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C HAP TE RPhysics HandBook
ALLEN
Units, Dimension, Measurements and
Practical Physics
Fundamental or base quantities Systems of Units
MKS CGS FPS MKSQ MKSA
The quantities which do not depend upon other
(i) Length Length Length Length Length
quantities for their complete definition are known (m) (cm) (ft) (m) (m)
as fundamental or base quantities. (ii) Mass Mass Mass Mass Mass
e.g. : length, mass, time, etc. (kg) (g) (pound) (kg) (kg)
(iii) Time Time Time Time Time
(s) (s) (s) (s) (s)
Derived quantities (iv) – – – Charge Current
The quantities which can be expressed in terms (Q) (A)
of the fundamental quantities are known as derived Fundamental Quantities in
quantities. S.I. System and their units
e.g. Speed (=distance/time), volume,
S.N. Physical Qty. Name of Unit Symbol
acceleration, force, pressure, etc. 1 Mass kilogram kg
2 Length meter m
Units of physical quantities 3 Time second s
4 Temperature kelvin K
The chosen reference standard of measurement
5 Luminous intensity candela Cd
in multiples of which, a physical quantity is 6 Electric current ampere A
expressed is called the unit of that quantity. 7 Amount of substance mole mol
e.g. Physical Quantity = Numerical Value × Unit
SI Base Quantities and Units
S I U n its
B a s e Q u a n tity
N am e Sym bo l D efin itio n
L e n gth m e te r m T h e m eter is the le ng th o f th e pa th tra vele d b y lig h t in
va cuum durin g a tim e in te rva l o f 1/(2 99 , 7 92 , 4 58 ) o f a
seco nd (1 9 8 3 )
M ass kilogra kg T h e kilogra m is e qual to th e m ass o f the in tern atio n al
m p roto typ e o f th e k ilogra m (a platin um -iridium a lloy
cy lin d e r) ke p t a t In tern atio n al B u rea u o f W eig h ts an d
M ea su re s, a t S e vres, ne ar P a ris, Fra nc e. (1 8 8 9)
T im e se co n d s T h e se co n d is th e dura tio n o f 9 , 1 9 2 , 6 31 , 77 0 pe rio ds
o f the rad iatio n c orre spo nd in g to th e tran sition b etw e e n
th e tw o h yp erfin e leve ls o f th e g ro u n d sta te of the
ce sium -1 3 3 a to m (1 9 6 7 )
E lectric C urren t am pe re A T h e a m p e re is th at c o n stan t cu rre nt w h ic h , if m a in ta in e d
node06\B0AI-B0\Kota\JEE(Advanced)\Leader\Phy\Sheet\Hand book (E+L)\Eng\01_Unit & Dimension.p65
in tw o straig h t pa ralle l c o n du cto rs o f infin ite le ng th , of
n egligible circular cro ss-section, an d p laced 1 m etre ap art
in vacuum , w o uld prod uce be tw e en these con du ctors a
fo rce e qual to 2 x 1 0 -7 N e w to n p e r m e tre o f le ng th .
(1 9 4 8 )
T h erm o d yn am ic kelvin K The k elvin, is th e fra ctio n 1 /2 7 3 .1 6 of the
T e m p era ture th erm ody n am ic tem p erature o f the triple p o in t of w a ter.
(1 9 6 7 )
A m ou nt o f m o le m ol T h e m ole is th e a m o unt o f substa nc e o f a sy stem , w h ic h
S ubsta n ce co n ta in s a s m an y elem e nta ry e ntitie s as the re a re a to m s
in 0.01 2 k ilo gram o f carb o n -1 2 . (1 9 7 1 )
L um inous ca nde la Cd T h e can dela is th e lum in o us in te nsity, in a give n dire ctio n,
Inte nsity o f a so u rce th at em its m o n oc h ro m a tic rad iatio n of
fre quen cy 5 4 0 x 1 0 1 2 h e rtz an d th at ha s a rad ia nt
inten sity in th at directio n of 1/ 6 8 3 w a tt pe r stera d ian
(1 9 7 9 ).
E 1
,Physics HandBook CH APTER
ALLEN
Supplementary Units Limitations of dimensional analysis
• Radian (rad) - for measurement of plane angle • In Mechanics the formula for a physical quantity
• Steradian (sr) - for measurement of solid angle depending on more than three other physical
quantities cannot be derived. It can only be checked.
Dimensional Formula
• This method can be used only if the dependency is
Relation which express physical quantities in terms of of multiplication type. The formulae containing
appropriate powers of fundamental units. exponential, trigonometrical and logarithmic
functions can't be derived using this method.
Use of dimensional analysis Formulae containing more than one term which
are added or subtracted like s = ut +½ at2 also
• To check the dimensional correctness of a given can't be derived.
physical relation
• To derive relationship between different physical • The relation derived from this method gives no
quantities information about the dimensionless constants.
• To convert units of a physical quantity from one • If dimensions are given, physical quantity may not
system to another be unique as many physical quantities have the same
a b c dimensions.
æM ö æL ö æT ö
n1u1= n2u2 Þ n2=n1 ç 1 ÷ ç 1 ÷ ç 1 ÷
è M2 ø è L 2 ø è T2 ø • It gives no information whether a physical quantity
is a scalar or a vector.
where u = MaLbTc
SI PREFIXES
The magnitudes of physical quantities vary over a wide range. The CGPM recommended standard prefixes for
magnitude too large or too small to be expressed more compactly for certain powers of 10.
Power of Power of
Prefix Symbol Prefix Symbol
10 10
PREFIXES 1018 exa E 10-1 deci d
USED FOR 10 15
peta P 10-2 centi c
DIFFERENT 10 12
tera T 10 -3
milli m
POWERS 10 9 giga G 10-6 micro m
OF 10 10 6
mega M 10 -9
nano n
10 3 kilo k 10-12 pico p
10 2 hecto h 10-15 femto f
1 - 18
10 deca da 10 atto a
P hysical quan tity Unit P hys ical q uantity U nit
node06\B0AI-B0\Kota\JEE(Advanced)\Leader\Phy\Sheet\Hand book (E+L)\Eng\01_Unit & Dimension.p65
A ngular acceleration rad s- 2 F requen cy h ertz
Mo ment o f in ertia kg – m 2 Resistan ce kg m 2 A- 2 s-3
Self in ductan ce h en ry Surface ten sio n n ew ton/m
UNITS
OF
Magnetic flux w eber Universal gas con stant joule K - 1 mo l-1
IMPORTANT
Po le str en gth A –m Dipole mo ment co ulo mb–meter
PHYSICAL
V iscosity po is e Stefan co n stant watt m - 2 K -4
QUANTITIES
Reactance oh m Perm ittivity o f free space (e0) co ulo mb 2 /N– m 2
P erm eability o f free space
Specific heat J/kg°C weber/A-m
(m0 )
Strength of m agnetic
n ewton A -1 m - 1 P lan ck's co nstan t jo ule –se c
field
A stronom ical dis tance Parsec En tro py J/K
2 E
, C HAP TE R
Physics HandBook
ALLEN
DIMENSIONS OF IMPORTANT PHYSICAL QUANTITIES
Physical quantity Dimensions Physical quantity Dimensions
Momentum M1 L 1 T –1 Capacitance M–1 L–2 T4 A 2
Calorie M1 L 2 T –2 Modulus of rigidity M1 L– 1 T –2
Latent heat capacity M L T– Magnetic permeability M L T – A–
0 2 2 1 1 2 2
Self inductance M L T – A– Pressure M L– T –
1 2 2 2 1 1 2
Coefficient of thermal conductivity M L T– K – Planck's constant M L T–
1 1 3 1 1 2 1
Power M L T– Solar constant M L T–
1 2 3 1 0 3
Impulse M1 L 1 T –1 Magnetic flux M1 L2 T –2 A– 1
Hole mobility in a semi conductor M– L T A Current density M L– T A
1 0 2 1 0 2 0 1
Bulk modulus of elasticity M L– T– Young modulus M L– T –
1 1 2 1 1 2
Potential energy M L T– Magnetic field intensity M L– T A
1 2 2 0 1 0 1
Gravitational constant M – L T– Magnetic Induction M T– A –
1 3 2 1 2 1
Light year M0 L1 T0 Electric Permittivity M– L– T A
1 3 4 2
Thermal resistance M–1 L–2 T3 K Electric Field M1L 1T–3A- 1
Coefficient of viscosity M1 L– 1 T–1 Resistance ML T – A–
2 3 2
SETS OF QUANTITIES HAVING SAME DIMENSIONS
S.N. Quantities Dimensions
1. Strain, refractive index, relative density, angle, solid angle, phase, distance
gradient, relative permeability, relative permittivity, angle of contact, Reynolds
[M 0 L 0 T 0]
number, coefficient of friction, mechanical equivalent of heat, electric susceptibility,
etc.
2. Mass or inertial mass [M 1 L 0 T 0 ]
3. Mom entum and impulse. [M 1 L 1 T – 1]
4. Thrust, force, weight, tension, energy gradient. [M 1 L 1 T – 2]
5. Pressure, stress, Young's modulus, bulk modulus, shear modulus, modulus of
rigidity, energy density. [M 1 L – 1 T – 2 ]
6. Angular momentum and Planck's constant (h). [ M 1 L2 T –1]
7. Acceleration, g and gravitational field intensity. [ M 0 L1 T –2]
node06\B0AI-B0\Kota\JEE(Advanced)\Leader\Phy\Sheet\Hand book (E+L)\Eng\01_Unit & Dimension.p65
8. Surface tension, free surface energy (energy per unit area), force gradient, spring
constant. [ M 1 L0 T –2]
9. Latent heat capacity and gravitational potential. [ M 0 L2 T –2]
10. Thermal capacity, Boltzmann constant, entropy. [ ML 2 T – 2K – 1 ]
11. Work, torque, internal energy, potential energy, kinetic energy, mom ent of force,
(q2 /C), (LI2 ), (qV), (V 2 C), (I 2 Rt),
V2
t , (VIt), (PV), (RT), (mL), (mc DT) [M 1 L 2 T – 2]
R
12. Frequency, angular frequency, angular velocity, velocity gradient, radioactivity
R 1
, ,
1 [M 0 L 0 T – 1]
L RC LC
13. ælö
12
æmö
1 2
æL ö
ç ÷ ,ç ÷ ,ç ÷ , (RC), ( LC ) , time [ M 0 L 0 T 1]
ègø è k ø èR ø
14. (VI), (I2 R), (V 2/R), Power [ M L 2 T – 3]
E 3
, Physics HandBook CH APTER
ALLEN
Gravitational constant (G) 6.67 × 10 –11 N m 2 kg –2
KEY POINTS
Speed of light in vacuum (c) 3 × 10 8 ms –1
• Trigonometric functions
SOME FUNDAMENTAL CONSTANTS
Permeability of vacuum (m 0 ) 4p × 10 –7 H m –1 sinq, cosq, tanq etc and their
Permittivity of vacuum (e 0 ) 8.85 × 10 –12 F m –1 arrangement s q are
Planck constant (h)
dimensionless.
6.63 × 10 –34 Js
Atom ic mass unit (am u) 1.66 × 10 –27 kg • Dimensions of differential
Energy equivalent of 1 amu 931.5 MeV é dny ù éyù
coefficients ê n ú = ê n ú
9.1 × 10 –31
kg º 0.511 ë dx û ë x û
Electron rest mass (m e )
MeV • Dimensions of integrals
Avogadro constant (N A ) 6.02 × 10 23 mol–1 é ydx ù = [ yx ]
ëê ò úû
Faraday constant (F) 9.648 × 10 4 C mol –1
Stefan–Boltzmann constant (s) 5.67× 10 –8 W m –2 K –4 • We can't add or subtract two
physical quantities of
Wien constant (b) 2.89× 10 –3 mK
different dimensions.
Rydberg constant (R ¥ ) 1.097× 10 7 m –1
• Independent quantities may
Triple point for water 273.16 K (0.01°C) be taken as fundamental
22.4 L = 22.4× 10 –3 m 3 quantities in a new system of
Molar volum e of ideal gas (NTP) units.
mol –1
PRACTICAL PHYSICS
Rules for Counting Significant Figures For example : 3.0 × 800.0 = 2.4 × 103
For a number greater than 1 The sum or difference can be no more precise than
• All non-zero digits are significant. the least precise number involved in the mathematical
• All zeros between two non-zero digits are operation. Precision has to do with the number of
significant. Location of decimal does not matter. positions to the RIGHT of the decimal. The more
• If the numbe is without decimal part, then the position to the right of the decimal, the more precise
terminal or trailing zeros are not significant. the number. So a sum or difference can have no
• Trailing zeros in the decimal part are significant. more indicated positions to the right of the decimal
For a Number Less than 1 as the number involved in the operation with the
Any zero to the right of a non-zero digit is significant. LEAST indicated positions to the right of its decimal.
All zeros between decimal point and first non-zero For example : 160.45 + 6.732 = 167.18 (after
digit are not significant. rounding off)
Significant Figures Another example : 45.621 + 4.3 – 6.41 = 43.5
All accurately known digits in measurement plus (after rounding off)
node06\B0AI-B0\Kota\JEE(Advanced)\Leader\Phy\Sheet\Hand book (E+L)\Eng\01_Unit & Dimension.p65
the first uncertain digit together form significant Rules for rounding off digits :
figure. 1. If the digit to the right of the last reported digit is
Ex. 0.108 ® 3SF, 40.000 ® 5SF, less than 5 round it and all digits to its right off.
1.23 × 10 ® 3SF, 0.0018 ® 2SF
-19 2. If the digit to the right of the last reported digit is
Significant Digits greater than 5 round it and all digits to its right off
The product or quotient will be reported as having and increased the last reported digit by one.
as many significant digits as the number involved 3. If the digit to the right of the last reported digit is a
in the operation with the least number of significant 5 followed by either no other digits or all zeros,
digits. round it and all digits to its right off and if the last
For example : 0.000170 × 100.40 = 0.017068 reported digit is odd round up to the next even
Another example : 2.000 × .0 × 10–3 = digit. If the last reported digit is even then leave it
0.33 × 107 as is.
4 E
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