Problem: Angular velocity is almost the same as linear velocity, except that we measure the rate of rotation in radians / second rather than measuring distance in degrees per second. This makes sense because different parts of the object might be moving at different speeds (imagine a merry-go-round, where someone near the center is rotating at the same rate (the same number of degrees per second, or revolutions per second, or radians per second), but moving more slowly than someone at the outer rim).
How long does it take for a uniformly accelerating disk to increase its angular velocity from 1 rad/s to 3.4 rad/s, if its angular acceleration is 2.5 radian per second per second?
Through what angle does the disk turn during this time?
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Solution: This situation is reasoned out in the same way as if we were measuring positions in meters and velocities in meters per second. In this case, though, we talk about angular positions in radians and angular velocities in radians / second.
The change in angular velocity is 2.4 radians/second. At the rate of 2.5 radians/second/second, it will require ( 2.4 radians/second) / ( 2.5 radians/second/second) = .9600 seconds.
During this time the average velocity of the object will be ( 1 radians/second + 3.4 radians/second)/2 = 2.2 radians/second. At this rate the angular displacement during .9600 seconds will be 2.112 radians.
Generalized Response: If the initial and final velocities were v0 and vf, and the acceleration a, we would easily see that the time `dt is related to these quantities by the definition of acceleration: aAve = (vf - v0) / `dt. This relationship is easily rearranged to give us `dt = (vf - v0) / a.
In the present case we have angular velocities `omega0 and `omegaf, and angular acceleration`alpha. The only difference is that whereas v0, vf and a were expressed in terms of meters and seconds, `omega and `alpha quantities are translational in radians and seconds. However the concepts are practically identical. The angular acceleration of an object is change in angular velocity divided by the time required for the change. Thus we have
`alphaAve = (`omegaf - `omega0) / `dt,
which we rearrange to obtain
`dt = (`omegaf - `omega0) / `alphaAve.
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