Dimensional Formula Analysis

In this article, you will learn about dimensions and the dimensional formula of physical quantities with the help of examples.

Let’s start with the definitions first.

What are the dimensions?

The dimension of a physical quantity is defined as the power to which the fundamental quantities are raised to express the physical quantity. For example, the dimension of velocity is 0, 1, -1 in Mass, Length, Time, respectively.

If this feels confusing at first, don’t worry, we will look into the calculations of dimensions later in this article.


What is the dimensional formula?

Dimensional Formula is defined as the expression of the physical quantity in terms of its base unit (fundamental quantity) with proper dimensions. For example,

Dimensional Formula of Velocity is [ M0 L1 T-1 ]

Here,

  • M, L, T are fundamental quantities Mass, Length and Time, respectively
  • 0, 1, -1 are the dimensions

In this article, we will be using 3 fundamental quantities: Mass, Length, and Time to derive dimensional formulas of different physical quantities.

But first, let’s look into the dimensional symbol of these fundamental quantities.

Dimensional Formula of 

  • Mass is [M]
  • Length is [L]
  • Time is [T]

How to calculate the dimensional formula of physical quantities?

Now that we know what the dimensional formula is, let’s see how we can compute the dimensional formula of a physical quantity.

1. What is the dimensional formula of Area?

The dimensional formula of Area is [ M0 L2 T0 ]. Let’s now see the derivation.


  Area = length * breadth

Dimensionally, we use [L] for both length and breadth. Hence,


  Area = [L] * [L]

Since there is no use of both Mass and Time, we can write 


  Area = [ M0 ] * [ L2 ] * [ T0 ]

Dimensional Formula of Area = [ M0 L2 T0 ]


2. What is the dimensional formula of Volume?

The dimensional formula of volume is [ M0 L3 T0 ]. Let’s start with the formula of volume.


  Volume = length * breadth * height

We know length, breadth, and height, all can be represented with the fundamental quantity [L], length. Hence,


  Volume = [L] * [L] * [L]

In volume, there is no use of Mass and Time so, we can write


  Volume = [ M0 ] * [ L3 ] * [ T0 ]

Dimensional Formula of Area = [ M0 L3 T0 ]


3. What is the dimensional formula of density?

The dimensional formula of density is [ M2 L-3 T0 ]. Let’s now use the formula of density to derive this.


  Density = mass / volume

          = mass / (length * breadth * volume)

We can use the fundamental quantity

  • [L] (Length) – for length, breadth, and volume
  • [M] (Mass) – for mass

Hence, our formula becomes


  Density = [M] / ([L] * [L] * [L]

          = [M] / [ L3 ]

          = [ M1 ] * [ L-3 ]

And, since there is no mention of time in density, we can write

Dimensional Formula of Density = [ M1 L-3 T0  ]

Note: We can calculate the dimension formula of a physical quantity when its relation with other physical quantities is known. In our examples, we are able to calculate the dimensional formula of density once we know how it is related to mass and volume.


Dimensional Formula of Physical Quantity

In the previous section, we learned to calculate the dimensional formula using the basic formula of physical quantities.

By following the same process, you can easily compute the dimensional formula of every physical quantity:

Dimensional formula of Speed/Velocity


  Speed = distance / time

        = [L] / [T]

        = [L] * [ T-1 ]

        = [ M0 L1 T-1 ]

Dimensional Formula of Acceleration


  Acceleration = velocity / time

               = [ M0 L1 T-1 ] / [T]

               = [ M0 L1 T-2 ]

Dimensional Formula of Momentum


  Momentum = mass * velocity

           = [M] * [ M0 L1 T-1 ]

           = [ M1 L1 T-1 ]

Dimensional Formula of Force


  Force = mass * acceleration

        = [M] * [ M0 L1 T-2 ]

        = [ M1 L1 T-2 ]

Dimensional Formula of Pressure


  Pressure = Force / area

           = [ M1 L1 T-2 ] / ( [L] * [L] )

           = [ M1 L-1 T-2 ]

Dimensional Formula of Work  


  Work = Force * distance

       = [ M1 L1 T-2 ] * [L]

       = [ M1 L2 T-2 ]

Dimensional Formula of Energy


  Energy = Work

         = [ M1 L2 T-2 ]

Dimensional Formula of Power


  Power = Work / time

        = [ M1 L2 T-2 ] / [T]

        = [ M1 L2 T-3 ]

Dimensional Formula and SI Unit

We can also derive SI units by using the dimensional formula and vice versa. Let’s see an example.

1. Dimensional formula to SI unit conversion

Dimensional Formula of density = [ M1 L-3 T0  ].

We know the SI unit of

  • [M] is kg (kilogram)
  • [L] is m (meter)
  • [T] is s (second)

Now, if we convert the formula to SI units, it will be [ kg1 m-3 s0 ]. Hence,

SI unit of density is kg m-3.

2. SI unit to dimensional formula conversion

SI unit of density is kg m-3 

Here, we know that 

  • kg is the SI unit of mass
  • m is the SI unit of length

So our formula will look like this [M L-3].

Since there is no mention of time here, we can use 0 for time.

Dimensional formula of density = [M1 L-3 T0 ]


Dimensional Formula Chart

In the following dimensional formula chart, you can find the dimensional formulas and SI units of various physical quantities:

Physical QuantityDimensional FormulaSI Unit
Area[ M0 L2 T0 ]m2
Volume[ M0 L3 T0 ]m3
Speed/Velocity[ M0 L1 T-1 ]ms-1
Acceleration[ M0 L1 T-2 ]ms-2
Momentum[ M1 L1 T-1 ]kg m s-1
Force[ M1 L1 T-2 ]kg m s-2
Pressure[ M1 L-1 T-2 ]kg m-1 s-2
Work/Energy[ M1 L2 T-2 ]kg m2 s-2
Power[ M1 L2 T-3 ]kg m2 s-3
Gravitational Constant[ M-1 L3 T-2 ]kg-1 m3 s-2
Impulse[ M1 L1 T-1 ]kg m s-1
Surface Tension[ M1 L0 T-2 ]kg s-2
Coefficient of Viscosity[ M1 L-1 T-1 ]kg m-1 s-1
Momentum of Inertia[ M1 L2 T0 ]kg m2
Angular Momentum[ M1 L2 T-1 ]kg m2 s-1
Torque/Couple[ M1 L2 T-2 ]kg m2 s-2
Frequency[ M0 L0 T-1 ]s-1
Dimensional Formula of Different Physical Quantities

Dimensional Equation

An equation containing physical quantities with dimensional formula is known as the dimensional equation. A dimensional equation is obtained by equating the dimensional formula on the right-hand side and the left-hand side of an equation. For example,

Dimensional equation of v = u + at is


[ M0 L1 T-1 ] = [ M0 L1 T-1 ] + [ M0 L1 T-1 ] * [ M0 L0 T1 ]

Principle of Homogeneity of Dimensional Equation

According to this principle, the dimensions of fundamental quantities on the left-hand side of an equation must be equal to the dimensions of the fundamental quantities on the right-hand side of that equation.

Let us consider three quantities A, B, and C such that C = A + B. Therefore, according to this principle, the dimensions of C are equal to the dimensions of A and B.

Let’s see an example. Here we have our earlier dimensional equation of v = u + at:


[ M0 L1 T-1 ] = [ M0 L1 T-1 ] + [ M0 L1 T-1 ] * [ M0 L0 T1 ]

1. Solving Right Hand Side


 RHS = [M0 L1 T-1] + [M0 L T-1] X [M0 L0 T]

     = [M0 L1 T-1] + [M0 L1 T0 ]

 RHS = [M0 L1 T-1 ] = LHS

Here, the right-hand side equation is now the same as the left-hand side equation. Hence, the principle of homogeneity of dimensional equations is proved.


What are the uses of dimensional equations?

Here are some of the uses of the dimensional equation:

  • We can use dimensional equations to check the correctness of physical relations.
  • It also helps us to derive the relation between various physical quantities.
  • We can convert the value of physical quantity from one system of units to another system using the dimensional equation.
  • It finds the dimension of constants in a given relation.

What are the limitations of dimensional analysis?

Following are the limitations of dimensional analysis.

  • We cannot get the information about the dimensional constant using the dimensional analysis.
  • If a quantity depends on more than three factors having dimension, We cannot derive the dimensional formula.
  • We cannot derive the formula containing trigonometric function, exponential functions, logarithmic function, etc.
  • We cannot develop the exact form of a relationship when there is more than one part in any relationship.
  • It gives no information whether a physical quantity is a scalar or vector.