# Is Angular Velocity Constant: When, Why And Detailed Facts

This article answers the question when is angular velocity constant? The rate of change of displacement is called as velocity.

In simple words it is the speed of an object with direction. Velocity is related to a linear motion, each and every molecule on the object moves with the same velocity. Angular velocity relates to the motion where in the object rotates on its axis.

## What is angular velocity?

The rate of vector rotation of any object around its axis is called as the angular velocity. Simply put, it is the measurement of how swiftly the angular position of an object changes.

From above discussion we can infer that angular velocity is the conjugate of velocity for angular motion. Angular velocity is to angular motion what velocity is to linear motion. We will study more about angular velocity in the sections below.

Image Credits: dnet based on raster version released under GFDL, Angular velocityCC BY-SA 3.0

## Angular velocity formula

As discussed above, angular position of an object changes with time in angular motion. The rate at which it changes is called as angular velocity. It can be given in terms of angular terms as well as it can be derived from terms used in linear motion also.

Angular velocity can be found using the linear velocity if the radius of motion is known. Let us see the different formulae of angular motion below-

In relation to terms of linear motion- V=rω

where,

r is the radius of angular trajectory followed by the object

V is the linear velocity of the object that is perpendicular to the circular path

ω is the angular velocity of the object

In relation to terms of angular motion- ω = dθ/dt

where,

theta is the angular displacement of the object

omega is the angular velocity of the object

dt is the instantaneous time considered

## When is angular velocity constant?

When an object is moving in a circle with same speed, it can be inferred that it is having a constant angular velocity.

Thus we can say that angular velocity of an object is constant when it is moving at the same speed in an angular motion. The revolution of Earth is an example constant angular motion. It takes 365 days to complete motion around the Sun.

## Why is angular velocity constant?

The reason for angular velocity to be constant is that the object in circular motion covers equal angular displacement all the time.

This happens when the object is having the same speed throughout the course of angular motion. There equal projections of displacement made by the object when it is moving with constant angular velocity.

## Angular velocity example

Angular velocity is defined for the objects in angular motion. We can see angular motion around us every where. It is a common thing to observe angular motions of objects.

Angular motions are important for our survival. Let us look at some examples of angular motion-

• Merry go round- A merry go round is a ride in which the children sit at the circumference of the ride, as the ride is switched on it starts rotating around its centre. The children also start rotating around the centre of rotation.
• Revolution of Earth– Earth revolves around the Sun with a time period of 365 days. This is an example of angular motion of Earth around the Sun.
• Rotation of Earth– Earth rotates around its axis (also called as axis rotation). The surface of the Earth can be said to be in angular motion.
• Toy top– A toy top rotates around its centre of rotation. It follows an angular motion.
• A ceiling fan– The blades of ceiling fan follow angular motion when the fan is switched on.
• A ballet dancer– A ballerina rotates with toes as her centre of rotation. The speed of rotation changes as she spreads and constricts her hand movements.
• A stone tied with thread- This is the most common example of angular motion. The stone experiences an angular motion when it is tied to a thread and rotated.
• A car or bike running on a circular path– The vehicles experience angular motion while running on a circular or curved path.
• Wheels of automobiles– The wheels of automobiles follow an angular motion. This motion is converted to linear motion of the vehicle.
• A giant wheel– The giant wheel moves in angular motion. The cabins are located at the circumference of the wheel. These cabins follow angular motion along with the wheel.

## Angular velocity types

The angular velocity are broadly classified into two types. We will discuss about these types with some examples from daily lives in the section given below.

Types of angular velocity are-

• Orbital angular velocity– This type of velocity exists in an object when the objects as a whole is revolving around a common centre of rotation. The example which explains this best is Earth revolving around the sun. The Earth as a whole follows a circular motion around the Sun.
• Spin angular velocity– This type of angular velocity exists for a object which spins or rotates itself around its centre. All the particles or molecules of the object revolve with the centre of object as centre of rotation. The best example to explain spin angular velocity is the rotation of Earth around its axis of rotation. The surface of Earth (including ground, river and crust) revolves around the centre of Earth.

## Constant angular velocity example

In the above sections we discussed about general examples of angular velocity. In this section we shall discuss about examples of constant angular velocity.

Constant angular velocity examples are given below-

• Revolution of Earth around Sun- As the Earth takes same amount of time to complete one revolution, we can say that it moves with constant angular velocity.
• Rotation of Earth around its axis of rotation– Earth takes 24 hours to complete one rotation, although the number is slightly decreasing but the change is very negligible. We can say that it follows a motion with constant angular velocity.
• Fan at same speed- When the regulator of the fan is untouched and the fan rotates at the same speed, we can say that the fan is rotating with constant angular velocity.
• Revolution of Moon around Earth- The Moon takes equal amount of time to complete one revolution around Earth, we can say that it follows a constant angular velocity motion.
• Wheels of a vehicle moving at a constant velocity– When the vehicle is moving at a constant velocity, it is obvious that the wheels are also rotating at constant speed or constant angular velocity.

The angular velocity will not change if there is little or no resistance to the angular motion. Without any resistance the objects can attain angular velocity motion easily.

## What happens to acceleration when angular velocity is constant?

The rate of change of angular velocity of an object is called as angular acceleration. As and when the magnitude of angular velocity changes, the magnitude of angular acceleration also changes.

Thus we can say that angular acceleration will change only if there is a change in angular velocity. If there is no change in angular velocity then the angular acceleration will be zero.

## Conversion of angular motion to linear motion

The conversion of one type of motion to another is a very useful tool which help us perform various tasks. Some commonly known examples where we can see angular motion converted to linear motion are listed below-

• Automobile– The circular motion of wheels are translated to linear velocity of the automobile.
• Rack and Pinion– The movement of rack and pinion is an example of inter conversion of linear motion to angular motion.
• Hammer throw– While the athlete rotates the hammer, the hammer follows angular motion. When the athlete releases the hammer, the hammer starts following a parabolic trajectory.
• Baller in cricket-The arms of baller follows an angular motion, the ball also copies this angular motion. It is then converted to linear motion when the ball is released.
• Hitting a golf ball– The tip of the golf stick follows an angular motion. The angular motion is converted to linear motion as the ball follows a parabolic path after getting hit.

Abhishek

Hi ....I am Abhishek Khambhata, have pursued B. Tech in Mechanical Engineering. Throughout four years of my engineering, I have designed and flown unmanned aerial vehicles. My forte is fluid mechanics and thermal engineering. My fourth-year project was based on the performance enhancement of unmanned aerial vehicles using solar technology. I would like to connect with like-minded people.