# Angular Velocity Formula

**Average angular velocity **is defined as the ratio of angular displacement to the time taken by the object to undergo the displacement. It is denoted by ω_{av}.

ω_{av}= angular displacement / time travel

Consider the movement of a particle on a circular path with a centre at O.

Let us assume that in time **Δt** the particle displaces by an angle of **Δθ**.

then the average angular velocity** ω _{av} = Δθ/ Δt **

Since Δθ is scalar, average angular velocity is a scalar quantity.

**Instantaneous angular velocity**

It is the limiting value of the average angular velocity of the object in a small time interval, as the time interval approaches zero. It is denoted by ω.

ω = Lt_{t -> 0 }Δθ/ Δt =dθ/dtAngular velocity is measured in radian per second and its unit is

rad/s.Since

dθis a vector, ω is also a vector.

**Relation between angular velocity and linear velocity.**

Linear velocity is the cross product of angular velocity and radius of the circular path.

v = ω x rwhere,

v = linear velocity,

ω = angular velocity &

r = position vector from the centre of the circular path (radius).

The terms time period and frequency are synonymous with angular velocity, therefore it’s necessary to define them alongside angular velocity.

**Time Period **

The time taken to complete one revolution or to displace by an angle of 2π radians is called time period and is denoted by **T**.

as angular velocity ω is equal to angular displacement/time, hence time period is related to angular velocity as

ω = 2π/**T**

T=2π/ω

**Frequency **

The frequency of an object is the number of revolutions completed in a second. It is denoted by the Greek letter **nu** (**ν**).

ν = 1/T = ω/2π

**Sample Problems**

**Problem 1: Calculate the angular velocity of the minute hand of a clock.**

The minute hand completes one rotation in 60 minutes.

=> it displaces by 2π radians in 60*60 seconds

=> ω = 2π/3600 = 1.74 * 10

^{-3}rad/sec.

**Problem 2: A ball is revolving in a circle of diameter 4 m with velocity 20 m/s find its angular velocity.**

radius of circle = diameter/2 = 4/2 = 2m

linear velocity = 20 m/s

angular velocity = v / r

ω = 20 / 2 = 10 rad/sec

**Problem 3: An object revolves in a horizontal circle of radius 12 m with a frequency of 4 hz. Find its linear speed.**

Linear speed v = ω * r

frequency (f) = ω/2π

=> ω = 2π * f

=> v = 2π * f * r

=> v = 301.59 m/s.

**Problem 4: An object has a constant angular velocity of 5 rad/s. What is its angular displacement in radians after 12 seconds?**

Angular velocity = angular displacement/ time

=> angular displacement = angular velocity * time

=> angular displacement = 5 * 12 = 60 radians.