Preference of Angular frequency over frequency for sinusoidal graphs

  1. 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 [itex]\nu[/itex], but the angular frequency [itex]\omega[/itex] = 2[itex]\pi[/itex][itex]\nu[/itex] so that the..."
    My question is, why are we prefering [itex]\omega[/itex] over [itex]\nu[/itex] 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. jcsd
  3. AlephZero

    AlephZero 7,298
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    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.
     
  4. so, according to you, we prefer [itex]\nu[/itex] over [itex]\omega[/itex]. o_O but the book says that we prefer [itex]\omega[/itex] over [itex]\nu[/itex]. [itex]\omega[/itex] contains 2[itex]\pi[/itex]
     
  5. AlephZero

    AlephZero 7,298
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    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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  6. ah, i see... Thanks a lot for the help, sir ^_^
     
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