Physical quantity, units and dimensions

Physical quantity: A quantity that can be measured by instrument, clearly defined and has proper units is called physical quantity. Physical quantities are classified as fundamental and derived quantities.

Fundamental units: The physical quantity which does not depend on any other physical quantity is called a fundamental physical quantity such as length; mass and time are called fundamental units.

Derived units: The units that can be obtained from fundamental units are called derived units.

System of units:

There are three systems of units.

In physics SI system is based on seven fundamental and two supplementary units.

(i) Fundamental units:

(ii) Supplementary units:

Unit & Dimensions & Vectors

(i) Meter:

The currently accepted definition of meter is the length of path travelled by light in vacuum in 1/299,792,458^th second.

(ii) Kilogram:

Kilogram is the fundamental unit of mass. It is defined as the mass of a specific cylinder of platinum - iridium kept at the International Bureau of Weights and Measures in Paris.

(iii) Second:

Second is the fundamental unit of time. It is defined as 86,400th part of a mean solar day. Second is accurately measured by an atomic clock.

(iv) Coulomb:

Coulomb is the fundamental unit of charge. It is defined as the charge required to obtain 9x10⁹ Newton of force between two equal charges separated at a distance of one meter in vacuum.

(v) Candle:

Candle is the fundamental unit of luminous intensity. It is defined as luminous intensity observed from a source of monochromatic light of frequency 540x1012 Hz, that has an intensity of 1/683 watt per steradian.

(vi) Kelvin:

Kelvin is the fundamental unit of temperature. It has value of zero where the molecular activity of gases cease.

(vii) Mole:

Mole is the fundamental unit of quantity of matter. It is defined as amount of substance of a system that contains as many elementary particle as there are in 0.012 kg of carbon-12 (C-12).

BASIC PHYSICAL QUANTITIES

Mechanical Physical Quantities (derived)

Electrical Physical Quantities (derived)

Applications of Dimensional analysis

(i) To find the unit of a physical quantity

Example-1 G = [M^-1L³T^-2]. Its SI unit is m³kg^-1s^-2 or Nm²kg^-2.

(ii) To convert a physical quantity from one system of units to another system of

units

n₁u₁ = n₂u₂ … (1)

( Where n_i and u_i are numerical constant unit and dimension in a particular system)

Example-2 Let us convert value of g (i.e. 9.8 m/s²) from SI system to CGS system

From eq. no. 1 [ n₁u₁]_{in SI} = [n₂u₂]_{in CGS}

[n₂]_CGS = ${[n_1]}_{inSI}\frac{{[u_1]}_{inSI}}{{[u_2]}_{inCGS}}$

= n₁ x$\left[\frac{L_1}{L_2}\right]\left[\frac{{T_1}^{-2}}{{T_2}^{-2}}\right]$

= 9.8 m/sec² x$\left[\frac{1m}{1cm}\right]\left[\frac{1se{c}^{-2}}{1se{c}^{-2}}\right]$

= 9.8 x $\left[\frac{100cm}{1cm}\right]\left[\frac{1se{c}^{-2}}{1se{c}^{-2}}\right]$

= 980

(iii) To check the correctness of a given physical relation

Based on principle of homogeneity, the dimensions on two sides must be same for a given relation.

Example-3 Check dimensionally

$F=\frac{m{v}^2}{r}where,[F]=[ML{T^{-}}^2];[v]=[L{T}^{-1}]$ Therefore, $LHS=[ML{T^{-}}^2]$

$RHS=\frac{[M]{[L{T}^{-1}]}^2}{L}=[ML{T}^{-2}]$

If dimensions are same on both sides then the relation is dimensionally correct otherwise incorrect.

(iv) To derive a relation

Example-4 Derive Planck's length in terms of G, c and h, where G is gravitation constant, c velocity of light and h is plank constant.

L= f(G, c, h), L = KG^xc^yh^z

[L] = [M^-1L³T²]^x [LT^-2]^y [ML²T^-1]^z

-x + y = 0, 3x + y + 2z = 1 and –2x – y – z = 0

$x=\frac{1}{2},y=\frac{-3}{2}andz=\frac{1}{2}$

Thus, L = $K\sqrt{\frac{Gh}{C^3}}$

If K = 1 then L =10^-35 m.

The importance of Plank's length is yet to be established.

Limitations of dimensional Analysis:

(i) The dimensional analysis cannot be applied to derive relations other than product of power functions, for example, s = ut + $\frac{1}{2}$....at² or y = y₀ cos wt and so on, cannot be derived directly.

(ii) The dimensional analysis cannot be applied to derive those relations that involve more than 3 unknowns, however, we can use them to check the correctness of a relation even if variables are more than 3.

(iii) Even if a physical quantity depends upon 3 quantities, out of which two have same dimension then dimensional analysis cannot be applied to derive such a formula but can be used to check the relation.

(iv) Numerical constants, trigonometric ratios and ratios which are dimensionless cannot be derived.

Physical quantities having same dimensions may not be the same. For example [ML²T^-2] is a dimensional relation for torque as well as work or energy.

Conversion Factors

(i) 1 A.U = 1.496x10¹¹m

(ii) 1X-ray unit = 10-¹³m

(iii) 1foot = 30.48 cm

(iv) 1Chandra Shekhar limit (CSL) = 1.4 times the mass of sun

(v) 1 metric Ton = 1000kg

(vi) 1pound = 0.4537kg

(vii) 1 atomic mass unit (a.m.u) = 1.67 x10^-27kg

(viii) 1shake = 10^-8kg

(ix) 1 year = 365.25d = 3.156x10⁷s

(x) 1 carat = 200mg

(xi) 1 bar = 0.1 M Pa = 10⁵Pa

(xii) 1curie = 3.7x10¹⁰s^-1

(xiii) 1 roentgen = 2.58 x 10^-4 C/kg

(xiv) 1quintal = 100kg

(xv) 1barn = 10^-28m²

(xvi) 1standard atmospheric pressure = 1.013x10⁵ Pa or N/m²

(xvii) 1mm of Hg = 133N/m²

(xviii) 1horse power = 746w

(xix) Gas constant, R = 8.36j/mol k = 8.36x10^-7erg/mol k = 2cal/mol

(xx) 1 Weber = 10⁸ maxwell

(xxi) 1 tesla = 1wb/m² = 10⁴ gauss

(xxii) 1amp turn/meter = oersted

(xxiii) 1electron volt (eV) =$4\pi \times 10^{-3}$ 1.6 x 10^-19J

(xxiv) 1calorie = 4.19J

(xxv) 1watt-hour = 3.6 x10³J

Example-5 The density of water is equal to

Solution: Ideally speaking, the examiner should specify the temperature in this question. This is because the density of water varies with temperature. It is maximum (10³ kg m^-3) at 4°C.

Example-6 One atmospheric pressure is equal to

Solution: 1 atmospheric pressure = 76 cm of Hg

= 76 x 13.6 x 981 dyne cm^-2

= 1.01 x 10⁶ dyne cm^-2 = 1.01 x 10⁵ N m^-2