Applications of dimensional analysis:-

Dimensional analysis analyses the correlations between physical quantities based on fundamental properties such as length, mass, time, and electric current, as well as units of measure such as miles VS kilometres or pounds VS kilogrammes. To put it another way, we study two sorts of physical quantities in Physics: basic and derived. Mass, length, amount of substance, time, luminous intensity, and electric current are the seven fundamental units. We get derived quantities when we combine two or more fundamental units. In dimensional analysis , the mass should be represented by M , Length should be represented by L , Time should be represented by T and so on.

Significance of Dimensional Analysis:-

1. The dimension of constants in a particular relation must be determined. It is possible to convert one type of unit into another type of unit.
2. An equation’s accuracy can be checked. Verifying the accuracy of any equation (i.e., the accuracy of various formulae).
3. To change the value of a physical quantity from one unit system to another. Different quantities can be combined to construct an equation.
4. To switch from one unit system to another. A physical quantity’s unit can be determined.
5. To ensure that a physical equation is correct. It is possible to discover the solution to a physical problem. To ensure that a given equation is dimensionally correct.
6. To calculate the relationship between different physical quantities. In a given system of units, determine the size and units of a physical quantity.
7. To determine the relationship between several physical quantities in a physical occurrence. To ensure that the physical relationship is accurate.

Uses of dimensional Analysis:-

1. To convert units of one system into the Units of another system:- The product of the numerical value of a physical quantity and its corresponding unit is a constant. For e.g. if the numerical values of a physical quantity p and n1 and n2 in two different systems and the corresponding units are u1 and u2 , then

   p=n1 u1=n2(u2)

• To check the correctness of the equation:- Every equation relating physical quantities in dimensional balance. It means that the dimensions of all the terms on both sides of a physical quantity must be the same. This is called the principle of homogeneity of dimension. The reason behind only similar quantities can be equated. 

• To establish the Relation among various Physical Quantities :- If we know the factors on which a given physical quantity may possibly depend , then , using dimensions , we can find a formula relating the quantity with those factors. For example , find the time period of a pendulum , Frequency of a stretched string etc.

Limitations of Dimensional Analysis

1. This method cannot be used to determine the value of dimensionless constants. The three dimensional quantities L, M, and T are used to create dimensional equations. However, if an unknown quantity is reliant on more than these three variables, we won’t be able to create dimensional equations for it. The dimensional equation of thermal conductivity, for example, cannot be represented solely in terms of L, M, and T because it is dependent on another quantity, temperature.
2. Furthermore, we cannot determine the value of a dimensionless quantity, i.e., a constant, using a dimensional procedure. If a physical quantity is dependent on more than three dimensions, it is impossible to derive a relationship or formula. As a result, equations involving addition and subtraction cannot be derived using the Dimensional technique. As a result, a dimensionally correct equation does not have to be true. A dimension of 1/T and 2/T, for example, are the same.
3. This method cannot be utilised if a quantity is dependent on trigonometric or exponential functions. Equations combining exponential and trigonometric functions cannot be solved using this method. It is impossible to create a formula that contains trigonometric, exponential, and logarithmic functions. It is impossible to deduce equations that use trigonometric, exponential, or logarithmic functions. This approach cannot be used to derive y = a cos(t – kx), for example.
4. If the physical quantity is dependent on more than three unknown variables, it cannot be employed. It doesn’t check whether anything is a scalar or a vector. If a physical quantity in mechanics is dependent on more than three physical quantities, this method cannot be used to generate a formula (mass, length, time).
5. It can be difficult to guess the components when deriving the relationship between two or more physical quantities in some instances. It can’t be used in a problem with more than three physical variables. It is impossible to derive a relationship from an equation with more than one element.

Conclusion:-

The study of the relationship between physical quantities based on their units and dimensions is known as dimensional analysis. It’s used to change one unit’s form into another. It is vital to keep the units the same when solving mathematical problems in order to solve the problem quickly. Understanding dimensions is crucial because it allows us to mathematically examine the nature of physical quantities. The essential idea behind dimensions is that we can only add or subtract quantities that have the same dimensions. Also, if two physical quantities have the same dimensions, they are equal. These fundamental concepts aid us in drawing a new relationship between physical quantities, similar to units.