## Monday, December 29, 2008

### Dimensions-Dimensional formulae.

Dimensions: Dimensions of a physical quantity are,the powers to which the fundamental units are raised to get one unit of the physical quantity.

The fundamental quantities are expressed with following symbols while writing dimensional formulas of derived physical quantities.

Mass →[M] ; Length→[L]; Time→[T]; Electric current →[I] ; Thermodynamic temperature →[K] ;Intensity of light →[cd] ; Quantity of matter →[mol] .

Dimensional Formula :Dimensional formula of a derived physical quantity is the “expression showing powers to which different fundamental units are raised”.

Ex : Dimensional formula of Force F →[$M^1L^1T^{-2}$]

Dimensional equation:When the dimensional formula of a physical quantity is expressed in the form of an equation by writing the physical quantity on the left hand side and the dimensional formula on the right hand side,then the resultant equation is called Dimensional equation.

Ex: Dimensional equation of Energy is E = [$M^1L^2T^{-2}$] .

Question : How can you derive Dimensional formula of a derived physical quantity.

Ans : We can derive dimensional formula of any derived physical quantity in two ways

i)Using the formula of the physical quantity : Ex: let us derive dimensional formula of Force .

Force F→ma ; substitute the dimensional formula of mass m →[M] ; acceleration →[$LT^{-2}$]

we get F → [M][$L T^{-2}$]; F →[$M^1L^1T^{-2}$] .

ii) Using the units of the derived physical quantity. Ex: let us derive the dimensional formula of momentum.

Unit of Momentum ( p ) → [$kg-m sec^{-1}$] ;

kg is unit of mass → [M] ; is unit of length → [L] ; sec is the unit of time →[T]

Substitute these dimensional formulas in above equation we get p →[$M^1L^1T^{-1}$].

Quantities having no units, can not possess dimensions: Trigonometric ratios, logarithmic functions, exponential functions, coefficient of friction, strain, poisson’s ratio, specific gravity, refractive index, Relative permittivity, Relative permeability. All these quantities nighter possess units nor dimensional formulas.

Quantities having units, but no dimensions : Plane angle,angular displacement, solid angle.These physical quantities possess units but they does not possess dimensional formulas.

Quantities having both units & dimensions : The following quantities are examples of such quantities.

Area, Volume,Density, Speed, Velocity, Acceleration, Force, Energy etc.

Physical Constants : These are two types

i) Dimension less constants (value of these constants will be same in all systems of units): Numbers, pi, exponential functions are dimension less constants.

ii)Dimensional constants(value of these constants will be different in different systems of units): Universal gravitational constant (G),plank’s constant (h), Boltzmann’s constant (k), Universal gas constant (R), Permittivity of free space($\in_0$) , Permeability of free space ($\mu_0$),Velocity of light (c).

Principle of Homogeneity of dimensions: The term on both sides of a dimensional equation should have same dimensions.This is called principle of Homogeneity of dimensions. (or) Every term on both sides of a dimensional equation should have same dimensions.This is called principle of homogeneity of dimensions.

Uses of Dimensional equations : dimensional equations are used i) to convert units from one system to another,

ii)to check the correctness of the dimensional equations iii)to derive the expressions connecting different physical quantities..

Limitations of dimensional method: The limitations of dimensional methods are

i)The value of dimensionless constants can not be calculated using dimensional methods,

ii)We can not analyze the equations containing trigonometrical, exponential and logarithmic functions using method of dimensions.

iii)If a physical quantity is sum or difference of two or more than two physical quantities, such physical quantities can not be derived with dimensional methods,

iv)If any equation having dimensional constants like, G, R etc can not be derived using dimensional methods,

v)If any equation is involving more than three fundamental quantities in it, such expressions can not be derived using dimensional methods.