Unit of Angular velocity

Question

Why is the angular velocity $\omega$ always written in $rad/sec$? Is there anything wrong if I write it in $degrees/sec$? If no, then why almost all the books have it as $rad/sec$??

Answer 1

$w $ is the angular velocity, not the angular displacement. You can write it in deg/ sec if you wish. The reason rad/sec are used is because the identities $\frac{d}{dx}\cos(x) = -\sin(x) $ and $ \frac{d}{dx}\sin(x) = \cos(x)$ only hold when x is measured in radians.

Answer 2

In principle it's a choice of unit, so you're free to do as you wish. However, rather than expressions like $\sin(\omega t)$ or $e^{i\omega t}$, you'll need to write $\sin(\frac{\pi \omega t}{180})$ and $e^{i\pi\omega t/180}$. Those extra factors will come into play when you take derivatives or perform integrals, as well as solve any differential equations, so before long I would be on my knees begging for radians back.

But, unless you're in a class where the instructor demands that you use radians per second (and don't get me wrong - if you were in my class, I would make that a requirement), then you're free to make your life as inconvenient as you'd like.

Answer 3

The radian is the standard unit of angular measure. Angular velocity is just the angle traversed by a particle or a body in unit time. You may give it any sensible unit which should obviously denote the angle traversed per unit time. Therefore you may use the unit $deg/s$.

The unit $rad/s$ is commonly used because it is an SI unit and the relations like $v=\omega r$ are derived for angular velocity in $rad/s$.

Answer 4

You can write angular velocity in any way you like, as long as it makes sense and you state the units. You can freely write it as degrees/second, rotations/hour, or anything along those lines. The reason why ω is in rad/sec is that it is much easier to do differentiation and integration with it(to find angular acceleration or angular displacement). Differentiating trigos with units other than radians will not work.