Dimensions
The dimensions of a derived quantity are defined as the powers which are raised on the units of fundamental quantities to represent the unit of the derived quantity, are known as the dimensions of that physical quantity.
When the dimensions of a physical quantity are expressed along with the fundamental quantities on which those are raised, it is called the dimensional formula of the physical quantity. Dimensional formula of the quantity is expressed as [MaLbTc], where a, b and c are respective dimensions of mass, length and time.
For example, the dimension formula for volume = [L3],
velocity = [LT – 1 ], power = [ML2T – 3 ] etc.
The equation formed by equating the physical quantity with its dimensional formula is called dimensional equation of the quantity.
Basic Concept of Dimensions
( In Hindi + English Mix)
Dimensions of Important Quantities
( In Hindi + English Mix)
DIMENSIONAL FORMULA OF COMMONLY USED QUNATITIES
Principle of Homogeneity of Dimensions:
“Every valid physical equation must be dimensionally homogeneous. That is, different parts of a valid physical equation must have the same dimensional formula.”
For example: S = ut + ½ at2
[S] = [M0L1To]
[ut ]= [M0L1T-1] T1 = [M0L1T0]
[½ at2] = [M0L1T-2] T-2 = [M0L1T0]
So, [S] = [ut] = [½at2 ]
The equation is dimensionally homogenous. So, it is a valid physical equation.
Principle of Homogeneity of Dimensions
(In Hindi + English Mix)
APPLICATIONS OF DIMENSIONAL EQUATION
The principal of homogeneity of the dimensional equation can be used for following applications.
- Conversion of one system of unit into another units
If n1 is numerical value of physical quantity with dimesions a, b and c for units of mass, length and as M1, L1 and T1, the numerical value of the same quantity, n2 can be calculated for different units of mass, length and time as M2, L2 and T2 respectively.
- To test the correctness of a physical equation of formula
The principal of homogeneity requires that the dimensions of all the terms on both sides of physical equation or formula should be equal if the physical equation or any derived formula is correct one.
- To derive a relation between different physical quantities in any physical phenomenon
If a physical quantity depends upon a number of parameters whose dimensions are known, the principal of homogeneity of dimensions can be used to derive the relations between any physical quantity and its dependent parameters.
Application of Dimensions
(In English + Hindi Mix)
LIMITATIONS OF DIMENSIONAL ANALYSIS
- From dimensional equation, the nature of physical quantities cannot be decided i.e., whether a given quantity is scalar or vector.
- The value of constant of proportionality cannot be determined.
- Relation among physical quantities having exponential, logarithimic or trigonometric functions cannot be established.
- Relation which depends on more than three physical quantities in mechanics and more than four physical quantities in current electricity and other parts, cannot be established using dimensional methods.
- The equations which involve two physical quantities of same dimensional formula cannot be derived using dimensional method.
