# Angular Speed Formula

Angular speed is the rate at which an object changes its angle (measured) in radians, in a given time period. Angular speed has a magnitude (a value) only.

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

ω = θ /t

ω = angular speed in radians/sec

θ = angle in radians (2π radians = 360 degrees)

t = time, sec

Angular speed and angular velocity use the same formula; the difference between the two is that Angular speed is a scalar quantity, while angular velocity is a vector quantity.

Angular Speed Formula Questions:

1) The earth rotates once on its axis every 24 hours. What is its angular speed?

Answer: The angle traversed, 1 rotation, means that θ = 2π. The time for this rotation, t = 24 hr. Time must be converted to seconds.

t = 24 hr x 60 min/hr x 60 sec/min = 86400 sec

ω = θ /t

ω = 2π/86400 sec

ω = 0.0000726 radians/sec = 7.26 x 10-5 rad/sec

2) At the state fair, you take your younger brother to ride the Ferris wheel. You notice that a sign says that the angular speed of the Ferris wheel is 0.13 rad/sec. How many revolutions will the wheel complete in 12 minutes?

Answer: The angular speed, ω = 0.13 rad/sec. The time, t = 12 min. Convert t = 12 min x 60 sec/min = 720 sec. Using the equation ω = θ /t , solve for θ .

ω = θ /t

ω t = θ

(0.13 rad/sec)(720sec) = θ

θ = 93.6 rad

θ = 93.6/ 2π revolutions

θ = 14.9 or ~15 revolutions

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