3. Dimensions

Dimensions

Each derived quantity requires proper power for fundamental quantities so as to represent it. The powers of fundamental quantities, through which they are to be raised to represent unit-derived quantity, are called dimensions. In other words, the dimensions of a physical quantity are the powers to which the base quantities (fundamental quantities) are raised to represent that quantity.

Thus, the dimensions of a physical quantity are the powers(or exponents) to which the fundamental units of length, mass, time, etc. must be raised to represent it or the dimension of the units of a derived physical quantity is defined as the number of times the fundamental units of length, mass, time, etc appear in the physical quantity.

A few examples are :

Some Physical quantities and their dimensions

Dimensional Analysis

Dimensional analysis is the practice of checking relations amongst physical quantities by identifying their dimensions and units of measurement. However, dimensional analysis is possible only if the dimensions of various terms on either side of the equation are the same. This rule is known as the principle of homogeneity of dimensions. The principle is based on the fact that two quantities of the same dimension only can be added, subtracted, or compared.

Also, dimensional analysis is an amazing tool for checking whether or not equations are dimensionally correct. It is also possible to use dimensional analysis to generate plausible equations if we know the quantities involved. Quantification of the size and shape of things can be done using dimensional analysis. The mathematical study of the nature of objects is possible today, thanks to dimensional analysis.

We have already learned to express most physical quantities in terms of basic dimensions. We will now learn about dimensional analysis and its applications with the help of fundamental quantities like mass, length, time, etc.

Principle of Homogeneity of Dimensions

The equations depicting physical situations must have the same dimensions. This principle is based on the fact that only two quantities of the same dimension can be added, subtracted, or compared.

The principle of Homogeneity states that “dimensions of each of the terms of a dimensional equation on both sides should be the same.”

As in the above equation dimensions of both sides are not the same; this formula is not correct dimensionally, so it can never be physically accurate.

An equation of the form x= a+b+c+....  is dimensionally correct if and only if the variables x, a, b, and c all have the same dimension. This principle may be applied to differential equations and integral equations, as well as to algebraic equations.

For example :

1.  F=mv^2/r^2

By substituting dimensions of the physical quantities in the above relations

[MLT^(-2) ]=[M] [LT^(-1) ]^2  / [L]^2   ;  we have [MLT^(-2)]=[MT^(-2)]

As in the above equation, dimensions of both sides are not  the same , the formula is not correct dimensionally , so it can never be physically accurate.

Note 

•  If  [M]^a [L]^b [T]^c   = [M]^x [L]^y [T]^z  then from the principle of homogeneity we have a=x, b=y and c=z.
• A dimensionally correct equation may not be physically correct, but a physically correct equation must be dimensionally correct.

Dimensional Analysis as Factor Label Method

Units can be converted from one system to the other. The method used for this is called the factor label method, unit factor method, or dimensional analysis.

In this method, a unit can be converted from one system to another by using a conversion factor that describes the relationship between units. It is based on the fact that the ratio of each fundamental quantity in one unit with their equivalent quantity in another unit is equal to one.

For example:

How many minutes are there in 5 hours?

Solution : 1 hour= 60 minutes , so 3 hours = 3×60=180 minutes

(Here the conversion factor from hours to minutes is 60)

We use conversion factors accordingly so that the answer comes in the desired unit and biased results are avoided. In this way, each fundamental quantity like mass, length, and time is converted into another desired unit system using the conversion factor.

Applications of Dimensional Analysis

Dimensional analysis is used to solve problems in real-life physics. We make use of dimensional analysis for five prominent reasons:

(i) Gravitational constant: According to Newton’s law of gravitation

           F=G (m1 m2)/r^2    or   G=(F r^2)/(m1 m2)

Substituting the dimensions of all physical quantities

        [G]= ([MLT^(-2)] [ L^2])/([M][M])= [M^(-1) L^3 T^(-2)]

(ii) Planck constant: According to Planck   E= hυ or  h=E/υ 

Substituting the dimensions of all physical quantities

         [h]= ([ML^2 T^(-2)])/([T^(-1)])= [ML^2 T^(-1)]

(iii) Coefficient of viscosity: According to Poiseuille’s formula

        dV/dt=(Πpr^4)/8ηl  or  n=(Πpr^4)/(8 (dV/dt))

Substituting the dimensions of all physical quantities

         [η]=([ML^(-1) T^(-2)][L^4])/([L][L^3 T^(-1)])=[ML^(-1) T^(-1)]

3. Converting a Physical Quantity from One System to the Other

The measure of a physical quantity is  nu= constant

If a physical quantity X has the dimensional formula [M^a L^b T^c] and if its derived units of that physical quantity in two systems are [〖M_1^a〗^ L_1^b 〖 T〗_1^c  ] and [M_2^a 〖 L〗_2^b  T_2^c  ]  respectively and n₁ and n₂  are the numerical values in the two systems respectively, then the system

Respectively.

n1[〖M_1^a〗^ L_1^b 〖 T〗_1^c  ]  = n2 [M_2^a 〖 L〗_2^b  T_2^c  ]

n2= n1[〖M_1^a〗^ L_1^b 〖 T〗_1^c  ] / [M_2^a 〖 L〗_2^b  T_2^c  ]

Where, M1, L1 and T1= basic units of mass, length, and time in the first (known) system

M2,  L2 and T2= basic units of mass, length, and time in the second (unknown) system

Example: conversion of Newton into dynes.

4. Checking the Dimensional Correctness of a Given Physical Relation

This application is based on the principle of homogeneity of dimensions. According to this rule, only those terms can be added or subtracted with the same dimensions.

If  X=A ±(BC)^2± √DEF

Then according to the principle of homogeneity

[X]=[A]=[(BC)^2]=[(√(DEF)])

If the dimensions of each term on both sides of an equation are the same, then the equation is dimensionally correct.

Example:     s= ut+(1/2)at^2

By substituting respective dimensions of the physical quantities in the above equation –

[L]=[LT^(-1)][T]-[LT^(-2)][T^2]   so [L]=[L]-[L]

As in the above equation, the dimensions of each term on both sides of an equation are the same; the equation is dimensionally correct. However, from equations of motion, we know that s= ut-(1/2)at^2

5. Tool for Research to Derive New Relations

Suppose one knows the dependency of a physical quantity on other quantities which is of the product type. In that case, using the dimensional analysis method, a relation between the quantities can be derived.

Derivation of stoke’s law using dimensional Analysis

The force of viscosity on an object in a fluid depends on the radius of the object, coefficient of viscosity and its velocity in the fluid.

From the dimensional analysis we have F= krηv, the value of  K=6Π

So viscous force F=6Πηrv

Derivation of centripetal force using dimensional analysis.

Centripetal force required to move an object in a circular path depends on its mass, velocity and radius of the circular path.