Angular Velocity Vs Linear Velocity: 3 Important Concepts

We already knew the concept of velocity. Now let us see the detailed comparison of angular velocity Vs linear velocity.

When items are traveling in a circular direction, the linear velocity and angular velocity may be seen. Let’s understand the comparison of angular velocity Vs linear velocity in detail.

Angular Velocity Vs Linear Velocity

It is possible to construct characteristics related to motion on a circle by looking at the relationship between an arc on a circle and the radian angle it subtends. Before getting into these two concepts of angular velocity and linear velocity, first, we need to understand the meaning of linear displacement and angular displacement.

Image Credits: “Merry Go Round – Palace Pier, Brighton” (CC BY 2.0) by Mark Wordy

We are already aware that angular displacement is defined as the angle drawn by a particle while moving in a circle. Because the displacement’s direction is parallel to the axis, angular displacement is represented by an axial vector in this case.

The displacements experienced by a particle while it travels in a circular motion are divided into two categories. These are as follows:

1. Linear Displacement which is along the circumference
2. Angular Displacement which is making an angle

If the radius of the circle is ‘r’ then the relationship between the linear displacement and angular displacement can be given as:

  s=rθ or  θ =s/r

Now, the linear velocity is defined as the rate of change of linear displacement. The linear displacement of a particle can be given as follow:

And When it comes to particles, their angular velocity is defined as the rate at which their angular displacement changes.

∴ In a simple sense, linear velocity refers to the rate at which the arc length varies over time, whereas angular velocity refers to the rate at which the angle around the central point changes over time.

Relation between Angular Velocity and Linear Velocity

Just now we have seen that the linear velocity for the particle is given as

And we know that  s=r 

r = the radius of the circle is constant

From the equation of angular velocity, we know that

Therefore the relationship between linear and angular velocity for a body moving in a uniform circular motion is given by : 

As stated in this equation, the linear velocity (v) of a particle is directly proportional to its distance from the center of the circular pathway and the angular velocity of the particle.

Comparative analysis of Angular Velocity and Linear Velocity

We may create a comparison based on our understanding of the terms of angular velocity and linear velocity, as illustrated in the table below:

Angular Velocity	Linear Velocity
The particle’s angular velocity is defined as the rate of change of angular displacement.	Linear velocity is defined as the rate at which linear displacement changes.
The angular velocity of a particle is measured along the circle’s axis. Throughout the circular motion, it remains constant.	In a circular motion, a particle’s linear velocity is along the circumference of the circle. It changes according to the location of the point on the circle.
Angular velocity is denoted by the symbol	Linear velocity is denoted by the symbol
The formula is given by:	The formula is given by:
It has the form of an axial vector. This is because of the fact that the particle’s displacement is directed in the direction of the circle’s centre axis.	It’s a vector quantity, that means it has both a magnitude and a direction associated with it.
The measuring unit of angular velocity is both degree and radians. Where the degree is dimensionless and radians is SI unit.	The measuring unit of linear velocity is m/s.

FAQ’s

Q. What is velocity?

Ans: The rapidity with which anything moves or acts is referred to as its velocity.

The words velocity and speed provide us with a sense of how quickly or slow an item is traveling in relation to our position.

We run into instances when we need to determine which of two or more things is traveling quicker than the other. This happens very frequently. The velocity of an item is defined as the rate at which its position changes in relation to a reference frame and over time. It can be measured in meters per second  (ms-1).

When a body’s velocity varies by a significant amount or in a certain direction, the body is said to be accelerating.

Q. What do you mean by angular velocity? Give examples.

Ans: It is the rate at which a particle spins around a central point.

The angular velocity of a particle is defined as the rate at which the angular displacement of the particle changes. The angular velocity of a particle is along the circle’s axis when an item moves in circles. In other words, the angular velocity is the constant quantity and is denoted by ‘ω’.

Examples of angular velocity are the Ferris wheel, Earth, Bus wheel, and Fan.

Q. What do you mean by linear velocity? Give examples.

Ans: Basically, it’s the pace at which a particle moves along a straight trajectory.

The linear velocity can be defined as the rate at which linear displacement changes over time. Angular and linear velocities are both present in everything that rotates or travels in a circle.

For example, consider riding a merry-go-round. A stone thrown off the edge of a spinning merry-go-round will not fall straight down. Instead, it will continue to go ahead at the same speed that the merry-go-round was traveling at when the stone was thrown. This is the pebble’s linear velocity.

Q.If an object is traveling the distance of 5 meters in the time of 2 seconds. Then what will be the linear velocity of that object?

Ans: Linear velocity can be calculated by using its simple formula.

Given: displacement = 5 meter, time = 2 seconds

Linear velocity:  =s/t 

∴=5/2 

∴=2.5 m/s 

∴ The linear velocity of the object is 2.5 m/s

Q. Is angular velocity and angular speed identical?

Ans: Both quantities have similarities and differences.

Similarities between them are their unit is radian/second. While they differ in the aspect of scalar and vector. Angular speed () is a scalar quantity whereas the angular velocity () is a vector quantity.

Q. Is linear velocity and linear speed identical?

Ans: While both velocity and speed seek to determine a moving object’s distance, they differ in some ways.

First, speed is a scalar quantity. Thus, expressing speed in m/s  simply shows magnitude. It says nothing about the object’s motion. However, linear velocity describes the direction in which something is moving.

This is due to the fact that velocity is a vector quantity, which indicates both the magnitude and the direction of the moving body.

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