# Preference of Angular frequency over frequency for sinusoidal graphs

• sbstratos79
In summary, the book prefers to use the angular frequency, represented by the symbol \omega, instead of the ordinary frequency, represented by the symbol \nu, when solving equations of motion involving vibrations and waves. This is because the math is much simpler when using radians instead of cycles. Additionally, the book only uses the factor of ##2\pi## once when defining \omega, making calculations easier.
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?

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.

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

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##.

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

## 1. Why is angular frequency used instead of frequency in sinusoidal graphs?

Angular frequency is used because it provides a more convenient way to describe the rate at which a sinusoidal graph oscillates. It is directly related to the frequency, but allows for easier calculations and comparisons.

## 2. What is the formula for converting between angular frequency and frequency?

The formula for converting between angular frequency (ω) and frequency (f) is ω = 2πf. This means that the angular frequency is equal to 2π times the frequency.

## 3. How is angular frequency related to the period of a sinusoidal graph?

The period of a sinusoidal graph is the amount of time it takes for one complete cycle. The relationship between angular frequency and period is ω = 2π/T, where T is the period. This means that the angular frequency is inversely proportional to the period.

## 4. Can angular frequency be negative?

Yes, angular frequency can be negative in some cases. This can occur when the sinusoidal graph is in the opposite direction of the positive direction, such as in a reflection or rotation.

## 5. What are the units of angular frequency?

The units of angular frequency are radians per second (rad/s). This unit is used because angular frequency is a measure of how many radians the graph rotates in one second.

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