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 = (θf – θi) / t
ω = angular velocity
θf = the final angle
θi = 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 θi = 0 degrees and moves to θf = 180 degrees from the point of origin. 180 degrees = 1/2 of a full revolution, so θf = (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 = (θf – θi) / 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 (θf – θi) = (150) x (2π rad). The time, t, is 1 min, or t = 60 s. Use the formula for angular velocity.
ω = (θf – θi) / t
ω = (150 x 2π ) rad / 60 s
ω = 300π rad / 60 s = 5π rad/s