# Preference of Angular frequency over frequency for sinusoidal graphs

1. ### sbstratos79

4
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?

2. ### AlephZero

7,248
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.

3. ### sbstratos79

4
so, according to you, we prefer $\nu$ over $\omega$. but the book says that we prefer $\omega$ over $\nu$. $\omega$ contains 2$\pi$

4. ### AlephZero

7,248
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##.

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5. ### sbstratos79

4
ah, i see... Thanks a lot for the help, sir ^_^