Angular Displacement And Angular Velocity: 7 Important Facts


In this article we are going to discuss 7 important facts that are related to angular displacement and angular velocity.

Let us first know about angular displacement and angular velocity. Suppose a body is rotating about an axis. Now the angle between its position of rest and its position of rotational motion is known as angular displacement of the body. The rate of change of angular displacement is known as angular velocity.

Hence we can say that both angular displacement and angular velocity are related to rotational motion. We usually denote angular displacement by θ and angular velocity by ω. The SI unit of angular displacement is radian and as we know that

Angular velocity ( ω) = change in angular displacement(∆θ) / change in time(∆t)                                           
= radian / second or radian.second⁻¹
expression of angular velocity

Angular displacement has a magnitude as well as a specific direction(either clockwise or anticlockwise),so it is a vector quantity. Similarly angular velocity also has a magnitude and a specific direction that is why it is also a vector quantity.

Suppose a body is moving around a fixed axis,it means that it is continuing its rotational motion. Now the angle subtended between its position of rest and position where it reached finally,is its angular displacement. If we divide this angular displacement by the total time taken by the body to reach the final position we will get angular velocity.

How is angular displacement related to angular velocity?

Suppose a body is moving around a fixed axis,it means that it is continuing its rotational motion.

Now the angle subtended between its position of rest and position where it reached finally,is its angular displacement. If we divide this angular displacement by the total time taken by the body to reach the final position we will get angular velocity.

Let us take the arc of the circle through which the body is moving is s and the radius of the circle is r. Hence

Angular displacement = arc of the circle/ radius of the circle
as s is small hence it is considered as a straight line i.e, normal sinθ = s/r                                                            
Here θ is a very small angle hence   sinθ ≈ θ                                                             
Hence                                            θ = s/r …(1)
expression of angular displacement

Similarly angular velocity can be written as,

Angular velocity = change in angular displacement/ change in time                                                                    
ω = ∆θ/∆t                                    
Here lim ∆t → 0 hence ∆θ/∆t → dθ/dt                    
Hence ,          ω = dθ/dt                                        
➡ ω = d/dt(s/r) [ putting the value of θ from equation (1) we get ]                                        
➡ ω = 1/r.(ds/dt)                                         
➡ ω = v/r [ as ds/dt = v = linear velocity]                                        
➡ v = ωr
relation between angular displacement and angular velocity

 ω = dθ/dt. This is the relation between angular displacement and angular velocity.

Can angular displacement and angular velocity be the same?

Angular displacement and angular velocity can be the same.

We know that angular velocity = ω = dθ/dt. If the initial angular displacement of a rotating body is 0 and the final angular displacement of that body is θ,then the change in angular velocity will be

= dθ= ( θ – 0) = θ and if the time is 0 when the angular displacement is 0 and the time is t=1 unit when the angular displacement is θ ,then dt = (t – 0) = (1-0) = 1 unit

Therefore ω = dθ/dt

                          = θ/1

                          = θ

                      ω = θ

Hence it is proved that change in angular displacement and angular velocity are the same.

How to find angular velocity from angular displacement?

The angular displacement and angular velocity hold the same relation as linear displacement and linear velocity.

So we calculate linear velocity by dividing it by time. Similarly we can obtain angular velocity by dividing angular displacement by time taken.

Suppose a body is moving around a fixed axis,it means that it is continuing its rotational motion. Now the angle subtended between its position of rest and position where it reached finally,is its angular displacement. If we divide this angular displacement by the total time taken by the body to reach the final position we will get angular velocity.

Angular displacement

Let us take the arc of the circle through which the body is moving is s and the radius of the circle is r. Hence Angular displacement = arc of the circle/ radius of the circle

as s is small hence it is considered as a straight line i.e, normal sinθ = s/r

                                                            Here θ is a very small angle hence   sinθ ≈ θ

                                                             Hence                                                 θ = s/r

Similarly angular velocity can be written as,

 Angular velocity = change in angular displacement/ change in time

                                               ω = θ/t

                      Here lim ∆t → 0 hence ∆θ/∆t → dθ/dt

                      Hence ,          ω = dθ/dt

                                           ➡ ω = d/dt(s/r) [ putting the value of θ from equation (1) we get ]

                                            ➡ ω = 1/r.(ds/dt) 

                                            ➡ ω = v/r               [ as ds/dt = v = linear velocity]

                                            ➡ v = ωr

Difference between angular displacement and angular velocity

The differences between angular displacement and angular velocity are written below:

      In terms of  Angular    displacement      Angular velocity
              Definition1. The angle between the position of rest of a body and its position of rotational motion is known as angular displacement of that body.1.The rate of change of angular displacement is known as angular velocity.
    Unit2.The SI unit of angular displacement is radian.2. The SI unit of angular velocity is radian/second.
  Mathematical formula3. The mathematical formula for angular displacement is :  θ = s/r3. The mathematical formula for angular velocity is ω = dθ/dt
            Dimension4. It is a dimensionless quantity.4. The dimension of angular velocity is [M⁰.L⁰.T⁻¹]
             Analogous to5. Angular displacement of rotational motion is analogous to the linear displacement of linear motion.5.  Angular velocity of rotational motion is analogous to the linear velocity of linear motion.  
difference between Angular displacement and Angular velocity

Angular displacement and angular velocity graph

The relation between angular displacement and angular velocity is 

      ω = dθ/dt

➡ dθ =  ω.dt

If the initial angular displacement of a rotating body is 0 and the final angular displacement of that body is θ,then the change in angular velocity will be

= dθ= ( θ – 0) = θ and if the time is 0 when the angular displacement is 0 and the time is t when the angular displacement is θ ,then dt = (t – 0) = t

Therefore

  dθ =  ω.dt

  θ =  ω x t

  ➡ θ =  ωt

Table for data of graph between angular displacement and angular velocity

              Time angular displacement (θ in rad)angular velocity ( ω in rad/s)
  t = constant = 5 s              15           1
  t = constant = 5 s              10           2
  t = constant = 5 s              15           3
  t = constant = 5 s              20          4
  t = constant = 5 s              25          5
table for data of graph between angular displacement and angular velocity
angular displacement and angular velocity
graph between angular displacement and angular velocity

Problem statements with solutions

  1. Mili is walking around a circular park whose radius is 70 m. if her linear displacement is 700 m then find out the value of her angular displacement.

Answer :

Radius of the circular park = r = 70 m

Linear displacement of mili = s = 700 m

Therefore her angular displacement,θ = s/r

                                                                       ➡ θ = 700/70 radian

                                                                       ➡ θ = 10 radian

Therefore the angular displacement of Mili is 10 radian.

  • 2. A car starts running around a circular pond. If initially it was at rest and then its final angular displacement becomes 100 radian after 20 seconds. What will be its angular velocity?

Answer :

Initial angular displacement, θ₁ = 0 rad

                Final angular displacement, θ₂ = 100 rad

              Change in angular displacement, ∆θ =  θ₂ –  θ₁ = (100 – 0) radian = 100 radian

                                    Change in time, ∆t = (20 -0) seconds = 20 seconds

Therefore, required angular velocity,  ω = θ/t

                                                                          ➡ ω = 100/20 rad/second 

                                                                          ➡ ω = 5 rad/ second

Therefore the required angular velocity of the car is 5 rad/second.

Conclusion

In this article our prime concern was to give a clear idea of angular displacement and angular velocity in a brief manner. So starting from the definition of both angular displacement and angular velocity we have tried to clarify every minute detail related to them. Other than these the difference between these two quantities, graph between angular displacement and angular velocity and numerical problems related to them have been discussed in this article.

Ankita Biswas

I am Ankita. I have done my B.Sc in physics honours and my M.Sc in Electronics. Currently I am working as physics teacher in a Higher Secondary School. I am very enthusiastic about high energy physics field. I love to write complicated physics concepts in understandable and simple words.

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