Are the magnitude of average angular velocity and the value of the average angular speed always same? If not then can you please give an example.
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2 Answers
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In general the average of velocities will not equal the average of speeds. Velocities are vectors for a reason - they can be positive or negative, while speeds can only ever be positive (or zero).
An example is a system which rotates clockwise for time $t$ and counterclockwise for time $t$. The angular velocity averages to zero, while the speed does not.
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Angular velocity, $\vec{\omega}$ is a vector quantity (axial vector) where as angular speed is a scalar quantity.
Are the magnitude of average angular velocity and the value of the average angular speed always same?
No. Their value may be different.
Example:
Suppose a bob doing circular motion (revolution) in vertical place with an angular velocity of 10 rad/s (anti-clockwise). It starts from horizontal (at some distance from origin on the X axis) and moves with the specified angular velocity, completes one revolution and comes back to the same point. Net angular displacement is zero, thus average angular velocity is also zero. But here average angular speed is not zero, it is equal to 10 rad/s since here we account for total angular distance travelled by the bob.
Distance $\not=$ Displacement
In the same way, Angular distance travelled $\not=$ Angular displacement
