Units and Measurement

Measurement of any physical quantity involves comparison with a certain basic, arbitrarily chosen, internationally accepted reference standard called unit.
The units for the fundamental or base quantities are called fundamental or base units.
The units of all other physical quantities can be expressed as combination of the base units, such units obtained for the derived quantities are called derived units.

SI Base Quantities and Units

	Base quantity	Name	Symbol
1.	Lenght	metre	m
2.	Mass	Kilogram	kg
3.	Time	second	s
4.	Electric current	ampere	A
5.	Temperature	kelvin	K
6.	Amount of substance	mole	mol
7.	Luminous intensity	candela	cd

The International System of Units

In CGS system the base units for length, mass and time are centimeter, gram and second respectively.
In FPS system the base units for length, mass and time are foot, pound and second respectively.
In MKS system the base units for length, mass and time are meter, kilogram and second respectively.

Significant Figures

All the non-zero digits are significant.
All the zeros between two non-zero digits re significant, no matter where the decimal point is, if at all.
If the number is less than 1, the zero(s) on the right of decimal point but to the left of the first non-zero digit are not significant.
The terminal or trailing zero(s) in a number without a decimal point are not significant.
The trailing zero(s) in a number with a decimal point are significant.
For a number greater than 1, without any decimal, the trailing zero(s) are not significant.
For a number with a decimal, the trailing zero(s) are significant.

Rules for Arithmetic Operations with Significant Figures

In multiplication or division, the final result should retain as many significant figures as are there in the original number with the least significant figures.
In addition, or subtraction, the final result should retain as many decimal places as are there in the number with the least decimal places.

Rounding off the Uncertain Digits

The preceding digit is raised by 1 if the insignificant digit to be dropped (the underlined digit in this case) is more than 5 and is left unchanged if the latter is less than 5.
If the preceding digit is even, the insignificant digit is simply dropped and, if it is odd, the preceding digit is raised by 1.

Dimensions of physical quantities

The dimensions of a physical quantity are the powers (or exponents) to which the base quantities are raised to represent that quantity.

Dimensional formulae and Dimensional Equations

An equation obtained by equating a physical quantity with its dimensional formula is called the dimensional equation of the physical quantity.

	Physical quantity	Dimentions	Unit
1.	Path length	[L]	m
2.	Displacement	[L]	m
3.	Velocity	[LT^-1]	ms^-1
4.	Speed	[LT^-1]	ms^-1
5.	Acceleration	[LT^-2]	ms^-2
6.	Angular speed	[T^-1]	radian/sec
7.	Centripetal acceleration	[LT^-2]	ms^-2
8.	Momentum	[MLT^-1]	kg m s^-1
9.	Force	[MLT^-2]	N
10.	Impulse	[MLT^-1]	kg m s^-1
11.	Static friction	[MLT^-2]	N
12.	Kinetic friction	[MLT^-2]	N
13.	Work	[ML²T^-2]	J
14.	Kinetic energy	[ML²T^-2]	J
15.	Potential energy	[ML²T^-2]	J
16.	Mechanical energy	[ML²T^-2]	J
17.	Spring constant	[ML⁰T^-2]	N m^-1
18.	Power	[ML²T^-3]	W
19.	Angular momentum	[ML²T^-1]	J s
20.	torque	[ML²T^-2]	N m
21.	Moment of inertia	[ML²T⁰]	kg m ²
22.	Gravitational Constant	[M^-1L³T^-2]	N m ² kg^-2
23.	Gravitational Potential Energy	[M¹L²T^-2]	J
24.	Gravitational potential	[M⁰L²T^-2]	j kg^-1
25.	Gravitational Intensity	[M⁰L¹T^-2]	m s ^-2
26.	Pressure	[M¹L^-1T^-2]	pascal(Pa)
27.	Density	[M¹L^-3T⁰]	kg m^-3
28.	Specific Gravity	No	No
29.	Co-efficient of viscosity	[M¹L^-1T^-1]	Pa s (poiselulles)
30.	Reynold's number	No	No
31.	Surface tension	[M¹L⁰T^-2]	N m^-1
32.	Amount of substance	[mol]	mol
33.	Celsius temperature	[K]	⁰C
34.	Kelvin absolute temprature	[K]	K
35.	Co-efficient of linear expansion	[K^-1]	K^-1
36.	Co-efficient of volume expansion	[K^-1]	K^-1
37.	Heat supplied to a system	[M¹L²T^-2]	J
38.	Specific heat	[M⁰L²T^-2K^-1]	J kg^-1 K^-1
39.	Thermal Conductivity	[M¹L¹T³K^-1]	J s ^-1 K ^-1
40.	Period	[T]	s
41.	Frequency	[T^-1]	s^-1
42.	Angular frequency	[T^-1]	s^-1
43.	phase constant	No	rad
44.	Force constant	[M¹ T ^-2]	N ^-1
45.	Wavelength	[M⁰L¹T^-0]	m
46.	Propagation constant	[L^-1]	m^-1
47.	Wave speed	[LT^-1]	ms^-1
48.	Beat frequency	[T^-1]	s^-1
49.	Vector area element	[L²]	m²
50.	Electirc field	[M¹L¹T^-3A^-1]	V m^-1
51.	Electric flux	[M¹L³T^-3A^-1]	V m
52.	Dipole moment	[M⁰L¹T¹A¹]	C m
53.	Linear Charge density	[M⁰L^-1T¹A¹]	C m^-1
54.	Suface Charge density	[M⁰L^-2T¹A¹]	C m^-2
55.	Volume Charge density	[M⁰L^-3T¹A¹]	C m^-3
56.	Potential	[M¹L²T^-3A^-1]	V
57.	Capacitance	[M^-1L^-2T^-4A²]	F
58.	Polarisation	[M⁰L^-2T¹A¹]	C m ²
59.	Dielectric constant	No	No
60.	Electric current	[A]	A
61.	Charge	[T A]	C
62.	Voltage (Electric potential difference)	[M¹L²T^-3A^-1]	V
63.	Electromotive force	[M¹L²T^-3A^-1]	V
64.	Resistance	[M¹L²T^-3A^-2]	Ω
65.	resistivity	[M¹L³T^-3A^-2]	Ω m
66.	Electrical conductivity	[M^-1L^-3T³A²]	S
67.	Drift speed	[LT^-1]	ms^-1
68.	Relaxation time	[T]	s
69.	Current density	[L^{-2 A}]	A m^-2
70.	Mobility	[M¹L³T^-4A^-1]	m² V ^-1 s ^-1
71.	Permeability of free space	[M¹L¹T^-2A^-2]	T m A ^-1
72.	Magnetic field	[M¹L⁰T^-2A^-1]	T (tesla)
73.	Magnetic moment	[L²A]	A m ²
74.	Torsion constant	[ML²T^-2]	N m rad ^-1
75.	Magnetic field Magnetic induction Magnetic flux density	[M¹L⁰T^-2A^-1]	T(tesla)
76.	Magnetic flux	[M¹L²T^-2A^-1]	W(weber)
77.	Magnetisation	[L^-1A]	A m ^-1
78.	magnetic intensity magnetic field strength	[L^-1A]	A m ^-1
79.	Magnetic susceptibility	No	No
80.	Relative magnetic permeability	[M¹L¹T^-2A^-2]	T m A ^-1
81.	EMF	[M¹L²T^-3A^-1]	Volt
82.	Mutual Inductance	[M¹L²T^-2A^-2]	Henry
83.	Self Inductance	[M¹L²T^-2A^-2]	Henry
84.	rms voltage	[M¹L²T^-3A^-1]	Volt
85.	rms current	[A]	A
86.	Inductive reactance	[M¹L²T^-3A^-2]
87.	Capacitive reactance	[M¹L²T^-3A^-2]
88.	Impedance	[M¹L²T^-3A^-2]
89.	Resonant frequency	[T	Hz
90.	Quality factor	No	No
91.	Power factor	No	No
92.	Planck's constant	[M¹L²T^-1	Js
93.	Stopping potential	[M¹L²T^-3A^-1]	V
94.	Work function	[M¹L²T^-2 ]	J or eV
95.	Threshold frequency	[T]	Hz
96.	de broglie wavelength	[L]	m
97.	Atomic mass unit	[M]	u
98.	Disintegration or decay constant	[T^-1]	s^-1
99.	half life	[T]	s
100.	mean life	[T]	s
101.	Activity of a radio-active sample	[T^-1]	Bq

DIMENSIONAL ANALYSIS AND ITS APPLICATIONS

Checking the Dimensional Consistency of Equations
Deducing Relation among the Physical Quantities

NCERT EXERCISES

Note : In stating numerical answers, take care of significant figures.

1.1 Fill in the blanks
(a) The volume of a cube of side 1 cm is equal to …..m³

Solution: Volume of a cube =(side)³ =(1cm)³ =(1×10−²m)³ =1×10⁻⁶m.

(b) The surface area of a solid cylinder of radius 2.0 cm and height 10.0 cm is equal to…(mm)²

Solution: Surface area = 2πr (h + r) = 2 × 3.14 × 20 (100 + 20) = 15072 mm².

Solution: 18 km/h = (18 × 1000)/3600 = 5 m/s.

(d) The relative density of lead is 11.3. Its density is ….g cm^–3 or ….kg m^–3.

Solution: density of lead = relative density of lead x density of water = 11.3 x 1 = 11.3 g cm^–3 = 11.3 x (10^-3 kg)x (10^-2m)^-3=11.3 x 10³ kg m^{– 3}