PHYSICS CHAPTER 4 Dimensional Analysis Complete Knowledge And Solution Class 11th

PHYSICS

        

                                       4

                    Dimensional Analysis

DIMENSIONS OF A PHYSICAL QUANTITY

The derived units of all the physical quantities can be suitably expressed in terms The derived units o Tne mental units of mass, length and time raised to some power)For example, of the if   units of mass, length and time are denoted by bracketed capital letters [M], [L] and (TI, then for area (=length x breadth), we have 

     area = [L] × [L] = [L]²

Further, as in expressing area, units of mass and time do not occur, we write 

  area = [M° L² T°]

The dimensions of area are zero in mass, two in length and zero in time. Similarly, for velocity, we have 

  Velocity = Distance / time = [L]/[T] = [L T¹]

         = [M°LT·¹]

   (· is the sign of - ) 

The dimensions of velocity are zero in mass, one in length and minus one in time. 

Hence the dimensions of a physical quantity are the powers to which the fundamental units of mass, length and time have to be raised in order to obtain its units. 

DIMENSIONAL FORMULA AND EQUATION

  The expression[M° L² T°] for area and [M°LT·¹] for velocity are said to be the dimensional formulae of area and velocity respectively. 

The dimensional formula of a physical quantity provides two important informations. For example, the dimensional formula [M° L T·¹]

for velocity tells that 

i) the unit of velocity depends upon the unit of length and time and is independent of the unit of mass. 

ii) in the unit of velocity, the powers of L and T i.e. the units of length and time are 1 and -1 respectively. 

Hence dimensional formula of a physical quantity may be defined as the expression that indicates which of the fundamental units of mass, length and time enter into the derived 4nit of that quantity and with what powers. 

  If we represent velocity by [v], then an equation such as

                  [v] = [M° L T¹]

Is knows as dimensional equation

   Thus, the equation obtained, when a physical quantity is equated with its dimensional formula, is known as dimensional equation. 

   In general, in the dimensional equation 

            [X] = [Ma Lb Tc ], 

right hand side represents the dimensional formula ot physical quantity X, whose dimensions in mass, length and time are a, b and c respectively. 

THE DIMENSIONAL FORMULAE OF A FEW PHYSICAL QUANTITIES AND THEIR SI UNITS 

   The dimensional formula of a physical quantity can be obtained by defining ts relation with other physical quantities and then expressing these physical quantities in terms of fundamental units of mass [MI, length [L] and time [T]

             In the following table, the dimensional formulae of a few physical quantities have beer deduced aier defining their relations with other physical quantities. The SI units of these quantities have also been given. 

DIFFERENT TYPES OF VARIABLES AND CONSTANT

From the study of dimensional formulae of physical quantities, we can divide them into four categories: 

i) Dimensional variables. The quantities like area, volume, velocity, force, etc POSsess dimensions and do not have a constant value. Such quantities are called Wat 1. A less b For e dimensional variables. 

ii) Non-dimensional variables. The quantities like angle, specific gravity, strain etc neither posses dimensions nor they have a constant value. Such quantities are called non - dimensional variables. 

iii) Dimensional constants.The quantities like gravitational constant, Planck's Constant, etc possess dimensions and also have a constant value. They are called dimensional constants. 

iv) Non-dimensional constants. The constant quantities having no dimen- Sions are called non-dimensional constants. These include pure numbers 1,2,3,4.. T, e (=2.718) and all trigonometrical functions. 

USE OF DIMENSIONAL EQUATION

The dimensional equations have got the following three uses 

1. To check the correctness of a physical equation. 

2. To derive the relation between different physical quantities involved in a physical phenomenon. 

3. To change from one system of units to another. 

TO CHECK THE CORRECTNESS OF A PHYSICAL RELATION AND THE PRINCIPLE OF HOMOGENEITY OF DIMENSIONS 

Checking the correctness of a physical relation (or equation) is based on the principle of homogeneity of dimensions. 

      According to this principle, the dimensions of the fundamental quantities (mass, length and time) are same in each and every term on either side of the physical relation. 

        To check the correctness of a given physical equation,the physical quantities on the two sides of the equation are expressed in terms of fundamental units of mass, length and time. If the powers of M, L and T on two sides of the equation are same, then the physical equation is correct and otherwise not. 

TO DERIVE THE RELATION BETWEEN VARIOUS PHYSICAL QUANTITIES

In derivinc the relation between the various physical quantities, again the principle of homogeneity of dimensional equation is used. 

  To derive a physical relation, we first of all explore the possible physical quantities upon which the given physical quantity may depend. By assuming the powers of the dependent physical quantities, the relation between the given physical quantity and the quantities on which it depends, is written. By making use of dimensional formulae of the physical quantities involved, the relation is expressed in terms of the fundamental units of mass, length and time. When the powers of M, Land T are equated on both sides of the dimensional equation, we get three equations from which the values of the three unknown powers can be calculated. Setting the values of the powers, the required relation is obtained. 

   

       CONVERSION OF ONE SYSTEM OF UNITS TO ANOTHER 

  The method of dimensional analysis can be used to obtain the value of a physical quantity in the some other system, when its value in one system is given

      As discussed earlier the measure of a physical quantity is given by

           X = n u, 

where u is for hore u is the size of the unit and n is the numerical value of the physical quantity the unit chosen. If 41 and u2 are units for measurement of the physical quantity in two systems and wO SYstems and n and ng are the numerical values of the physical quantity for the two units, then 

            n1 u1 = n2 u2

If a, b and care the dimensions of the physical quantity in mass, length and time, then 

   Here, M, Li, T1 and M2, L2 and T, are the units of mass, length and time in the two systems. Therefore, 

                 

         

LIMITATIONS OF DIMENSIONAL ANALYSIS 

The method of dimensional analysis provides simple and quick solutions to so many Physical problems. However, it has a few limitations also, which are as explained below:

1. This method does not enable us to determine the value of the constant of  proportionality, which may be a pure number or a dimension less ratio. The value of the constant has to be determined experimentally or by some other method. 

2.This method cannot be used to derive the relations, such as S = ut+a, -u=2aS, etc by the usual method. Such relations are called composite relations (the relation, which have two or more than two terms on one side of the relation) and are derived in parts. Even while deriving such a relation in part it does not tell about the nature of sign connecting the various terms in the relation

3. This methods fails to derive a relationship, when a physical quantity depends upon more than three physical quantities. This is due to the fact that we can get at the most three equations by comparing the powers of three fundamental units, which according to the theory of algebraic equations, can be used to find the values of three unknown quantities only.  the relation. 

4. Since many physical quantities, though different, have the same dimensions eg. modulus of elasticity and pressure; momentum and impulse), the equality of dimensions of the two sides of an equation is a necessary but not a sufficient condition for the correctness of a physical relation. 

5. The method of dimensional analysis leads to incorrect result, if the physical quantities on which the given physical quantity depends are not rightly chosen. The proper choice of physical quantities requires a sound background of the subject. 

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