Dimensional Formula of Angular Velocity

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Dimensions

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 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 [MaLbTc]


Dimensional Formula 

The dimensional formula of any physical quantity is that expression which 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)


Dimensional Equation 

The equation obtained by equating a physical quantity with its dimensional formula is called a dimensional equation. 


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 magnitude of a physical quantity remain 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. Path of the angular velocity is vertical to the plane of rotation, and this can be easily demonstrated by right hand thumb rule.

In mathematical form angular velocity is written as:

ω = \[\frac{dӨ}{dt}\]

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


The Dimension of Angular Velocity

The dimensional formula of angular velocity is given by, [M0 L0 T-1]

Where, standard unit mass is represented as M, length by L, and time by T.


Derivation of the Dimensional Formula of Angular Velocity

From the above definition we can derive the formula of angular velocity:

The dimensional formula of a separate entity is:

The dimension of time = [M0 L0 T1] . . . equation(1)

The dimension of angular displacement =  [M0 L0 T0] . . . equation(2)

So by multiplying equation (1) and equation(2) we will get the dimension of angular velocity:

Angular velocity = Angular displacement × [Time]-1

Therefore dimension of angular velocity =  [M0 L0 T0] × [M0 L0 T1]-1 = [M0 L0 T-1]

Therefore we can write the dimension of angular velocity as  [M0 L0 T-1].

FAQ (Frequently Asked Questions)

1. Define the Dimension Formula.

Answer: Dimensions of the physical quantity are the power to which the base quantities are raised to represent that quantity. The dimensional formula of any physical quantity is that expression which 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 ( ).

2. Explain a few limitations of Dimension Formulas.

Answer: Some of the limitations of the dimension formula are given below:

  • 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. A formula containing more than one term which is added or subtracted likes s = ut+ ½ at2 also cannot be derived.

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

3. Write a few sets that have the same Dimension Formula.

Answer: Some of the sets having the same dimensional formula that are discussed below:

  • Strain, refractive index, relative density, distance gradient, relative permeability, angle of content.

  • Mass and inertia.

  • Momentum and impulse.

  • Thrust, force, weight, tension, energy gradient.

  • Angular momentum and Planck’s constant.

  • Surface tension, surface area, force gradient, spring constant.

  • Latent heat and gravitational potential.

  • Thermal capacity, Boltzman constant, entropy.

4. Explain the term Angular Velocity?

Answer: 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 in the vertical direction to the plane of rotation, and this can be easily demonstrated by right-hand thumb rule.