Unit and Dimensions

1. Measurement of any physical quantity involves comparison with a certain basic, arbitrary chosen, widely accepted reference standard called Unit. 

Mathematically, a measure of a quantity Q = nu, where u is the size of the unit, and n is the numerical value of the given measure.  

2. Fundamental quantities: Fundamental quantities are the base quantities. There are 7 fundamental quantities: 

(i) Length 

(ii) Mass

(iii) Time

(iv) Electric Current

(v) Thermodynamic Temperature

(vi) Amount of substance

(vii) Luminous Intensity.

3. Derived quantities: These quantities are formed using fundamental quantities like density, volume, force, etc.

4. Length: Unit is metre (m). Meter is defined as the length of the path traveled by light in vacuum during a time interval of $\frac{1}{299792458}$ part of a second. 

 1 fermi  = $1f =   10^{-15} m$

 1 angstrom =  $1A = 10^{-10} m$

 1 nano-metre = $1nm = 10^{-10}m$

 1 micro-metre = $1\mu m = 10^{-6}m$

 1 mili-metre = $1mm = 10^{-3} m$

 1 Astronomical unit = $1AU  = 1.496 \times 10^{11}m$

 1 light-year = $1ly = 9.46 \times 10^{11} m$

 1 parsec = $3.08 \times 10^{16}m$

5. Mass: Unit is Kilogram(kg). The mass of a cylinder made of platinum-iridium alloy kept at the International Bureau of Weights and Measures is defined as 1 kg.

6. Time: Unit is second(s). One second is the time taken by 9 192 631 770 oscillations of the light (of a specified wavelength) emitted by a cesium-133 atom.

7. Electric Current: Unit is Ampere. If equal currents are maintained in the two wires so that the force between them is $2 x 10^{-7}$ newton per meter of the wires, the current in any of the wires is called 1 A

8. Thermodynamic Temperature: Unit is Kelvin(K). The fraction $\frac{1}{273.16}$ of the thermodynamic temperature of the triple point of water is called 1 K.

9. Amount of the Substance: Unit is mole(mole). The amount of a substance that contains as many

elementary entities as there is the number of atoms in 0.012 kg of carbon-12 is called a mole. 

10. Luminous Intensity: Unit is Candela(cd). The SI unit of luminous intensity is 1 cd which is the luminous intensity of a blackbody of surface area $\frac{1}{600 000} m^{2}$ placed at the temperature of freezing, platinum, and at a pressure of 101,325 ${N/m^{2}}$, in the direction perpendicular to its surface. 

11. Dimensions: Dimensions are the powers to which fundamental quantities are raised to represent that quantity. It is represented by using a square bracket. 

Physical Quantities	Dimensions
Distance, Length, Displacement	$[M^0LT^0]$
Velocity, Speed	$[M^0LT^{-1}]$
Acceleration	$[M^0LT^{-2}]$
Force	$[MLT^{-2}]$
Linear momentum, Impulse	$[MLT^{-1}]$
Torque, Work, Kinetic Energy, Potential Energy, Energy,	$[ML^2T^2]$
Power	$[ML^2T^{-3}]$
Pressure, Stress, Modulus of Elasticity	$[ML^{-1}T^{-2}]$

12. Principle of homogeneity of Dimensions: A correct dimensional equation must be homogeneous i.e. dimensions on both sides are the same. 

13. Use of Dimension: To convert a unit from one system to another system, To find the relation between various physical parameters and to check whether the formula is dimensionally correct or not.

Example1:  Find the dimension of the constants a and b in Van Der Wall Equation

i.e. $(P + \frac{a}{V^2})(V-b) = RT$         

Solution:  Using principle of homogeneity,  

Dimension of b  = Dimension of V (volume) = $[{ L^3 }]$

Dimension of P (pressure) = Dimension of $(\frac{a}{V^{2}})$                               

Dimension of a = dimension of $PV{^2}$  = $[ML^{-1}T^{-2}] [L^{3}]^{2}$= $[ML^{5}T^{-2}]$          

 

Example 2: The value of the gravitational constant is G = $6.67 * 10^{-11}$ $Nm{^2}kg^{-2}$. Convert it into a system based on kilometer, tonne and hour as base units.  

Solution: Dimnsional formula of  G is $[M^{-1}L^{3}T^{-2}]$

$n_2 = n_1 [\frac{M_1}{M_2}]^{-1}[\frac{L_2}{L_1}]^{3} [\frac{T_2}{T_1}]^{-2}$  

$n_1 = 6.67 * 10^{-11}, M_1 = 1 kg, M_2 = 1 tonne = 1000kg,$

$T_1 = 1s, T_2 = 3600s, L_1 = 1m and L_2 = 1000m$

$n_2 = 6.67*10^{-11}[\frac{1}{1000}]^{-1}[\frac{1}{1000}]^{3} [\frac{1}{3600}]^{-2} = 8.64 * 10^{10}$

Example 3: The frequency f of a stretched string depends upon the Tension (T), length (l) and the linear mass density $/mu$. Find the relation for frequency. 

Solution: Let frequency depends on T, l, and $\mu$ as follow:

                           $f = kT^{a}l^{b}{\mu ^{c}}$                      where k is constant.

            

writing dimension formula of both sides,

$[M^{0}L^{0}T^{-1}]$ = $[MLT^{-2}]^{a}[L]^{b}[ML^{-1}]^{c}$  =  $[M^{a+c}L^{a+b-c}T^{-2a}]$

Comparing dimensions on both sides, 

                                     a + c  =  0

                                a + b - c  =  0

                                        -2a  =  -1

solving these we get,  a = $\frac{1}{2}$, b = -1 and c = $\frac{-1}{2}$

so relation will be,   $f = \frac{k}{l} \sqrt {\frac{T}{\mu}}$ 

Video Lecture:

Fundamental and Derived Quantities, Dimensions, How to find dimension of any physical Quantity, Formula validation by dimensions, Deriving relation between physical quantities Unit conversion Watch video