Angular Velocity Formula: Definition, Unit, Equation and Solved Examples

The Angular Velocity Formula is ω = Δθ / Δt, where ω is angular velocity, Δθ is angular displacement and Δt is the time taken.
For circular motion, angular velocity can also be written as ω = 2π / T, ω = 2πf and ω = v / r.

The Angular Velocity Formula helps students calculate how fast an object rotates or moves around a circular path. It is used when angular displacement, time, frequency, time period, linear velocity or radius is given. The SI unit of angular velocity is radians per second, written as rad/s.

In Class 11 Physics, angular velocity is an important part of rotational motion and circular motion. CBSE, ICSE, state board, JEE foundation and NEET foundation questions often ask students to find angular velocity from time period, frequency, angular displacement or linear speed.

Key Takeaways

• Angular Velocity Formula: ω = Δθ / Δt.
• SI Unit: The angular velocity unit is rad/s.
• Time Period Formula: ω = 2π / T.
• Frequency Formula: ω = 2πf.
• Linear Relation: ω = v / r, where v is linear velocity and r is radius.

Angular Velocity Formula Structure 2026

What is Angular Velocity Formula?

The Angular Velocity Formula gives the rate of change of angular displacement with respect to time. It tells how quickly an object rotates around a fixed point or axis.

Formula:

ω = Δθ / Δt

Where:

• ω = angular velocity
• Δθ = angular displacement
• Δt = time interval
• θ is measured in radians
• t is measured in seconds

The SI unit of angular velocity is:

rad/s

Angular velocity is commonly represented by the Greek letter omega, written as ω.

Angular Velocity

Angular velocity is the rate at which angular displacement changes with time. In simple words, it tells how fast an object is rotating.

For example, a fan blade, spinning wheel, rotating Earth, moving clock hand and turning bicycle wheel all have angular velocity.

If an object covers a larger angle in less time, its angular velocity is higher. If it covers a smaller angle in more time, its angular velocity is lower.

Formula:

ω = Δθ / Δt

This is the basic omega formula used in rotational motion.

Angular Velocity Equation

The angular velocity equation depends on the information given in the question. The most common equation is based on angular displacement and time.

Basic equation:

ω = Δθ / Δt

If the object completes one full rotation, the angular displacement is:

Δθ = 2π radians

So, for one complete revolution:

ω = 2π / T

Where:

• T = time period
• 2π radians = one full revolution

If frequency is given:

ω = 2πf

Where:

• f = frequency
• frequency means number of rotations per second

Angular Velocity Formula Physics

In Physics, angular velocity is used in rotational motion and circular motion. It connects angular displacement, time period, frequency and linear velocity.

Important formulas:

ω = Δθ / Δt

ω = 2π / T

ω = 2πf

ω = v / r

v = rω

Where:

• ω = angular velocity
• θ = angular displacement
• T = time period
• f = frequency
• v = linear velocity
• r = radius

These formulas are used in circular motion numericals, rotating bodies and mechanical systems.

Angular Speed Formula

Angular speed formula gives the magnitude of angular velocity. It measures how fast an object rotates, without considering direction.

Formula:

ω = θ / t

or

ω = Δθ / Δt

Where:

• ω = angular speed
• θ = angular displacement
• t = time taken

Angular speed is a scalar quantity, while angular velocity is a vector quantity. In many school-level numericals, the same formula is used for both when direction is not discussed.

Angular Displacement Formula

Angular displacement is the angle covered by a rotating object. It is usually measured in radians.

Angular displacement formula from angular velocity:

θ = ωt

or

Δθ = ωΔt

Where:

• θ or Δθ = angular displacement
• ω = angular velocity
• t or Δt = time

From the basic angular velocity formula:

ω = Δθ / Δt

Rearranging:

Δθ = ωΔt

This formula is useful when angular velocity and time are given.

Angular Velocity Unit

The SI unit of angular velocity is radians per second.

Unit:

rad/s

This comes from the formula:

ω = Δθ / Δt

Since angular displacement is measured in radians and time is measured in seconds:

Angular velocity unit = radian / second

So:

ω = rad/s

Other units may include revolutions per second or revolutions per minute, but rad/s is the standard SI unit.

Radians Per Second

Radians per second is the standard unit used to measure angular velocity. One complete revolution is equal to 2π radians.

Important conversion:

1 revolution = 2π radians

If an object completes 1 revolution in 1 second:

ω = 2π rad/s

If an object completes f revolutions per second:

ω = 2πf rad/s

This is why frequency and angular velocity are connected by the formula:

ω = 2πf

Time Period Angular Velocity Formula

Time period is the time taken by an object to complete one full revolution.

For one revolution:

Angular displacement = 2π radians

Time taken = T

So:

ω = 2π / T

Where:

• ω = angular velocity
• T = time period
• 2π = angular displacement for one full revolution

Example:

If a wheel completes one rotation in 2 seconds, then:

ω = 2π / T

ω = 2π / 2

ω = π rad/s

The angular velocity is π rad/s.

Frequency and Angular Velocity Formula

Frequency is the number of complete rotations made per second. It is represented by f.

Formula:

ω = 2πf

Where:

• ω = angular velocity
• f = frequency
• 2π radians = one complete revolution

If frequency increases, angular velocity also increases.

Example:

If a fan rotates with frequency 5 Hz, then:

ω = 2πf

ω = 2π × 5

ω = 10π rad/s

The angular velocity is 10π rad/s.

Linear Velocity and Angular Velocity Formula

Linear velocity and angular velocity are related in circular motion. Linear velocity tells how fast a point moves along the circular path, while angular velocity tells how fast the angle changes.

Formula:

v = rω

So:

ω = v / r

Where:

• v = linear velocity
• r = radius of circular path
• ω = angular velocity

If the linear speed increases, angular velocity increases. If the radius increases while speed remains the same, angular velocity decreases.

Example:

If v = 20 m/s and r = 5 m, then:

ω = v / r

ω = 20 / 5

ω = 4 rad/s

The angular velocity is 4 rad/s.

Rotational Motion Formula

Angular velocity is one of the main rotational motion formulas. It is used with angular displacement, angular acceleration and time.

Important rotational motion formulas:

ω = Δθ / Δt

v = rω

ω = 2π / T

ω = 2πf

If angular acceleration is involved:

ω = ω₀ + αt

Where:

• ω = final angular velocity
• ω₀ = initial angular velocity
• α = angular acceleration
• t = time

For basic school-level circular motion questions, ω = Δθ / Δt, ω = 2π / T, ω = 2πf and ω = v / r are the most used formulas.

Difference Between Angular Velocity and Linear Velocity

Angular velocity and linear velocity describe motion in circular paths, but they measure different quantities.

A point farther from the centre has greater linear velocity for the same angular velocity.

How to Find Angular Velocity

To find angular velocity, first check which values are given in the question.

Case 1: When angular displacement and time are given

Use:

ω = Δθ / Δt

Example:

If Δθ = 12 rad and Δt = 4 s:

ω = 12 / 4

ω = 3 rad/s

Case 2: When time period is given

Use:

ω = 2π / T

Example:

If T = 5 s:

ω = 2π / 5 rad/s

Case 3: When frequency is given

Use:

ω = 2πf

Example:

If f = 4 Hz:

ω = 2π × 4

ω = 8π rad/s

Case 4: When linear velocity and radius are given

Use:

ω = v / r

Example:

If v = 30 m/s and r = 10 m:

ω = 30 / 10

ω = 3 rad/s

Solved Examples on Angular Velocity Formula

Angular Velocity Formula questions usually ask students to calculate angular velocity from angular displacement, time period, frequency or linear velocity.

Example 1: Find angular velocity when angular displacement is 20 rad in 5 s

Given:

Δθ = 20 rad

Δt = 5 s

Formula:

ω = Δθ / Δt

Substitute:

ω = 20 / 5

ω = 4 rad/s

Answer:

The angular velocity is 4 rad/s.

Example 2: Find angular velocity when time period is 4 s

Given:

T = 4 s

Formula:

ω = 2π / T

Substitute:

ω = 2π / 4

ω = π / 2 rad/s

Answer:

The angular velocity is π/2 rad/s.

Example 3: Find angular velocity when frequency is 6 Hz

Given:

f = 6 Hz

Formula:

ω = 2πf

Substitute:

ω = 2π × 6

ω = 12π rad/s

Answer:

The angular velocity is 12π rad/s.

Example 4: Find angular velocity when linear velocity is 40 m/s and radius is 8 m

Given:

v = 40 m/s

r = 8 m

Formula:

ω = v / r

Substitute:

ω = 40 / 8

ω = 5 rad/s

Answer:

The angular velocity is 5 rad/s.

Example 5: Find linear velocity when angular velocity is 10 rad/s and radius is 3 m

Given:

ω = 10 rad/s

r = 3 m

Formula:

v = rω

Substitute:

v = 3 × 10

v = 30 m/s

Answer:

The linear velocity is 30 m/s.

Common Mistakes in Angular Velocity Formula

Many Angular Velocity Formula mistakes happen when students use degrees instead of radians. Angular velocity in SI units must be written in rad/s.

Important checks:

• Use radians for angular displacement.
• Use seconds for time.
• Write angular velocity unit as rad/s.
• Use ω = Δθ / Δt when angular displacement and time are given.
• Use ω = 2π / T when time period is given.
• Use ω = 2πf when frequency is given.
• Use ω = v / r when linear velocity and radius are given.
• Convert one full revolution into 2π radians.

For circular motion questions, check whether the question gives frequency, time period or radius before choosing the formula.

Applications of Angular Velocity Formula

The Angular Velocity Formula is useful in Physics, engineering, astronomy and daily-life rotational motion. It helps describe how fast objects rotate.

Main applications:

• It calculates the rotation of wheels.
• It is used in circular motion questions.
• It helps study rotating fans and turbines.
• It is used in planetary motion and satellite motion.
• It helps relate linear speed with circular motion.
• It is used in engines, gears and machines.
• It supports rotational motion numericals in Class 11 Physics.