Dimensional Formula

DIMENSIONAL FORMULA

The dimensional formula of any physical quantity is the formula that tells which of the fundamental units have been used for the measurement of that physical quantity.

How is dimensional formula written for a physical quantity?

(1) The formula of the physical quantity must be written. The quantity must be on the left-hand side of the equation.

(2) All the quantities on the right-hand side of the formula must be written in terms of fundamental quantities like mass, length and time.

(3) Replace mass, length and time with M, L and T, respectively.

(4) Write the powers of the terms.                                                 

CHARACTERISTICS OF DIMENSIONS

(1) Dimensions do not depend on the system of units.

(2) Quantities with similar dimensions can be added or subtracted from each other.

(3) Dimensions can be obtained from the units of the physical quantities and vice versa.

(4) Two different quantities can have the same dimension.

(5) When two dimensions are multiplied or divided, it will form the dimension of the third quantity.

DIMENSIONAL ANALYSIS

The dimensional formula can be used to

(1) Check the correctness of the equation.

(2) Convert the unit of the physical quantity from one system to another.

(3) Deduce the relation connecting the physical quantities.

Units and Dimensions of a Few Derived Quantities:-

Physical Quantity

Unit

Dimensional Formula

Displacement

m

M0L1T0

Area

m2

M0L2T0

Volume

m3

M0L3T0

Velocity

ms-1

M0L1T-1

Acceleration

ms-2

M0L1T-2

Density

Kg m-3

M1L-3T0

Momentum

Kg ms-1

M1L1T-1

Work/Energy/Heat

Joule (or) Kg m2/sec2

M1L2T-2

Power

Watt (W) (or) Joule/sec

M1L2T-3

Angular Velocity

rad s-1

M0L0T-1

Angular Acceleration

rad s-2

M0L0T-2

Moment of Inertia

Kg m2

M1L2T0

Force

Newton (or) Kg m/sec2

M1L1T-2

Pressure

Newton/m (or) Kg m-1/sec2

M1L-1T-2

Impulse

Newton sec (or) Kg m/sec

M1L1T-1

Inertia

Kg m2

M1L2T0

Electric Current

Ampere (or) C/sec

QT-1

Resistance/Impedance

Ohm (or) Kg m2/sec C2

ML2T-1Q-2

EMF/Voltage/Potential

Volt (or) Kg m2/sec2 C

ML2T-2Q-1

Permeability

henry/m (or) Kg m/C2

MLQ-2

Permittivity

Farad/m (or) sec2C2/Kgm3

T2Q2M-1L-3

Frequency

Hertz (or) sec-1

T-1

Wavelength

m

L1

Dimensional Analysis Explained

The study of the relationship between physical quantities with the help of dimensions and units of measurement is termed dimensional analysis. Dimensional analysis is essential because it keeps the units the same, helping us perform mathematical calculations smoothly.

Unit Conversion and Dimensional Analysis

Dimensional analysis is also called Factor Label Method or Unit Factor Method because we use conversion factors to get the same units. To help you understand the stated better, let’s say you want to know how many metres make 3 km?

We know that 1000 metres make 1 km,

Therefore,

3 km = 3 × 1000 metres = 3000 metres

Here, the conversion factor is 1000 metres.

Using Dimensional Analysis to Check the Correctness of Physical Equation

Let’s say that you don’t remember whether

  1. time = speed/distance, or
  2. time = distance/speed

We can check this by making sure the dimensions on each side of the equations match.

Reducing both the equations to its fundamental units on each side of the equation, we get

However, it should be kept in mind that dimensional analysis cannot help you determine any dimensionless constants in the equation.


PRINCIPLE OF HOMOGENEITY

According to the principle of homogeneity of dimensions, all the terms in a given physical equation must be the same.

Ex. s = ut + (½) at2

Dimensionally

[L] = [LT-1.T] + [LT-2. T2] [L] = [L] + [L]

Applications of Dimensional Analysis

Dimensional analysis is a fundamental aspect of measurement and is applied in real-life physics. We make use of dimensional analysis for three prominent reasons:

  • To check the consistency of a dimensional equation
  • To derive the relation between physical quantities in physical phenomena
  • To change units from one system to another

Limitations of Dimensional Analysis

Some limitations of dimensional analysis are:

  • It doesn’t give information about the dimensional constant or while deriving the formula, the proportionality constant cannot be found.
  • The formula containing trigonometric function, exponential functions, logarithmic function, etc. cannot be derived.
  • It gives no information about whether a physical quantity is a scalar or vector.
  • This method cannot be used if the physical quantity depends on more parameters than the number of fundamental quantities.