Angular Velocity

What is Angular Velocity?

Angular velocity tells us how fast something is rotating. It shows how quickly an object moves around a circular path. In simple words, angular velocity means the rate at which an object turns or spins around a point.

If you have ever watched a fan or a spinning wheel, you’ve seen angular velocity in action.

Angular Velocity Definition

In physics, angular velocity is the rate of change of angular displacement with respect to time. It measures how fast an angle is changing when something rotates.

Mathematically:

Angular Velocity (ω) = Angular Displacement (θ) / Time (t)
ω = θ / t

Where:

• ω (omega) = Angular velocity
• θ (theta) = Angular displacement in radians
• t = Time in seconds

Units of Angular Velocity

The most common unit of angular velocity is:

• Radian per second (rad/s)

Other units you may see:

• Degrees per second
• Revolutions per minute (RPM)

SI Unit:

• The SI unit of angular velocity is radian/second.

Angular Speed vs Angular Velocity

Many people confuse angular speed and angular velocity. Let’s clear that up.

So, angular velocity is a vector, meaning it has both magnitude and direction.

Instantaneous and Average Angular Velocity

Instantaneous angular velocity is the angular velocity at a specific moment in time.

Average angular velocity is calculated over a time interval.

Formula:
ω = (θ₂ - θ₁) / (t₂ - t₁)

Uniform Circular Motion and Angular Velocity

In uniform circular motion, the angular velocity stays the same throughout. This happens when an object moves around a circle at a constant speed.

Relation Between Angular and Linear Velocity

Linear velocity is the speed of something moving along a straight path, while angular velocity deals with circular paths.

The relationship is given by:

v = r × ω

Where:

• v = Linear velocity
• r = Radius of the circular path
• ω = Angular velocity

So, the larger the radius, the faster the object moves in a straight line, even if the angular speed stays the same.

Angular Displacement and Angular Velocity

Angular displacement is the angle through which an object has moved. When angular displacement increases over time, we get angular velocity.

So, angular velocity is how fast the angular displacement is happening.

How to Calculate Angular Velocity

Use the formula:
ω = θ / t

Example:
If a wheel rotates radians in 2 seconds, then:
ω = / 2 = rad/s

To convert RPM to rad/s:
ω = (2π × RPM) /

Constant Angular Velocity

When an object rotates at a steady rate, it has constant angular velocity. This often happens in motors, fans, or wheels turning without speeding up or slowing down.

Direction of Angular Velocity

Angular velocity is a vector. The direction is given by the right-hand rule:

• Curl your fingers in the direction of rotation.
• Your thumb points in the direction of the angular velocity.

Clockwise motion = negative direction
Counterclockwise motion = positive direction

Yes, angular velocity can be negative depending on the direction.

Applications of Angular Velocity

You can find angular velocity in:

• Ceiling fans
• Gyroscopes
• Rotating wheels and gears
• Satellite motion
• Planetary orbits
• CD/DVD players
• Car tires

Angular Velocity in Everyday Life

• The Earth has an angular velocity as it spins on its axis.
• A fidget spinner rotating in your hand shows angular velocity.
• Bicycle tires while riding.
• Washing machines during the spin cycle

Angular Velocity in Physics (Class 11 & Class 12)

In Class 11 and 12 Physics, angular velocity is a key topic in:

• Rotational motion
• Mechanics
• Kinematics of circular motion

Students learn how angular velocity connects to torque, acceleration, and moment of inertia.

Summary Table of Key Concepts