Angular velocity and its relation with linear velocity

Before determining the relation between linear velocity and angular velocity, let’s understand the meaning of each of these concepts. When an object moves in a linear fashion i.e. in a straight line, we can find linear velocity by dividing the covered distance by time. 

For instance, if Rahul traveled 100 meters in 5 minutes, it means that Rahul is traveling with a velocity (linear) of 20 meters per minute. However, things get different when an object moves in an angular fashion i.e. rotates. As per the definition of angular velocity, the speed with which a rotating object changes its angular position is known as angular velocity. In simpler terms, when an object is in circular motion, the rate of difference in displacement as per the time is called angular velocity.  Alternatively, we can say that angular velocity of an object rotating around a center point is the rate at which it changes the displacement (angular) with respect to its origin. 

By the above definitions, we can find the mathematical relation between linear velocity and angular velocity. As per the definition of linear velocity, we can find it using the below given formula:

Linear velocity  v= ΔS/Δt

Here S is the displacement of object

t is the time duration of the motion.

Angular velocity v=Δθ/ΔT

Here is angular displacement of the object

t is the time duration of angular motion.

A body moves in a circular motion with radius r. The relation between linear and angular displacement can be given by the below formula:

S =rθ or θ = S/r

Relation Between Angular Velocity and Linear Velocity 

From the above  equations, we have 

S = rθ 

And v = Δ(S)/Δt 

If we substitute the value of S we get the below equation that helps us define the relation between linear velocity and angular velocity:

v = Δ(rθ)/Δt 

= r (Δθ/Δt)

However, as per the definition of angular velocity, Δθ/Δt =. Therefore, we can substitute this value in the above equation. 

Finally we have,

v = r

Here, is the speed of rotation of the object and r is the object’s linear velocity which will be zero at the center. When the object moves away from the position of center, there is an increase in its linear velocity. 

It should be noted that an object’s linear velocity will be the highest at the circumference of the circle in which it is rotating. When we draw a tangent from that particular point on the circumference, we get the linear velocity’s direction as well. 

When an object moves in a circular motion, the relation between linear velocity and angular velocity can be mathematically defined as below:

=vr

Key points to remember from the relation between linear and angular velocity

• As per this equation, linear velocity i.e. v is directly proportional to its angular velocity and to the distance between the center and the object

• Though the linear velocity can vary according to the position of the circle, angular velocity remains the same irrespective of the object’s position on the circle
• The linear velocity between any position on the circumference and the center is minimal

Angular Velocity to Linear Velocity 

The magnitude of an object’s (rotating in circle) linear velocity can be connected to its angular velocity by the below equation:

=vr

This relationship or equation is true to any object that has a rigid body. 

State How Angular Velocity and Linear Velocity Are Different?

The differences among the angular velocity and linear velocity are explained in the below table:

Linear Velocity	Angular Velocity
Linear Velocity has magnitude and we can also find its direction. Therefore, it is considered as a vector.	Angular Velocity moves an object along the axis of the circular path. Therefore, it is considered as an axial vector.
As the displacement of the object is linear, meters/seconds is its unit of measurement.	As the displacement of the object is angular, the angular velocity of an object is measured in radians.
A car moving on a straight path is an example of linear velocity. Similarly, a ball moving straight on a slope acquires a linear velocity.	A racing car that moves along a circular track is an example of angular velocity. When you drop a ball on a roulette, the spinning of the roulette gives rise to angular velocity. Similarly, the giant wheel moving in circular motion is also an example of angular velocity.
‘v’ is used to denote linear velocity.	‘’ or omega is used to express Angular velocity.
An object moving in a circular path  has both angular velocity and linear velocity. However, its linear velocity differs from one point on the circle to another.	Angular Velocity does not change with change in the position of an object moving in a circular path.
The pace at which linear displacement occurs is equal to its linear velocity.	The pace at which angular displacement occurs is equal to its angular velocity.

Conclusion 

The relation between linear velocity and angular velocity can be studied in this article. Angular speed and angular velocity are similar to linear speed and linear velocity respectively.  The relation between linear velocity and angular velocity can be mathematically defined w=v/r.