Angular Velocity Formula

Angular Velocity is a measure of how quickly an object moves through an angle. It is the change in angle of a moving object (measured in radians), divided by time. Angular velocity has a magnitude (a value) and a direction.

Angular velocity = (final angle) - (initial angle) / time = change in position/time

ω = (θ_f - θ_i) / t

ω = angular velocity

θ_f = the final angle

θ_i = the initial angle

t = time

Δθ = short form for 'the change in angle'

Angular Velocity Formula Questions:

1) The second hand of a clock takes 30 seconds to move through an arc of 180 degrees. What is the angular velocity?

Answer: The second hand starts at θ_i = 0 degrees and moves to θ_f = 180 degrees from the point of origin. 180 degrees = 1/2 of a full revolution, so θ_f = (0.5 x 2 π). (360 degrees is 2π rad). The time it takes for the second hand to move through 180 degrees is 30 seconds, so t = 30 s. We can now calculate the angular velocity.

ω = (θ_f - θ_i) / t

ω = (0.5 x 2π ) rad / 30 s

ω = (π) rad / 30 s = 3.14 rad / 30 s

ω = 0.105 rad/s

2) You tour a jewelry-polishing facility. Their polishing wheel moves at 150 revolutions per minute. What is it's angular velocity?

Answer: Each revolution is 2π rad. The angle (θ_f - θ_i) = (150) x (2π rad). The time, t, is 1 min, or t = 60 s. Use the formula for angular velocity.

ω = (θ_f - θ_i) / t

ω = (150 x 2π ) rad / 60 s

ω = 300π rad / 60 s = 5π rad/s