In the simple harmonic oscillator model, we talk about the angular frequency and I understand that this tells something about the instantaneous angular velocity of the object. For example, a simple pendulum. But what does this mean when we apply this concept to an object attached to a spring that behaves as a simple harmonic oscillator? There is no instantaneous circular motion for the object, so how can we relate the angular frequency to a system like this?


The basic equation for the simple harmonic oscillator is $\frac{d^2x}{dt^2} =- \omega^2 x$. One solution that works here is $x=x_0 e^{i\omega t}$. Notice how the complex conjugate, $x^* = x_0^* e^{-i\omega t}$ is also a valid solution. The reason for this is because, while the body may be oscillating in one dimensional motion, it can be thought of as going either clockwise or counterclockwise in circle in the complex plane, where its angular velocity is either $i\omega$ for counterclockwise or $-i\omega$ for clockwise.

  • $\begingroup$ so simply, the graphical representation of the motion behaves in the circular way right? $\endgroup$ – Kosala Sep 20 '16 at 21:54
  • $\begingroup$ Not sure what you mean by "the circular way". It behaves like the real part of a circle in the complex plane. In fact, the function $x(t)$ is actually $Re(z)$ where $z = x + iy$, so it is the real part of a complex circle. $\endgroup$ – hebetudinous Sep 21 '16 at 3:02

Angular velocity

Imagine a point $P$ going round on a circle with centre point $O$. If the speed of rotation is constant then we call $\omega$ the angular velocity (aka angular frequency$^*$):


If we project the points $O$ and $P$ onto an $y$-axis then$^{**}$:

$$|O'P'|=y=|OP|\sin\theta=R\sin \omega t$$

where $R$ is the radius of the circle (i.o.w. $|OP|$). $P'$ performs a harmonic oscillation about $O'$ (the point where $y=0$).

Other types of harmonic oscillators have the general Newtonian equation of motion:

$$u''(t)+\omega^2 u(t)=0$$

which leads to the same type of kinematic equation. By analogy we also call $\omega$ the angular velocity, even if there's no originating circular motion (and its projection) involved. $u$ may be any oscillating quantity (position, pressure, electrical potential etc).

$^*$ Strictly speaking the term angular frequency should be reserved for $\omega=2\pi f$, with $f$ the angular frequency or frequency for short.

$^{**}$ We also assume that at $t=0$, $P'$ coincides with $O'$, to avoid introducing a phase angle for (simplicity's sake).

  • $\begingroup$ thanks for your explanation. Does this mean that the motion of the spring mimics the motion of "P"? $\endgroup$ – Kosala Sep 21 '16 at 0:03
  • $\begingroup$ Yes, exactly. If the parameters are the same then the mathematical description of both oscillations is the same. For a simple mass/spring system for example: $\omega^2=\frac{k}{m}$. The value of "R" will depend on the initial spring displacement. $\endgroup$ – Gert Sep 21 '16 at 0:25
  • $\begingroup$ It mimicks the motion of $P'$, the projection of $P$ onto an axis. $\endgroup$ – Gert Sep 21 '16 at 0:32

In some way it all starts with a lack of letters in the English and Greek alphabet.
In this case it is the use of omega $\omega$ for two entirely different things - angular speed/velocity and angular frequency.

A body whose acceleration is directed towards a point and is proportional to the distance from that point is said to undergo simple harmonic motion and as an equation that is often written as $\ddot a=-\omega^2 x$.
The constant of proportionality $- \omega^2$ is chosen to ensure that it is always negative.

When such motion is analysed it is found that the motion is periodic with a period $T$.
Another way of stating this is to say that the frequency of the motion is $f = \frac 1 T$.

So you can write the equation for the motion as something like $x = x_o \sin (\frac{2 \pi}{T}t)$ or $x = x_o \sin (2 \pi f t)$

This might be thought as a rather cumbersome way of writing the equation so why not introduce a parameter omega with $\omega = \frac{2 \pi}{T}$ or $\omega = 2 \pi f$.

Now this omega has the units of radians per second whereas the frequency, $f$, has units of $s^{-1}$ or hertz but it is something being counted in a unit of time and that something is angle in radians.
So to differentiate it from frequency it is called angular frequency.

So your spring-mass system is undergoing linear oscillatory motion but a parameter used to describe the motion of your spring-mass system happens to be measured in radians which is a measure of angle.

In rotational motion the parameter angular speed is used which is the rate of change of angle with time and often the symbol, omega $\omega$ is used for this parameter although quite a number of authors now used $\Omega$ to differential angular speed from angular frequency both of which are measured in radians per second.

This ambiguity as to what $\omega$ stands for is compounded when simple harmonic motion is likened to the projection of the position of an object undergoing circular motion onto a diameter.
It just so happens that for this example numerically $\omega = \Omega$.

I believe that if two different symbols were used, $\omega$ for angular frequency and $\Omega$ for angular speed, then there would not be this confusion about a spring-mass system which is undergoing vertical oscillation at the same time is also thought to be going round in circles.


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