Average Angular Velocity Formula
The angular velocity of a rotating object is the rate at which the angular coordinate changes with respect to time. The angular coordinate is the angle of the object relative to a certain coordinate system, and is usually represented with the Greek letter θ ("theta"). The average angular velocity is the change in the angular coordinate θ, expressed in radians, divided by the change in time. The angular velocity is a vector that points in the direction of the axis of rotation. The magnitude of the angular velocity is given by the formula below. The unit of angular velocity is .
= average angular velocity, ()
= change in angular coordinate (radians)
= change in time (s)
= initial angular coordinate (radians)
= final angular coordinate (radians)
t1 = initial time (s)
t2 = final time (s)
Average Angular Velocity Formula Questions:
1) The pendulum of a large grandfather clock swings between angles of and . It swings from one angle to the other in 1.00 seconds. What is the average angular velocity of the pendulum as it swings from one angle to the other?
Answer: The change in time between the two angular coordinate values is given as . The initial angular coordinate is , and the final angular coordinate is . The average angular velocity can be found using the formula:
Between the initial and final angles, the average angular velocity of the clock's pendulum was , which is approximately .
2) A powerful fan blade spins at an average angular velocity of . How long does it take, in seconds, to rotate ?
Answer: The change in the angular coordinate requested in this problem is . The first step to answering this problem is to convert from degrees to radians. A full circle consists of , or . The change in the angular coordinate is therefore:
The time required for the fan to rotate by this amount can be found by rearranging the average angular velocity formula:
The time required for the fan to rotate is only .