## Table of Contents

- What is Angular Velocity?
- Angular Velocity Formula
- How to Find Angular Velocity
- How Angular Velocity is Used
- Lesson Summary

What is angular velocity? Find out with an angular velocity definition and examples. Learn how to find angular velocity using the angular velocity formula.
Updated: 07/27/2021

- What is Angular Velocity?
- Angular Velocity Formula
- How to Find Angular Velocity
- How Angular Velocity is Used
- Lesson Summary

To understand the concept of angular velocity, it is important to first know what **rotation** is. Think of a spinning top or a windmill. These objects are rigid bodies and rotate about a fixed axis. A **rigid body** is one that rotates with all of it locked together, without any change in the shape of the body, and a **fixed axis** is an axis that does not move. In the case of the spinning top, the axis of rotation passes through the body of the top itself; for the windmill, the blades rotate around an axis that passes through the rotor. In both cases, the objects are exhibiting rotational motion. Some other examples of bodies exhibiting rotational motion are:

- A spinning ice skater
- The rotation of the earth
- A rotating ceiling fan

The velocity associated with rigid bodies as they exhibit rotation about a fixed axis is called **angular velocity**. Angular velocity is generally represented by the Greek letter **omega** ({eq}\mathbf{\omega} {/eq}).

Angular velocity in rotational motion is analogous to **linear velocity** in translational motion. When a body travels in a straight line, the change in its linear position per unit time gives the linear speed; when the direction of motion is also considered, the linear velocity of the body is defined. Similarly, when a body exhibits rotational motion, the change in its angular position per unit time along with the direction of motion gives the angular velocity of the body.

The angular velocity and linear velocity of a body can be related to each other. If a body is rotating about a fixed axis with an angular velocity of {eq}\omega {/eq}, the linear speed {eq}v {/eq} of any point within the body at a distance {eq}r {/eq} from the axis of rotation is given by

{eq}v=\omega r {/eq}.

Before we look at the formula for angular velocity, let's first understand the more fundamental concepts of angular position and angular displacement. Consider the figure below.

A rigid body is rotating about the z-axis with angular velocity {eq}\omega {/eq}. Every particle of the rigid body rotates about the z-axis and traces a circle lying on a plane perpendicular to the axis of rotation (z-axis) with its center on the axis. Now, consider a rigid body particle at a distance {eq}r {/eq} from the axis. Its angular position with respect to a reference line (x-axis) is {eq}\theta {/eq}. As time passes, the particle rotates about the z-axis and after time {eq}\Delta t {/eq}, its angular position becomes {eq}\theta' {/eq}. Thus, the angular displacement of the particle, {eq}\Delta \theta =\theta '-\theta {/eq}.

The angular velocity is defined as the angular displacement per unit time. Therefore, the angular velocity of the particle

The angular position (and angular displacement) is measured in units of radians (rad). A radian is a measure of an angle. It is defined as the ratio of an arc of a circle to the radius of the circle. Thus, 1 rad is the angle for which the length of the arc is equal to the radius of the circle.

Another common measure of angles is degrees ({eq}^{\circ} {/eq}). The conversion of the unit of angles from radians to degrees and from degrees to radians is pretty simple. If we go around an entire circle, the length of the arc traversed is equal to the circumference of the circle, i.e., {eq}2\pi r {/eq}. So, the number of radians in the whole circle is,

Now, we know that the total angle of a circle in degrees is {eq}360^{\circ} {/eq}.

The table below shows some measures of angles in radians and degrees.

Angular velocity is the angular displacement of a body per unit of time. As radians (rad) and seconds (s) are the SI units of angular displacement and time, respectively, the SI unit of angular velocity is **radian per second (rad/s)**. Angular velocity is also sometimes expressed in terms of **revolution per second (rev/s)**. Another unit of angular velocity is one that was commonly used when vinyl records were in, the unit of **revolution per minute (rev/min)**, commonly termed **rpm**.

We know, {eq}1\: rev=2\pi \: rad {/eq}. Thus, {eq}1\: rev/s=2\pi \: rad/s {/eq}.

Again, {eq}1\:s=\frac{1}{60}\:min {/eq}. Thus, {eq}1\: rpm=\frac{2\pi}{60} \: rad/s {/eq}.

Now that we've learned the formula and units of angular velocity, let's look at a few interesting examples.

Consider a Ferris wheel at a carnival. If the Ferris wheel completes one rotation in {eq}3\:min {/eq}, what is its angular velocity in rad/s and rpm?

We have,

Here, {eq}\Delta \theta=2\pi {/eq} (one full rotation) and {eq}\Delta t=3\: min=\left ( 3\times 60 \right )\: s=180\: s {/eq}.

Thus, the angular velocity of the Ferris wheel in rad/s is,

Now,

A mountain bike has wheels of diameter {eq}61\:cm {/eq}. An amateur biker rides for {eq}1.22\:m {/eq} to test out the bike and finds that it takes him {eq}2\:s {/eq}. What is the angular velocity of the wheel of the bike?

We have, angular velocity

Here, {eq}arc=1.22\:m=122\:cm {/eq}, {eq}radius=61/2\:cm {/eq}, and {eq}t=2\:s {/eq}.

What should the minimum angular velocity of a race car be so that it has a speed of {eq}150\:mph {/eq} or above when racing along a circular track of radius {eq}0.3\: miles {/eq}?

We have

Here {eq}v=150\:mph {/eq} and {eq}r=0.3\: miles {/eq}.

Thus, the race car should have a minimum angular velocity of {eq}500\:rad/h {/eq}.

You might remember how at the beginning of the lesson we said that angular velocity involves both magnitude and direction. In the above examples, the angular velocity we determined was actually the magnitude of angular velocity. The direction of the angular velocity vector can be determined by the **right-hand rule**.

According to the **right-hand rule**, if you curl your right hand around the axis of rotation of a rotating body with your fingers pointing in the direction of rotation, the direction of your extended thumb in the direction of the angular velocity vector. Thus, here, the rigid body rotates around the direction of the angular velocity vector.

To understand the relationship between **torque** and angular velocity, let us first understand what torque signifies. The torque or moment of force is the force involved in the rotation of an object. Torque in rotational motion is equivalent to force in translational motion. Consider a lever that rotates about a fixed point. To rotate this lever, you need to apply a force at any point along the length of the lever; this force that causes the rotational motion of the lever is called the torque. The magnitude of torque is the product of the amount of force applied and the perpendicular distance between the point of application of force and the fixed point about which the lever rotates.

As mentioned earlier, torque is similar to simple force. The analogous nature of torque and force is evident from the formulae of power in rotational and translational motions.

By definition, power {eq}P {/eq} is the work done {eq}W {/eq} in unit time {eq}t {/eq}. In mathematical form,

Now, the work done can be expressed as the product of force {eq}F {/eq} and displacement {eq}d {/eq}; further, the displacement per unit time gives the velocity {eq}v {/eq}.

As the relation between linear velocity {eq}v {/eq} and angular velocity {eq}\omega {/eq} is given by {eq}v=\omega R {/eq}, the expression of power in rotational motion becomes {eq}P=F\omega R {/eq}.

We can now bring in the torque {eq}\tau {/eq} here (which is the product of force and perpendicular distance). Thus, we have

{eq}P=\tau \omega {/eq}.

This shows how the torque and angular velocity are related, and this expression is analogous to

{eq}P=Fv {/eq},

where the torque {eq}\tau {/eq} is replaced by simple force {eq}F {/eq} and angular velocity {eq}\omega {/eq} is replaced by linear velocity {eq}v {/eq}.

In rotational motion, **angular acceleration** {eq}\alpha {/eq} is the change in angular velocity {eq}\omega {/eq} in unit time {eq}t {/eq}. In mathematical form,

The kinetic energy associated with rotational motion is referred to as the **rotational kinetic energy**. It depends on the angular velocity {eq}\omega {/eq} and moment of inertia {eq}I {/eq} of the rotating body.

When a body of mass {eq}m {/eq} rotates about a fixed point which is at a distance {eq}r {/eq} from it, the body feels an outward force acting on. This force is called the **centrifugal force**, and it acts away from the fixed point about which the body is rotating. The centrifugal force is given by

As the Earth rotates about its axis, the circulating air in the atmosphere gets deflected towards the right in the Northern Hemisphere and towards the left in the Southern Hemisphere. This deflection of air currents due to the rotational motion of the Earth is called the **Coriolis Effect** and the force associated with it is called the Coriolis Force. The magnitude of this force is given by

A pulley is a small wheel with a grooved rim on which a thread or rope can be attached. When a heavy load is fixed to one end of the rope, the load can be easily pulled up by exerting a downward force on the other (free) end of the rope attached to the pulley. More than one pulley can be appropriately arranged to form a **system of pulleys**, which makes moving the heavy load even easier.

As the pulley wheel is rotating, it has an associated angular velocity {eq}\omega {/eq}. The angular velocity associated with the pulley can be found out by equating the potential and kinetic energies of the system, as follows.

Substituting the expressions for the three energies, we have

Wrapping up this lesson, let's look back at some important concepts that were discussed. **Rotation** can be defined as a **rigid body** moving about a **fixed axis**, wherein all the particles of the body are locked in place. The velocity associated with the rotational motion of a body is called the **angular velocity**. The angular velocity of a rotating body is defined as the change in its angular displacement per unit of time. Angular velocity is an important concept in mechanics, and it determines several physical quantities, such as **angular acceleration**, **rotational kinetic energy**, **centrifugal force**, **Coriolis force**, and the motion of a **system of pulleys**.

Here are some highlights of the lesson:

- Angular velocity {eq}\omega {/eq} has several associated formulae that can be used in different contexts: {eq}\omega=\frac{\Delta \theta }{\Delta t}=\frac{arc}{radius}\times \frac{1}{t}=\frac{v}{r} {/eq}

- Angular velocity {eq}\omega {/eq} is related to
**linear velocity**{eq}v {/eq} by the formula {eq}v=\omega r {/eq}

- The SI unit of angular velocity is
**rad/s**, and its other units are**rev/s**and**rev/min**(or**rpm**)

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Additional Activities

Angular velocity describes the rate at which an object rotates and the direction it rotates. It is standard convention for the counterclockwise direction to be the positive direction. Let's practice some angular velocity problems. For constant angular velocity the ratio of the angular displacement (θ) to the time of motion is the angular velocity.

1. A ball on a string is rotating clockwise and it covers 20 radians in 5 seconds. What is its angular velocity?

2. A merry-go-round is rotating counterclockwise. It makes 10 revolutions in 23 seconds. What is its angular velocity?

3. How many revolutions does a turntable rotate through if it is moving at 32 rad/s and has been turning for 4 minutes?

1.

- θ = 20 rad
- t = 5 sec
- ω = ?

ω = θ / t

ω = 20 rad / 5 s

ω = 4 rad/s

2.

- θ = 10 rev x (2π rad) / (1 rev) = 62.83 rad
- t = 5 s
- ω = ?

ω = θ / t

ω = 62.83 rad / 5 s

ω ≈ 12.57 rad/s

3.

- θ = ?
- ω = 32 rad/s
- t = 4 min x (60 seconds) / (1 min) = 240 s

ω = θ / t

32 rad/s = θ / 240 s

θ = 7680 rad

7680 rad x (1 revolution / 2π radians) = 1223 revolutions

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