# What is the difference between angular velocity and angular frequency in angular SHM?

A body free to rotate about a given axis can make angular oscillation. This angular oscillation are called Angular simple harmonic motion,

In derivation,

Ohm = thetawcos(wt+ phi)

Where omega is angular velocity, w is angular frequency and theta is maximum angular displacement . I am really confused bewteen two!

As w is angular velocity in circular, rotational dynamics.. etc, but now in SHM, what is the need to use the same variable for angular frequency as that of angular velocity that we use in different concepts?

According to me, Angular frequency w is total angular displacement in one oscillation per unit Period to achieve this. And angular velocity (omega) is any angular displacement per unit time to go there .

Is that thinking is correct? Or done it wrongly?

In circular motion we user the Greek letter $$\omega$$ (omega) to represent angular velocity, so the angle $$\theta$$ travelled through at time $$t$$ is

$$\theta = \omega t$$

Typically $$\omega$$ is in radians per second, so the time taken to complete one rotation is

$$\displaystyle t_{rot} = \frac {2 \pi} {\omega}$$

and the reciprocal of this is the frequency of rotation

$$\displaystyle f_{rot} = \frac {\omega} {2 \pi}$$

When describing linear SHM we use $$\omega$$ to denote an angular velocity in phase space rather than in real space. So the displacement $$x$$ at time $$t$$ is

$$x(t) = A \sin (\omega t + \phi)$$

where $$A$$ is the amplitude of the motion and $$\phi$$ is its phase, so that $$x(0) = A \sin (\phi)$$. The time taken to complete one oscillation is the time taken to complete one rotation in phase space

$$\displaystyle t_{osc} = \frac {2 \pi} {\omega}$$

and the frequency of oscillation is the reciprocal of this time:

$$\displaystyle f_{osc} = \frac {\omega} {2 \pi}$$

With linear SHM, linear velocity is:

$$\displaystyle v(t) = \frac {dx}{dt} = A \omega \cos (\omega t + \phi)$$

With linear SHM there is no danger of confusion between $$x$$ and $$v$$ in physical space and $$\omega$$ and $$\phi$$ in phase space. However, with rotational SHM we often re-use the same notation as in linear SHM, but now the left hand side of the equation is angular displacement $$\theta$$ instead of linear displacement $$x$$. So we might write

$$\theta (t) = A \sin (\omega t + \phi)$$

Note that in this expression $$\theta$$ represents an angle in physical space, whereas $$\omega t$$ and $$\phi$$ represent angles in phase space. Confusion may arise if we want to write an expression for angular velocity and we naively write

$$\displaystyle \omega (t) = \frac {d \theta}{dt} = A \omega \cos (\omega t + \phi)$$

This is incorrect (or, at least, very confusing) because we are using $$\omega$$ to represent different things on either side of the equation. On the left hand side it represents an angular velocity in physical space, which varies with time, and on the right hand side it represents an angular velocity in phase space, which is constant in SHM.

To avoid this confusion we could either use a different label for the angular velocity in phase space e.g.

$$\omega(t) = A \Omega \cos (\Omega t + \phi)$$

or we could change the way we denote angular velocity in physical space e.g.

$$\dot {\theta}(t) = A \omega \cos (\omega t + \phi)$$

what is the need to use the same variable for angular frequency as that of angular velocity that we use in different concepts?

Honestly as long as you know what you are talking about, you can take up any variable as per your convenience. It does not matter.

If it is bothering you so much, then use english letter "W" for angular velocity and Greek letter $$\omega$$ for angular frequency or vice versa.