Saturday, September 6, 2014

Dimensional formula for a physical quantities and its Uses

Dimensional formula is a representation of the physical quantity in terms of fundamental quantities. By writing dimensional formula we are identifying how fundamental quantities are raised to different powers to get a new physical quantity. The powers of the fundamental physical quantities are called as dimensions.

We represent dimensional formulas in terms of length, mass and the time. Writing dimensional formula for any physical quantities is a easy task once if you know the concept behind that physical quantity. If you know the dependence of the physical quantity on the other physical quantities, with can write the dimensional formula quite easily. In dimensional formula Length shown with L, mass is shown with M and the time is shown with T.

Though many of the physical quantities consists of only this three fundamental quantities, it is quite possible that some of the physical quantities can have the representation of SI system fundamental physical quantities in the dimensional formula.

Some of the physical quantities and the way of writing their dimensional formulas are explained in a typical example below.

Different physical quantities may have the same dimensional formulas. We can only get a rough idea by looking at the physical quantities dimensional that what it is made up of in terms of fundamental quantities.We cannot judge everything about physical quantity just by looking at the dimensional formula alone. If that is the case learning about dimensional form itself is sufficient to interpret the entire physics itself.

Different physical quantities of quite different nature sometimes can have the same dimensional formula.For example work and torque are having the same dimensional formulas but the nature of the physical quantities is quite different.

It is quite possible that some of the physical quantities won't have any dimensional formulas.Examples are like angle, strain and coefficient of friction.

Principal of homogeneity:

Only physical quantities of the same nature having the same dimensions can be added, subtracted or can be equated. It is as simple as we need to add one velocity with another velocity to get another velocity. We cannot add velocity and displacement like we cannot add water and kerosene and get something productive.

It simply means the terms of the both sides of the dimensional equation shall have the same dimensions.

Applications of dimensional analysis:

  1. Dimensional formulas can be used to convert the physical quantities numerical value  from one system of units to other system of units.
  2. We can check the correctness of a given equation basing on dimensional analysis.
  3. We can derive the relation between different physical quantities using the dimensional analysis.
Explanation for the applications of dimensional analysis:

Here let us discuss how can we convert a physical quantities numerical value from one system to other. In a simple example here we would like to convert one jowl of energy is how many erg ?

In solving this problem we have simply taken a known relation to others that is discussed in the previous topic physical quantities under units. Larger the value of the unit smaller its multiplication factor.

In this step we would like to analyse how can we check the correctness of a given equation using dimensional analysis.In using this application we're taking a basic concept into consideration which is called as principal of homogeneity. The concept simply says that dimensional formula of the LHS side of a given equation shall be equal to its RHS side.

Here we are discussing another example of using the same concept. The concept is to check the relation between physical quantities using the principle of homogeneity.Whenever a equation is written in physics will be generally taking it for granted that it is satisfying the principle of homogeneity. By using the same concept of the below problem is solved.

Finally we would like to find the relation between physical quantities using the dimensional analysis. Here also we are going to use the same principle of homogeneity. When it is given that the physical quantity depends on some other physical quantities,we will simply write a proportional relation and equate dimensions of each fundamental physical quantities on the both the sides of the equation.

Limitations on the use of dimensional formulas:

  1. we cannot calculate the value of the proportionality constants using the dimensional analysis.
  2. Formulas containing non-algebraic functions like trigonometric functions and exponential functions cannot be derived basing on the dimensional analysis.
  3. To derive the relation between the physical quantities at least we need to have a rough idea of dependence of the physical quantities ,otherwise we cannot obtain the relation.
  4. We cannot derive a equation which contains two or more than two terms in the right-hand side of the equation using the dimensional analysis.

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