Preference of Angular frequency over frequency for sinusoidal graphs

[sbstratos79]

Quote from 'The Physics of Vibrations and Waves by H.J.Pain': "However when we solve the equation of motion we shall find that the behaviour of x with time has a sinusoidal or cosinusoidal dependence, and it will prove more appropriate to consider not $\nu$, but the angular frequency $\omega$ = 2$\pi$$\nu$ so that the..."
My question is, why are we prefering $\omega$ over $\nu$ just because of the fact that the graph will be sinusoidal/cosinusoidal? Does it make the calculations somehow easy, or has it got some other purpose?

 

[AlephZero]

The math works out MUCH easier if the trig functions are measured in radians. Otherwise, there are far too many factors of $2\pi$ to keep track of.

 

[sbstratos79]

so, according to you, we prefer $\nu$ over $\omega$. but the book says that we prefer $\omega$ over $\nu$. $\omega$ contains 2$\pi$

 

[AlephZero]

The way I interpret your notation, $\nu$ means the number of complete cycles of the oscillation per unit time, and $\omega$ means the number of radians per unit time.

The book has used $2\pi$ once in its definition or explanation of what $\omega$ is. You won't find $2\pi$ again when you use $\omega$.

For example if the displacement of something is $A \cos \omega t$, its velocity is $-A\omega \sin \omega t$ and its acceleration is $-A\omega^2 \cos \omega t$. No factors of $2\pi$.

If you have a mass m on a spring of stiffness, the oscillation frequency is $\omega = \sqrt{k/m}$. For a simple pendulum, $\omega = \sqrt{g/l}$. Again, no factors of $2\pi$.

 

[sbstratos79]

ah, i see... Thanks a lot for the help, sir ^_^