 # Angular Velocity

Angular velocity is an important physics concept that is applied to objects that move along a circular path. In this article, we will explore the definition of angular velocity and the relationship between angular velocity and linear velocity.

## What is Angular Velocity?

Angular velocity defines the speed of an object that revolves or rotates relative to another point. It explains the degree of swiftness in the change of orientation or the angular position with time. Angular velocity is known by various terminologies such as rotational velocity or angular frequency vector.

Angular velocity is defined as “The rate of velocity at which a particle or an object rotates around a centre or a specific point in a given time”.

Or

In simple words, angular velocity is defined as “The amount of change of angular displacement of the particle at a given period of time”.

Angular velocity is represented by $$\omega \ or\ \Omega$$.

Angular velocity is measured in angle per unit time or radians per second (rad/s). Hence, the SI unit of angular velocity is radians per second. It is a vector quantity. It is represented as:

$$\omega = \frac{d\Theta }{dt}$$

Where,

• dθ is the change in angular displacement
• dt is the change in time t

Note: The positive value indicates anti-clockwise rotation, and the negative value indicates clockwise direction.

### Types of Angular Velocity

There are two types of angular velocity, they are:

• Orbital angular velocity
• Spin angular velocity

#### Orbital Angular Velocity

Orbital Angular Velocity explains how fast a point object can revolve about a fixed origin, i.e. the time rate of change of its angular position relative to the origin. Example: Consider a particle moving in the plane. The orbital angular velocity of the particle is given by the rate at which the position vector relative to a chosen origin “sweeps out” angle. It is the measure of the fastness of a point object that revolves around a fixed origin.

#### Spin Angular Velocity

Spin angular velocity measures how fast a rigid body can rotate with respect to its centre of rotation. Spin angular velocity is independent of the choice of origin. We know that angular velocity ω is analogous to linear velocity v. Let us learn the relation between angular and linear velocity.

## Relation Between Angular Velocity and Linear Velocity

When an object/particle completes one revolution, the angle covered by the radius vector is 360 degrees or 2π radians. If T is the time taken to complete one revolution, the angular velocity of the particle is given by:

ω= θ/t = 2 π/T .

For n revolutions per second, then ω=2π(1/T) = 2πn

where n = 1/T is the frequency of revolution.

Consider the figure shown below. Assume object P moving along the circumference of a radius r with linear velocity v and angular velocity ω. When the object moves from P to Q in time dt and dθ be the angle swept by the radius vector. Let us consider ds to be the arc covered by P when moving along the circle. Therefore, PQ=ds.

The angular displacement d θ is expressed as dθ = ds/r.

ds=vdt

dθ/dt=v/r

Angular velocity ω = v/r or v =ω r

In vector notation,

$$\vec{v}=\vec{\omega }*\vec{r}$$

To understand the concept of angular velocity in detail, refer to the video below. ## Angular Velocity Practice Problem

Calculate the angular velocity of a 0.500 m radius bus tyre when the bus travels at 10.0 m/s (about 36 km/h).

Solution

Given:

v = 10.0 m/s

The radius of the tyre is given to be r = 0.500 m

We know that v= rω

or

ω= v/r

ω= 10.0 /0.500 = 20.0 rad/s

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## Frequently Asked Questions – FAQs

### What is Angular Acceleration?

The time rate of change of angular velocity is known as angular acceleration.

### What is Orbital Angular Velocity?

Orbital Angular Velocity explains how fast a point object can revolve about a fixed origin, i.e. the time rate of change of its angular position relative to the origin.

### What are the types of angular velocity?

There are two types of angular velocity, they are:

• Orbital angular velocity
• Spin angular velocity

### Define angular velocity.

The amount of change of angular displacement of the particle at a given period of time gives the angular velocity.