# Angular Velocity formula — definition, examples

The angular velocity formula describes how quickly an object rotates or revolves. It is relative to another point. Also, there is also another definition. It says how quickly an object’s angular position or orientation changes over time. Again, angular velocity is classified into two types. They are orbital angular velocity and spin angular velocity.

The spin angular velocity of a stern body is the rate at which it rotates. It is with respect to its centre of rotation. Orbital angular velocity is the rate at which a point object revolves around a fixed origin. In other words, the time rate at which its angular position changes with respect to the origin.

In general, we express the angular velocity in terms of angle per unit time, such as radians per second.

Moreover, we express the SI unit of angular velocity as radians/sec. Furthermore, the radian has a dimensionless value of unity.

Also, we list the SI units of angular velocity as 1/sec.

We use the symbol omega (ω, sometimes Ω) to represent angular velocity.

Positive angular velocity denotes counterclockwise rotation. On the other hand, negative denotes clockwise rotation.

Thus, it is very important to learn the angular velocity formula.

## What is angular velocity?

Angular velocity is very simple.

Angular velocity is a vector quantity.

It is very crucial to learn the angular velocity formula.

We define it as the rate of change of angular displacement. It specifies an object’s angular speed or rotational speed. Additionally, it specifies the axis about which the object rotates.

Moreover, we refer to angular velocity as the amount of change in the particle’s angular displacement. It is measured over a given period of time.

However, the track of the angular velocity vector is vertical to the plane of rotation. It is in the direction indicated by the right-hand rule.

So, we define angular velocity as the rate of velocity. It is measured when an object or particle rotates around a centre or a specific point. This is measured in a given time period.

Moreover, we also refer to it as rotational velocity.

Angle per unit time, or radians per second (rad/s), is the unit of measurement for angular velocity.

Angular acceleration is the rate of change of angular velocity.

Let us now look at the relationship between angular velocity and linear velocity. Also,  angular displacement and angular acceleration in greater depth.

## Angular velocity formula

Note that it is very important to learn the angular velocity formula.

There are three formulas that we can use to find the angular velocity of an object.

#### 1st formula of angular velocity

This angular velocity formula comes from its definition. It is the rate of change of the position angle of an object with respect to time. So, in this way the formula is

w = θt

w = refers to the angular velocity

θ = refers to the position angle

t = refers to the time

#### 2nd formula of angular velocity

In the second angular velocity formula, we recognize that θ (theta) is given in radians, and the definition of radian measure gives theta = s / r. Also, we can put theta in the first angular velocity formula. This will give us

w = (s / r) / t

On further simplification we get

w = s / (rt)

s = refers to the arc length

r = refers to the radius of the circle

t = refers to the time taken

#### 3rd formula of angular velocity

The third angular velocity formula comes from distinguishing that we can rewrite the second formula as

w = s / (rt)

w = (s / t) (1 / r)

Now recall that s / t is linear velocity. Hence, we can rewrite it as

w = v (1 / r) = v / r

w = is the angular velocity

v = linear velocity

r = is the radius of the circle

## Dimension of angular velocity

The dimensional formula of angular velocity is given by,

[M0 L0 T-1]

Where,

M = Mass

L = Length

T = Time

## Derivation of the dimension of angular velocity

So, the derivation is:

Angular Velocity = Angular displacement × [Time]-1 . . . . (1)

The dimensional formula of Angular displacement = [M0 L0 T0] . . . . (2)

And, the dimensions of time = [M0 L0 T1] . . . (3)

Thus, on substituting equation (2) and (3) in equation (1) we get,

Angular Velocity = Angular displacement × [Time]-1

Or, v = [M0 L0 T0] × [M0 L0 T1]-1 = [M0 L0 T-1]

Therefore, the angular velocity is dimensionally represented as [M0 L0 T-1].

## What is the right hand rule?

The right hand rule is very crucial.

The right hand rule is used to determine the direction of angular velocity. Consider a spinning disc.  Consider a pole passing through the disc’s centre at the axis of rotation. Using the right hand rule, your right hand would grasp the pole with four fingers. It will point in the direction of rotation. Furthermore, your thumb is perpendicular to your other fingers and pointing straight out in the axis.

## What is angular velocity of earth?

Time taken by the earth to complete 1 revolution = t = 24 hours = 24*60*60 = 86400 secs

So, frequency of the revolution of the earth = f = 1/t = (1/86400) sec-1

Hence, angular velocity of earth = ω = 2πf = (2π/86400) rad/sec

## What is angular velocity unit?

We measure Angular velocity in radians per unit time. Or, usually per second. However, we measure linear velocity, in length per unit time. Both are measured differently.

Again, we define the rate of change of an angle as angular velocity.

We represent this by symbols:

ω=

Here an angular rotation takes place in time t.

The angular velocity increases as the rotation angle increases in a given amount of time.

So, radians per second is the SI unit of angular velocity.

However, we can also measure it in other units. For example, degrees per second, degrees per hour, etc.

The symbol omega (Ω or ω) is commonly used to represent angular velocity.

So, it is very important to learn the angular velocity formula.

## Angular velocity equation

It is very important to learn the angular velocity formula. It is also crucial to understand the expression.

First, see that you are talking about “angular” anything. If it is velocity or some other physical quantity, you are talking about travelling in circles. Or portions thereof because you’re dealing with angles.

Read Also: Polynomial Remainder Theorem | Formula

In geometry or trigonometry, the circumference of a circle is equal to its diameter. The diameter is multiplied by the constant pi, or πd. The value of pi is approximately 3.14159.

Also, we more commonly express it in terms of the radius r of a circle. Radius is half the diameter. So, the circumference of it is 2πr.

## Angular velocity expression

We continue our discussion from the above paragraph.

Furthermore, you have probably learned somewhere along the way that a circle has 360 degrees (360°).

Suppose, you move a distance S along a circle. Then we denote the angular displacement by θ. Which is equal to S/r.

Thus, the result of one full revolution is 2πr/r. Thus, it only leaves us with 2π.

That is, angles less than 360° can be expressed in terms of pi, or in radians.

Now, combine all of this information. You can express angles, or portions of a circle, in units other than degrees:

1 radian equals (360°/2π) = 57.3°

Suppose you know that a particle is moving in a circular path. It moves with a velocity v at a distance r. Also, note that it moves from the centre of the circle. In addition to this, you know that the direction of v is always perpendicular to the radius of the circle. So, you can calculate the angular velocity.

ω =

Here, omega (ω) is the Greek letter Angular velocity units are radians per second. We also know this unit as “reciprocal seconds”. It is because v/r equals m/s divided by m or s-1. It indicates that radians are a unitless quantity.

## Angular velocity and circular motion

Only uniform circular motion means that it is a motion in a circle at constant speed.

And thus, constant angular velocity. It was discussed in Uniform Circular Motion and Gravitation. Remember that we define angular velocity as the time rate of angle change θ:

ω=

where θ is the rotational angle that we have displayed in Figure 1.

Rotation Angle and Angular Velocity also defined the relationship between angular velocity and angle velocity. It is:

v = rω

or, ω =

Here, r denotes the radius of curvature (as shown in Figure 1).

The sign convention considers the counterclockwise direction to be positive. Also, the clockwise direction is negative.

This diagram depicts uniform circular motion as well as some of its good quantities.

Look at these examples. When a skater pulls in her arms. A child starts up a merry-go-round from rest. Or, a computer’s hard disc slows to a halt when turned off. Here, angular velocity is not constant. There is an angular acceleration in all of these cases.

Thus, there is ω which changes.

Moreover, the greater the angular acceleration, the faster the change occurs.

We define the rate of change of angular velocity as angular acceleration.

We express Angular acceleration in equation form as follows:

α=

Where  is the angular velocity change and  is the time change.

If ω rises, then α is a positive value. If ω decreases, then α is a negative value.

## Angular Velocity Direction

A point on a rotating object constantly changes direction. So, tracking the angular velocity direction is difficult.

Thus, the rotating object’s axis is the only point where the object has a fixed direction.

The Right Hand Rule determines the direction of angular velocity. It uses the axis of rotation. It is very important to learn the angular velocity formula.

## Angular velocity and linear velocity

Angular velocity is the rate at which the angular position of a rotating body changes. We can define the angular velocity of a particle as the rate at which it rotates around a centre point. That is the time rate of change of its angular displacement. It is relative to the origin.

We define linear velocity as the rate of change of displacement. It is in respect to time when an object moves along a straight path. We define linear velocity as a vector quantity. Let us look at angular velocity in detail in this article.

## What is Angular velocity? example

• Suppose a race car is travelling in a circular path or track. Additionally, it travels 1 lap or 2π radians in 8 minutes. Then calculate the angular velocity of the car.

Solution:

Now let’s put the values in the first formula to get the answer

w = θ / t

w = 2pi / 8 = pi / 4

So, the angular velocity of the race car is pi / 4 radians per minute.

• Now consider another race car travelling along a circular track at 100 kilometres per hour. Also, the radius of the track is 0.4 km. Now, find the angular velocity of the car.

Solution:

We can solve this with the help of the third formula

w = v / r

w = 100 / 0.4 = 250 radians per hour.

So, the angular velocity of the car is 250 radians per hour.

## Revolutions per minute uses

We also use revolutions per minute to describe the rate at which a circular object, such as a wheel, spins.

Also, one revolution is equal to one complete rotation or spin about a centre point. Suppose, a wheel makes one complete rotation about its centre in a minute. So, we say the wheel rotates at a rate of one revolution per minute, or 1 rpm.

Again, it has a rotation rate of one revolution per minute or one rpm. Also,  it takes one minute for the second hand of a clock to complete one complete revolution around its centre.

## Angular Velocity to RPM Conversion

One revolution is 360 degrees. Also, there are 60 seconds per minute.

So, we can convert angular velocity in degrees per second to revolutions per minute. You can do this by multiplying the angular velocity by 1/6.

Moreover, if the angular velocity is 6 degrees per second, the rpm is 1 revolution per minute. It is because 1/6 multiplied by 6 equals 1.

## RPM to Angular Velocity Conversion

One revolution is 360 degrees. Also, there are 60 seconds per minute. So, we can convert revolutions per minute to angular velocity in degrees per second. We can do it by multiplying the rpm by 6.

Read Also: Derivative of ln x – Natural Logarithm

Thus, if the rpm is one revolution per minute, the angular velocity in degrees per second is six. It is because six multiplied by one equals six.

## What is the formula for velocity?

The velocity equation or formula is similar to the speed equation or formula.

So, to calculate velocity, do the following.

Divide the distance by the time it takes to travel the same distance. Then multiply by the direction.

## What is the formula for angular momentum?

p equals m*v.

We define angular momentum (L) as the distance of an object from a rotation axis. It is then multiplied by the linear momentum.

L = r*p or L = mvr.

## What is the formula for angular velocity?

We define it as the change in angle of a moving object (measured in radians) divided by time.

So, there is a magnitude (a value) and a direction to the angular velocity. (360 degrees equals 2 rad.) The second-hand takes 30 seconds to rotate through 180 degrees, so t = 30 s. So, now, you can calculate the angular velocity.

## What is the angular speed example?

The angular velocity, denoted by w, is the rate at which this angle changes with respect to time. Angular Velocity A Ferris wheel, for example, may rotate pi / 6 radians per minute. As a result, the angular velocity of the Ferris wheel is pi / 6 radians per minute.

## Angular velocity FAQs

#### 1.   What exactly is angular velocity?

We define linear velocity as the time rate at which an object rotates or revolves around an axis. Again, it is very important to learn the angular velocity formula.

#### 2.   What is the significance of angular velocity as a vector?

Angular velocity has both magnitude and direction. So, it is a vector quantity.

#### 3.   Is it possible for angular velocity to be negative?

Positive angular velocity denotes counterclockwise rotation, while negative denotes clockwise rotation.

#### 4.   Is the angular velocity constant everywhere?

Every point on an object rotating about an axis has the same angular velocity. On the other hand, there are points further away from the axis of rotation. It moves at a different tangential velocity than points closer to the axis of rotation.

#### 5.   What is the angular velocity’s direction?

The plane of rotation is always perpendicular to the direction of angular velocity.

#### 7.   When you look at a wall clock, what is the direction of the angular velocity vector of the hour hand?

We know that the hour hand rotates clockwise. So, the angular velocity vector is into the wall according to the right-hand rule.

#### 8.   Why is angular velocity a vector?

Because it has both magnitude and direction, angular velocity is a vector quantity.

#### 9.   How do you calculate linear velocity?

We measure linear velocity in metres per second.

#### 10. Which formula expresses the relationship between linear velocity and angular velocity?

The relationship between linear velocity and angular velocity is vi = ωri

#### 11.  Where can you use angular velocity?

We usually use it in physics.

#### 12.  What is the symbol of angular velocity?

The symbol for it is omega. We usually denote it by ω or Ω.