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 $rad/sec$??
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The arc length is $s = r \theta$ when using radians. Take the time derivative of this $v = r \omega$ to get to $\frac{\rm rad}{\rm sec}$. – John Alexiou Nov 10 '17 at 13:34
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1Actually, your question is : "Why do we use radians for angular units" and the answer is readily available online, See Here For Example – John Alexiou Nov 10 '17 at 13:37
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Frankly, all the confusion about degrees vs radians goes away if you just consider the symbol ° to refer to the real number $\pi/180$ (and then do everything in radians). – Emilio Pisanty Nov 10 '17 at 19:38
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$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.
David Reed
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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.
J. Murray
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Not sure how did you come with $e^i\omegat$?? Can you explain a bit? – Piano Land Nov 10 '17 at 15:14
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It's called a complex exponential, and it's an expression that comes up over and over and over again in math, physics, and engineering applications. My point was to demonstrate that if you want to use degrees rather than radians, you need to include the $\pi/180$ conversion factor over and over and over every single time, and it would very quickly become infuriating. – J. Murray Nov 10 '17 at 15:47
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I am curious to know the proof of $e^{i\omega t}$. How can I get one?? – Piano Land Nov 11 '17 at 03:25
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I don't know what you mean by "proof". If you google "complex exponential" you'll see some examples. – J. Murray Nov 11 '17 at 03:30
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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 $rad/s$.
Abhinav Dhawan
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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.
See Jian Shin
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