Dimensional Formula of Angular Velocity

Dimensions of the physical quantity are the power to which the base quantities are raised to represent that quantity. Dimensions of any given quantity tell us about how and in which way different physical quantities are related. Finding dimensions of different physical quantities has many real-life applications and is helpful in finding units and measurements. Imagine a physical quantity X which depends mainly on base mass(m), length(L) and time(T) with their respective powers, then we can represent dimensional formula as M^aL^bT^c.

In the study of physics, angular velocity is known by various terms such as rotational velocity or angular frequency vector. The concept of angular velocity is briefly explained in the Class 11 NCERT book, Chapter 7 called systems of particles and rotational motion. To understand the concept of angular velocity it is extremely important to learn about rotation rate and rotation motion. Students who want to understand this complex concept in a simplified version can go through the study notes provided by Vedantu, these study notes are a product of extensive research, the Vedantu’s team has done a critical analysis of the previous year question papers and they have made the study material based on the important questions that can be asked in the exams.

Angular velocity is considered as a vector quantity which can be explained as the rate of change of angular displacement. This explains the specific angular speed also referred to as the rotational speed of an object. In other words, a change in angular displacement of an object at a particular time is called angular velocity. The path of the angular velocity is vertical to the plane of rotation, and this can be easily demonstrated by the right-hand thumb rule.

Mathematically, angular velocity can be represented as-

ω = dӨ/dt,

Where, dθ = the change in angular displacement,

 dt = the change in time t.

The Dimensional Formula of Angular Velocity is as Follows-

The dimensional formula of any physical quantity is that expression that represents how and which of the base quantities are included in that quantity. 

It is written by enclosing the symbols for base quantities with appropriate power in square brackets i.e ( ).

E.g: Dimension formula of mass is: (M)

The Dimension of Angular Velocity is given by,

[M⁰L⁰T^-1]

Here,  standard unit mass = M, 

length = L, and 

time = T.

How is the dimensional formula of angular velocity derived?

From the above definition, we can derive the formula of angular velocity.

The dimensional formula of a separate entity can be represented as-

dimension of time = [M⁰L⁰T¹] . . . equation(1)

dimension of angular displacement =  [M⁰L⁰T^-1] . . . equation(2)

multiplying equation (1) and equation(2), we can find the dimension of angular velocity:

Angular velocity = Angular displacement ×Time-1

Therefore,

dimension of angular velocity = [M⁰L⁰T⁰] × [M⁰L⁰T¹]^-1

= [M⁰L⁰ T^-1]

Key Concepts related to Angular Velocity are as Follows-

• Centre of mass

• The motion of the centre of mass

• Linear momentum of a system of particles

• Vector product of two vectors

• Angular velocity and its relation with linear velocity

• Torque and angular momentum

• Equilibrium of a rigid body

• Moment of inertia

• Theorems of perpendicular and parallel axes

• Kinematics of rotational motion about a fixed axis

• Dynamics of rotational motion about a fixed axis

• Rolling motion

Application of Dimensional Analysis 

1. To convert a physical quantity from one system of the unit to the other

It is based on a fact that the magnitude of a physical quantity remains the same whatever system is used for measurement i.e magnitude = numeric value(n) multiplied by unit (u) = constant

n₁u₁= n₂u₂

2. To check dimensional correctness of a given physical relation

If in a given relation, the terms of both sides have the same dimensions, then the equation is dimensionally correct. This concept is best known as the principle of homogeneity of dimensions. 

3. To derive a relationship between different physical quantities 

Using the principle of homogeneity of dimension, the new relation among physical quantities can be derived if the dependent quantities are known. 

Limitation of This Method

1. This method can be used only if dependency is of multiplication type. The formula containing exponential, trigonometric, and logarithmic functions can not be derived using this method. The formula containing more than one term which is added or subtracted likes s = ut + ½ at² also cannot be derived.

2. The relation derived from this method gives no information about the dimensionless constants. 

Angular Velocity

It is considered as a vector quantity and defined as the rate of change of angular displacement which tells us about the specific angular speed or rotational speed of an object and also the axis about which the object is rotating. It can also be said that the change in angular displacement of the particle at that given period of time is called angular velocity. The path of the angular velocity is vertical to the plane of rotation, and this can be easily demonstrated by the right-hand thumb rule.

In mathematical form angular velocity is written as:

ω = dӨ/dt

Where dθ is considered as the change in angular displacement and dt is considered as the change in time t.