Unit, Dimension and Error Analysis

1.1 Physical Quantities : Different quantities needed to describe the physical phenomenon or object are called physical quantities. Examples: speed, length, mass, density, etc.

The physical quantities are divided in two groups.

1. Fundamental quantities: Those physical quantities which are independent to any other physical quantities. Eg. Mass, length, time, etc.

2. Derived quantities: Those physical quantities which are dependent on other         physical quantities and obtained by multiplying and dividing the fundamental quantities are called derived quantities. Eg. Density, velocity, acceleration, force, etc.

1.2 Units: The physical quantities are measured by comparing with some standard measurement of same kinds are called units. The units are divided in two groups.

1. Fundamental units: Units of fundamental quantities are called fundamental unit. These are independent to any other units.

2. Derived units: Units of derived quantities are called derived unit. These units are  obtained by multiplying and dividing the fundamental units.

System of measurement:

1.     CGS system: The system of measurement in which 3 fundamental quantities mass, length and time are measured in gm, cm and s respectivily. All other derived quantities are also measured in terms of these units of measurements.

2.     MKS system: The system of measurement in which 3 fundamental quantities mass, length and time are measured in kg, m and s respectivily. All other derived quantities are measured in terms of these units of measurement is called MKS system.

3.     FPS system: : The system of measurement in which 3 fundamental quantities mass, length and time are measured in pound, foot and second respectivily. All other derived quantities are measured in terms of these units of measurement is called FPS system.

4.     International system units (SI): SI is an abbreviation “Le systeme International d’ unites.” Which is French and equivalent of international system of units. It is used widely throughout the world in which seven different quantities are introduced as fundamental quantities and their units as fundamental units.

	Quantity	Unit	Symbol
1.	Mass	Kilogram	Kg
2.	Length	Meter	M
3.	Time	Second	S
4.	Temperature	Kelvin	K
5.	Electric Current	Ampere	Amp
6.	Luminous Intensity	Candela	Cd
7.	Amount of Substance	Mole	Mol

Two more quantities are introduced as supplementary quantities and their units as supplementary unit.

1.	Plane angle	Radian	Rad
2.	Solid angle	Steradian	Sr

Prefixes of power ten:

Prefix	Abbreviation	Power of ten
Atto	A	-18
Femto	F	-15
Pico	P	-12
Nano	N	-9
Micro	µ	-6
Milli	M	-3
Centi	C	-2
Deci	D	-1
Kilo	K	3
Mega	M	6
Giga	G	9
Tera	T	12
Peta	P	15
Exa	E	18

Some useful practical units:

1.     Astronomical unit (AU): Average distance between centre of earth and centre of sun.

2.     Par Sec used to measure long distance and represents a parallactic second. One par sec is the distance at which an arc of 1 AU long subtends an angle of 1῞.

         l =1AU

          ∴ 1 Par sec = 3.1 × 10¹⁶m

 3.     Light year (ly):- Distance traveled by light is vacuum in one year.

∴ 1ly = 9.46 × 10¹⁵m

4.     1 inch = 2.54 cm

1 foot = 30.48 cm

1 yard = 91.44 cm

1 mile = 1.609 × 10³

1 nautical mile = 1.852 × 10³m

1 angstrom (1A⁰) =× 10^-10m

1 Fermi = 1 femtometer = 10¹⁵m

For area:

1 barn = 10^-28m²

1 acre = 4047 m²

1 hectare = 10⁴m²

For mass:

1 tonne or metric ton = 1000 kg

1 quintal = 100 kg

1 slug = 14.57 kg

1 lb = 0.4536

1 chandra shekhar limit (CSL) = 1.4 times mass of sun

1 amu (atomic mass Unit) = 1.67× 10^-27kg

For time:

1 shake = 10^-8sec

1 solar year = 365.25 days

For pressure:

1 bar = 1 atmosphere pressure = 10⁵N/m²

1 torr = 1 mm of Hg = 133 N/m²

1 atmospheric pressure = 760 mm of Hg = 760 torr.

Dimension: The power of fundamental quantity involved in any physical quantity iis called dimension of that physical quantity. The representation of physical quantity in terms of fundamental quantities involved in it is called dimensional formula of that phsical quantity. Three fundamental quantities mass, length and time are represented by [M], [L] and [T] respectively. All other derived quantities are also expressed in terms of these representations.

Eg. Force = ma = kgm/s² = [MLT^-2]

The dimension of mass is 1, length is 1 and time is -2 in force and force and the representation [MLT^-2] is called dimensional formula of force.

Dimensional formula of some quantities:

SN	Physical quantity	Formula	Dimensional formula	Unit
1.	Density	Mass Volume	[ML-3]	Kgm-3
2.	Specific Gravity	Density of body Density of water at 4°C	Dimensionless	-------------
3.	Linear momentum	M × v	[MLT-1]	Kgms-1
4.	Impulse	F×t	[MLT-1]	Ns
5.	Pressure	F/A	[ML-1T-2]	Nm-2
6.	Universal Gravitational Constant	G = Fr2 m1m2	[M-1L3T-2]	Nm2/Kg2
7.	Work	F × d	[ML2T-2]	Nm
8.	Moment of force	F × r	[ML2T-2]	Nm
9.	Power	w/t	[ML2T-3]	W
10.	Surface tension	F/l	[ML0T-2]	Nm-1
11.	Surface energy	Energy	[ML2T-2]	J
12.	Force Constant	K= F/x	[ML0T-2]	Nm-1
13.	Upthrust	Force	[MLT-2]	N
14.	Stress	F/A	[ML-1T-2]	Nm-2
15.	Strain	e/L	Dimensionless	---------
16.	Modulus of elasticity	Stress Strain	[ML-1T-2]	Nm-2
17.	Radius of gyration		[L]	M
18.	Moment of inertia	Mr²	[ML2T0]	Kgm2
19.	Angle	θ = l/r	Dimensionless	Rad
20.	Angular acceleration	α = θ/r	[T-2]	rad/s2
21.	Angular velocity	ω= θ/t	[T-1]	rad/s
22.	Angular momentum	L= Iω	[ML2T-1]	Kgm2s-1
23.	Torque	T=Iα	[ML2T-2]	Nm
24.	Frequency	f = 1/t	[T-1]	s-1or Hz
25.	Velocity gradient	Dv Dx	[T-1]	s-1
26.	Rate of flow	V/t	[L3T-1]	m3s-1
27.	Planck’s constant	h = E/f	[ML2T-1]	Js
28.	Mass per unit length	M/l	[ML-1]	Kgm-1
29.	Specific latent heat	L=Q/m	[L2T-2]	JKg-1
30.	Thermal conductivity	K=Q/{t.A(dθ/dl)}	[MLT-3K-1]	wm-1K-1
31.	Universal gas Constant	R =  PV Nt	[ML2T-2K-1mol-1]	Jmol-1K-1
32.	Boltzman Constant	k = R NA	[ML2T-2K-1]	JK-1
33.	Entropy	S=dQ/T	[ML2T-2K-1]	JK-1
34.	Charge	Q=It	[AT]	C
35.	Electric dipole Moment	(q.2l)	[ALT]	Cm
36.	Current density	J=I/A	[AL-2]	Am-2
37.	Permittivity	ℇ0 = cd         A	[M-1L-3T4A2]	Fm-1
38.	Magnetic flux	Φ = BA	[M1L2T-2A-1]	Wb
39.	Permeability	μ = 2BR/I	[M1L1T-2A-2]	Hm-1
40.	Specific heat Capacity	S = Q/(mΔθ)	[L2T-2K-1]	J/kg-k
41.	Capacitance	c=4πε0r	[M-1L-2T4A2]	F
42.	Resistance	R=V/I	[M1L2T-3A-2]	Ω
43.	Magnetic field Induction	B = F/(Il)	[MT-2A-1]	T
44.	Inductance	L = E/(dI/dt)	[M1L2T-2A-2]	H
45.	Resistance  Capacitance	(RC)	[T]	S
46.	Inductance Resistance	(L/R)	[T]	S
47.	√(inductance×capacitance)	√(LC)	[T]	S

Dimensionless quantities:

Strain, Coefficient of friction, Mechanical equivalent of heat, Poisson’s ratio, Angle and solid angle, Relative density, Refractive index, Emissivity, Magnetic susceptibility, Dielectric constant, Relative permeability, Relative permittivity, Loudness.

Error Measurement :

Error in a combined equation X = K  (a^mbⁿ/c^p)

(i)                Maximum or permissible Relative error

(Δx/x)= m(Δa/a) + n(Δb/b)+ p(Δc/c)

(ii)             Maximum or permissible percentage error­­­­­

(Δx/x) × 100% = {m(Δa/a)+ n(Δb/b)+ p(Δc/c)} × 100%

Where (Δa/a),(Δb/b) and (Δc/c) are errors in measuring a, b and c.

Note: (-ve is not considered in error i.e -p(Δc/c) for denominator term.)