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 ωav.
ωav = 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 ωav = 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 = Ltt -> 0 Δθ/ Δt = dθ/dt
Angular velocity is measured in radian per second and its unit is rad/s.
Since dθ is 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 r
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.
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
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
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.