation mein units ko numbers ke saath le kar chalo aur unhe symbols ki tarah cancel hone do: numerator ke hours denominator ke hours ko cancel karte hain, isliye $(k m / h) \times h = k m$ . Dimensions bhi aise hi behave karte hain, $[M L T^{- 2}] / [M] = [L T^{- 2}]$ . Units ko is tarah handle karna unhe convert karne ka sabse safe tareeka bhi hai: har unit ko nayi units mein uski value se replace karo, phir simplify karo.

Example

Pressure ko CGS units mein convert karna

Atmospheric pressure lagbhag $1.0 \times 10^{5} P a$ hai. Ise pressure ki CGS unit, dyne per square centimetre, mein likho, jahan $1 d y n = 1 g c m s^{- 2}$ .

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Yahi idea ek general recipe deta hai. Maan lo kisi quantity ke dimensions $[M^{a} L^{b} T^{c}]$ hain aur ek system mein uska numerical measure $n_{1}$ hai, jiski mass, length aur time ki units $M_{1}$ , $L_{1}$ aur $T_{1}$ hain. Har base unit par $n_{1} u_{1} = n_{2} u_{2}$ ((1.1)) lagaane par, doosre system mein uska measure hai

Systems ke beech convert karna

\begin{aligned} n_{2} & = n_{1} {(\frac{M_{1}}{M_{2}})}^{a} \\ \times {(\frac{L_{1}}{L_{2}})}^{b} {(\frac{T_{1}}{T_{2}})}^{c} \end{aligned}

(1.7)

1.6.1 Equations ki dimensional consistency check karna

Paanch kilograms plus teen seconds kisi bhi cheez ke aath nahi hote. Sum, difference ya equality sirf ek hi kism ki quantities ke beech matlab rakhti hai, isliye sahi physical equation ke har term ke dimensions same hone chahiye.

PrinciplePrinciple of homogeneity of dimensions

Physically meaningful equation mein dono sides ke har term ke dimensions same hote hain.

Principle of homogeneity lagaana jaldi ho jaata hai. Constant acceleration $a$ se chal rahi kisi body ke liye ek student ko $v^{2} = u^{2} + 2 a s$ yaad hai. Har term ek squared speed honi chahiye, $[L^{2} T^{- 2}]$ . Terms $v^{2}$ aur $u^{2}$ saaf taur par hain, aur $[a s] = [L T^{- 2}] [L] = [L^{2} T^{- 2}]$ , isliye equation pass ho jaati hai. Agar usne $v = u + a t^{2}$ likha hota, to aakhri term ke dimensions $[L]$ hote jabki baakiyon ke $[L T^{- 1}]$ , aur equation fail ho jaati (Fig. 1.15).

3/3 $v = u + a t^{2}$ mein aakhri term $[L]$ hai, $[L T^{- 1}]$ nahi: inconsistent, isliye yeh zaroor galat hai.

Fig. 1.15Equation ke har term ke dimensions same hone chahiye.

Equation v squared barabar u squared plus 2 a s likhi hai, har term ek box mein. Doosre step mein teeno terms ke neeche dimensions L squared T to the minus 2 dikhte hain, boxes green ho jaate hain aur label consistent padhta hai. Teesre step mein neeche ek doosri equation, v barabar u plus a t squared, dikhti hai; uske pehle do terms ke dimensions L T to the minus 1 hain lekin aakhri ke dimensions L hain, isliye uska box dashed aur orange hai aur label inconsistent padhta hai.

Dimensions check karna asal mein units check karna hai, lekin aise form mein jo ek saath har system mein kaam karta hai: tum metres mein socho ya feet mein, grams mein ya kilograms mein, length $[L]$ hi hai. Koi conversion factors nahi chahiye, isliye yeh kisi bhi naye result ka sabse jaldi wala pehla test hai.

Kuch quantities ke koi dimensions hote hi nahi. Do ek jaisi quantities ka ratio ek pure number hai: angle (arc ÷ radius), strain (extension ÷ original length), relative density (ek density ÷ doosri). Isi tarah, $\sin x$ , $\log x$ ya $e^{x}$ jaise function ka argument dimensionless hona chahiye. Aise functions apne argument ki powers ke sums se bante hain, jaise $e^{x} = 1 + x + x^{2} / 2 + \dots$ , aur $x$ , $x^{2}$ aur $x^{3}$ ko tabhi add kiya ja sakta hai jab $x$ ek pure number ho. Isliye $y = A \sin ω t$ mein $ω t$ ke koi dimensions nahi hain, to $[ω] = [T^{- 1}]$ .

Is test ki ek kamzori hai: pure-number factors ise dikhte hi nahi. Equations $v^{2} = u^{2} + 2 a s$ , $v^{2} = u^{2} + a s$ aur $v^{2} = u^{2} + 4 a s$ teeno pass ho jaati hain, haalaanki sirf pehli sahi hai. Test mein fail hona prove karta hai ki equation galat hai; pass hona sirf yeh dikhata hai ki equation sahi ho sakti hai.

Example 1.3NCERT Example 1.3 · p. 8

Behte liquids ke liye ek relation check karna

Pipe mein smoothly beh rahe liquid ke liye ek book $P + \frac{1}{2} ρ v^{2} + ρ g h = constant$ deti hai, jahan $P$ pressure hai, $ρ$ density, $v$ speed, $g$ acceleration due to gravity aur $h$ height. Kya yeh relation dimensionally consistent hai?

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Example 1.4NCERT Example 1.4 · p. 9

Sound ki speed ke formulas ko rule out karna

Sound ek gas mein speed $v$ se chalti hai jo gas ke pressure $P$ aur uski density $ρ$ par depend karti hai. Paanch formulas suggest kiye gaye hain, jahan $γ$ ek pure number hai:

(a) $v = \sqrt{P / ρ}$
(b) $v = \sqrt{ρ / P}$
(c) $v = P / ρ$
(d) $v = \sqrt{γ P / ρ}$
(e) $v = \sqrt{P / ρ} + P / ρ$

Sirf dimensions se tum kaun se rule out kar sakte ho?

Pehle khud try karo. Phir solution ek-ek step karke kholo.

1.6.2 Physical quantities ke beech relation nikaalna

Dimensions sirf check karne se zyada kar sakte hain. Maan lo hume experiment ya ek achhe physical guess se pata hai ki result kin quantities par depend karta hai. Maan lo result un quantities ki powers ka product hai, ek dimensionless constant se guna, aur maang karo ki dono sides ke dimensions same hon. Mechanics mein isse teen equations milti hain, M, L aur T ke liye ek-ek, isliye yeh method zyada se zyada teen unknown powers tay kar sakta hai.

Example 1.5NCERT Example 1.5 · p. 9

Kheenchi hui string par wave ki speed

Sitar ki string ko chhedo to ek wave uske saath daudti hai. Maan lo wave ki speed $v$ string ke tension $F$ , uske mass per unit length $μ$ , aur uski length $l$ par depend karti hai. Dimensions se relation ka form nikaalo.

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Neeche wala exponent finder tumhe yeh method paanch aur situations par try karne deta hai. Powers ko tab tak badlo jab tak dimensions match na ho jaayein, aur dhyaan do ki kaun si quantities matter hi nahi karti.

InteractiveExponent finder

Ek menu paanch situations deta hai: kheenchi hui string par wave, oonchai se gira patthar, spring par uchhalta mass, gehre paani par lehrein aur tez ball par air drag. Har ek ke liye teen quantities unke dimensions ke saath listed hain, aur buttons har power ko aadhe-aadhe steps mein badhaate ya ghataate hain. Ek table M, L aur T ki chahiye wali powers ko tumhare product ki powers se compare karta hai, current formula dikhta hai, aur ek message batata hai ki ab bhi kya alag hai ya match confirm karta hai aur us quantity ka naam batata hai jo hat jaati hai. Ek button powers ki teen equations dikhata hai.

Fig. 1.16Ek situation chuno, phir har quantity ki power ko tab tak badhao ya ghatao jab tak dono sides ke dimensions same na ho jaayein.

Dimensional analysis powerful hai, lekin iski saaf limits hain:

Yeh dimensionless constants nahi nikaal sakta, jaise upar wala $k$ , $\frac{1}{2} m v^{2}$ mein $\frac{1}{2}$ ya kisi period mein $2 π$ .
Yeh same dimensions wali quantities mein fark nahi kar sakta, jaise work aur torque, dono $[M L^{2} T^{- 2}]$ .
Yeh sirf product-of-powers forms dhoondhta hai. Yeh $s = u t + \frac{1}{2} a t^{2}$ jaise terms ke sum ke liye fail ho jaata hai, aur $\sin$ , $\log$ ya $e^{x}$ wale relations ke liye bhi.
Mechanics mein yeh zyada se zyada teen unknown powers tay kar sakta hai, kyunki compare karne ke liye sirf teen base dimensions hain.
Yeh nahi bata sakta ki shuru mein kaun si quantities matter karti hain; uske liye physical samajh ya experiment chahiye.

Recap

Summary aur key points

Measurement ek quantity ko ek chune hue standard, uski unit, se compare karta hai; result numerical measure times unit hota hai, aur $n_{1} u_{1} = n_{2} u_{2}$ (Section 1.1).
Base quantities ki base units hoti hain; baaki saari units unse derive hoti hain, aur sab milkar ek system of units banati hain (Section 1.1).
SI mein saat base units hain (m, kg, s, A, K, mol, cd), jo 2019 se constants of nature ki exact values se define hoti hain; CGS, FPS aur MKS pehle ke systems the (Section 1.2).
Plane angle $θ = s / r$ radians mein naapa jaata hai aur solid angle $d Ω = d A / r^{2}$ steradians mein; dono dimensionless hain (Plane angle aur solid angle).
Kuch non-SI units, jaise litre, hectare aur electronvolt, abhi bhi use hoti hain; prefixes aur unit symbols fixed rules follow karte hain (Section 1.2).
Significant figures reliable digits plus pehla uncertain digit hain; unki sankhya instrument par depend karti hai, unit par nahi (Section 1.3).
Scientific notation, $a \times 10^{b}$ , aakhri zeros ka confusion door karta hai; order of magnitude sabse paas wali power of ten ka exponent hai (Section 1.3).
Products aur quotients sabse kam significant figures rakhte hain; sums aur differences sabse kam decimal places rakhte hain (Section 1.3.1).
Aadhe se zyada chhoda gaya hissa aakhri digit ko badha deta hai aur aadhe se kam use waisa hi chhod deta hai; exact 5 sirf odd digit ko badhata hai. Intermediate steps mein ek extra digit rakho (Section 1.3.2).
Product ya quotient mein relative errors add hote hain; subtraction significant figures ki sankhya ko kaafi ghata sakta hai (Section 1.3.3).
Dimensions base quantities ki woh powers hain jinse koi quantity banti hai; force ka dimensional formula $[M L T^{- 2}]$ hai (Section 1.4, Section 1.5).
Units aur dimensions ordinary algebra follow karte hain, jisse unit conversion aasaan ho jaata hai (Section 1.6).
Principle of homogeneity se valid equation ke har term ke dimensions same hote hain; dimensionally galat equation galat hai, lekin consistent equation bhi galat ho sakti hai (Section 1.6.1).
Dimensional analysis product-of-powers relations nikaal sakta hai, lekin dimensionless constants, terms ke sums ya same dimensions wali quantities ka fark nahi (Section 1.6.2).

Flash cards · 10

Yaad rakhne wale dimensional formulae

1 / 10

Apna NCERT exercise question dhoondo

Hamare practice questions ncert.ai ke liye likhe gaye hain aur NCERT exercises ko follow karte hain. Apni textbook ke question se match karne wala question dhoondo; uska answer uske saath hai.

Tumhari NCERT textbook mein	Yahan practice question
NCERT Exercise 1.1	Practice Q1
NCERT Exercise 1.2	Practice Q2
NCERT Exercise 1.3	Practice Q3
NCERT Exercise 1.4	Practice Q4
NCERT Exercise 1.5	Practice Q5
NCERT Exercise 1.6	Practice Q6
NCERT Exercise 1.7	Practice Q7
NCERT Exercise 1.8	Practice Q8
NCERT Exercise 1.9	Practice Q9
NCERT Exercise 1.10	Practice Q10
NCERT Exercise 1.11	Practice Q11
NCERT Exercise 1.12	Practice Q12
NCERT Exercise 1.13	Practice Q13
NCERT Exercise 1.14	Practice Q14
NCERT Exercise 1.15	Practice Q15
NCERT Exercise 1.16	Practice Q16
NCERT Exercise 1.17	Practice Q17

Quick check: MCQs

1.1 Introduction3 questions
Plane angle aur solid angle4 questions
1.3 Significant figures4 questions
1.3.3 Arithmetic calculations ke results mein uncertainty nikaalne ke rules4 questions
1.4 Physical quantities ke dimensions3 questions
1.5 Dimensional formulae aur dimensional equations3 questions
1.6.2 Physical quantities ke beech relation nikaalna4 questions

Glossary · 15

Key terms (English + Hindi)

base quantityमूल राशिmool raashi: SI mein independent chuni gayi saat quantities mein se ek, jaise length, mass aur time.Chapter mein pehli baar yahan aaya →

derived unitव्युत्पन्न मात्रकvyutpann maatrak: Base units ko multiply aur divide karke bani unit, jaise $m s^{- 1}$ ya newton.Chapter mein pehli baar yahan aaya →
dimensional equationविमीय समीकरणvimiya sameekaran: Aisi equation jo quantity ko uske dimensional formula ke barabar rakhti hai, jaise $[F] = [M L T^{- 2}]$ .Chapter mein pehli baar yahan aaya →
dimensional formulaविमीय सूत्रvimiya sootra: $[M L T^{- 2}]$ jaisa expression jo dikhata hai ki kaun si base quantities, kin powers ke saath, milkar ek quantity banati hain.Chapter mein pehli baar yahan aaya →
dimensionsविमाएँvimaayen: Woh exponents jo har base quantity par hote hain jab physical quantity ko base quantities ke product ki tarah likha jaata hai.Chapter mein pehli baar yahan aaya →

least countअल्पतमांकalpatamaank: Sabse chhoti value jo koi instrument seedhe padh sakta hai, jaise school ruler par 1 mm.Chapter mein pehli baar yahan aaya →

order of magnitudeपरिमाण की कोटिparimaan ki koti: Value ke sabse paas wali power of ten ka exponent: $a \times 10^{b}$ ka order of magnitude $b$ hai agar $a \leq 5$ , aur $b + 1$ agar $a > 5$ .Chapter mein pehli baar yahan aaya →

principle of homogeneityविमीय समांगता का सिद्धांतvimiya samaangata ka siddhaant: Valid physical equation mein add, subtract ya equate hone wale har term ke dimensions same hote hain.Chapter mein pehli baar yahan aaya →

radianरेडियनradian: Plane angle ki SI unit: circle ke centre par woh angle jiska arc radius ke barabar ho; $θ = s / r$ .Chapter mein pehli baar yahan aaya →
relative errorआपेक्षिक त्रुटिaapekshik truti: Value ki uncertainty divided by value; 100 se multiply karne par yeh percentage error hai.Chapter mein pehli baar yahan aaya →

scientific notationवैज्ञानिक संकेतनvaigyanik sanketan: Number ko $a \times 10^{b}$ ki tarah likhna, jahan $1 \leq a < 10$ ; $a$ ka har digit significant hai.Chapter mein pehli baar yahan aaya →
significant figuresसार्थक अंकsaarthak ank: Measured value ke woh digits jo reliably pata hain, plus pehla uncertain digit.Chapter mein pehli baar yahan aaya →
steradianस्टेरेडियनsteradian: Solid angle ki SI unit: sphere ke centre se nikla woh cone jo uski surface ka $r^{2}$ area kaatta hai; $Ω = A / r^{2}$ .Chapter mein pehli baar yahan aaya →
system of unitsमात्रकों की पद्धतिmaatrakon ki paddhati: Base units ka poora set, unse bani derived units ke saath, jaise CGS, FPS, MKS ya SI.Chapter mein pehli baar yahan aaya →

unitमात्रकmaatrak: Kisi quantity ki ek fixed amount, jis par poori duniya sehmat hai, aur jisse usi quantity ki doosri amounts compare ki jaati hain.Chapter mein pehli baar yahan aaya →

Units and Measurement: Class 11 Physics Chapter 1 Hinglish mein

Is chapter mein aap kya seekhenge

1.1 Introduction

1.2 International system of units

Plane angle aur solid angle

1.3 Significant figures

1.3.1 Significant figures ke saath arithmetic operations ke rules

1.3.2 Uncertain digits ko round off karna

1.3.3 Arithmetic calculations ke results mein uncertainty nikaalne ke rules

1.4 Physical quantities ke dimensions

1.5 Dimensional formulae aur dimensional equations

1.6 Dimensional analysis aur uske applications

1.6.1 Equations ki dimensional consistency check karna

1.6.2 Physical quantities ke beech relation nikaalna

Summary aur key points

Practice questions answers ke saath (NCERT “Exercises”)

Apna NCERT exercise question dhoondo

Quick check: MCQs

Key terms (English + Hindi)

Tum chapter ke end tak pahunch gaye