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```Dimensions of Physical Quantities

Table of Content

Introduction to Dimensions of Physical Quantities

Definition of Dimensions of Physical Quantities

Rules for writing dimensions of a physical quantity

Benefits of Dimensions

Limitations of Dimensions

Dimension Table

Introduction to Dimensions of Physical Quantities

The nature of physical quantity is described by nature of its dimensions. When we observe an object, the first thing we notice is the dimensions. In fact, we are also defined or observed with respect to our dimensions that is, height, weight, the amount of flesh etc. The dimension of a body means how it is relatable in terms of base quantities. When we define the dimension of a quantity, we generally define its identity and existence. It becomes clear that everything in the universe has dimension, thereby it has presence.

Image 1: Dimensions are responsible in defining shape of an object

Definition of Dimensions of Physical Quantities

The dimension of a physical quantity is defined as the powers to which the fundamental quantities are raised in order to represent that quantity. The seven fundamental quantities are enclosed in square brackets [ ] to represent its dimensions.

Examples

Dimension of Length is described as [L], the dimension of time is described as [T], the dimension of mass is described as [M], the dimension of electric current is described as [A] and dimension of the amount of quantity can be described as [mol].Adding further dimension of temperature is [K] and that dimension of luminous intensity is [Cd]

Consider a physical quantity Q which depends on base quantities like length, mass, time, electric current, the amount of substance and temperature, when they are raised to powers a, b, c, d, e, and f. Then dimensions of physical quantity Q can be given as:

It is mandatory for us to use [ ] in order to write dimension of a physical quantity. In real life, everything is written in terms of dimensions of mass, length and time. Look out few examples given below:

1. The volume of a solid is given is the product of length, breadth and its height. Its dimension is given as:

Volume = Length × Breadth × Height

Volume = [L] × [L] × [L] (as length, breadth and height are lengths)

Volume = [L]3

As volume is dependent on mass and time, the powers of time and mass will be zero while expressing its dimensions i.e. [M]0 and [T]0

The final dimension of volume will be [M]0[L]3[T]0 = [M0L3T]

2. In a similar manner, dimensions of area will be [M]0[L]2[T]0

3. Speed of an object is distance covered by it in specific time and is given as:

Speed = Distance/Time

Dimension of Distance = [L]

Dimension of Time = [T]

Dimension of Speed = [L]/[T]

[Speed] = [L][T]-1 = [LT-1] = [M0LT-1]

4. Acceleration of a body is defined as rate of change of velocity with respect to time, its dimensions are given as:

Acceleration = Velocity / Time

Dimension of velocity = [LT-1]

Dimension of time = [T]

Dimension of acceleration will be = [LT-1]/[T]

[Acceleration] = [LT-2] = [M0LT-2]

Fundamental Quantity
Dimension

Length
L

Mass
M

Time
T

Temperature
K

Electric Current
A

Luminous Intensity
Cd

Amount of substance
mol

5. Density of a body is defined as mass per unit volume, and its dimension are given as:

Density = Mass / Volume

Dimension of mass = [M]

Dimension of volume = [L3]

Dimension of density will be = [M] / [L3]

[Density] = [ML-3] or [ML-3T0]

6. Force applied on a body is the product of acceleration and mass of the body

Force = Mass × Acceleration

Dimension of Mass = [M]

Dimension of Acceleration = [LT-2]

Dimension of Force will be = [M] × [LT-2]

[Force] = [MLT-2]

Rules for writing dimensions of a physical quantity

Force, [F] = [MLT-2]

Velocity. [v] = [LT-1]

Charge, (q) = [AT]

Specific heat, (s) = [L2T2K-1]

Gas constant, [R] = [ML2T-2K-1 mol-1]

We follow certain rules while expression a physical quantity in terms of dimensions, they are as follows:

Dimensions are always enclosed in [ ] brackets

If the body is independent of any fundamental quantity, we take its power to be 0

When the dimensions are simplified we put all the fundamental quantities with their respective power in single [] brackets, for example as in velocity we write [L][T]-1 as [LT-1]

We always try to get derived quantities in terms of fundamental quantities while writing a dimension.

Laws of exponents are used while writing dimension of physical quantity so basic requirement is a must thing

If the dimension is written as it is we take its power to be 1, which is an understood thing

Plane angle and Solid angle are dimensionless quantity, that is they are independent of fundamental quantities

Benefits of Dimensions

Before writing dimensions of a physical quantity, it is must know a thing to understand why do we need dimensions and what are benefits of writing a physical quantity. Benefits of describing a physical quantity are as follows:

Describing dimensions help in understanding the relation between physical quantities and its dependence on base or fundamental quantities, that is, how dimensions of a body rely on mass, time, length, temperature etc.

Dimensions are used in dimension analysis, where we use them to convert and interchange units

Dimensions are used in predicting unknown formulae by just studying how a certain body depends on base quantities and up to which extent

It makes measurement and study of physical quantities easier

We are able to identify or observe a quantity just because of its dimensions

Dimensions define objects and their existence

Limitations of Dimensions

Besides being a useful quantity, there are many limitations of dimensions, which are as follows:

Dimensions can’t be used for trigonometric and exponential functions

Dimensions never define exact form of a relation

We can’t find values of certain constants in physical relations with the help of dimensions

A dimensionally correct equation may not be the correct equation always

Dimension Table

It consumes a lot of time while deriving dimensions of quantities. So in order to save time, we learn some basic dimensions of certain quantities like velocity, acceleration, and other related derived quantities. For Example, suppose you’re asked to find dimensions of Force and you remember dimension of acceleration is [LT-2], you can easily state that the dimension of force as [MLT-2] as force is the product of mass and acceleration of a body.

The table below depicts dimensions of several derived quantities which one can use directly in problems of dimension analysis.

Quantity
Formulae
Dimensions

Velocity
Displacement/Time
[LT-1]

Acceleration
Velocity/Time
[LT-2]

Work done
Force × displacement
[ML2T-2]

Pressure
Force/ Area
[ML-1T-2]

Power
Work done/ Time
[ML2T-3]

Density
Mass / Volume
[ML-3T0]

Area
[L2]

Watch this Video for more reference