[Physics] Spring pendulum – why is it possible to use this equation

Question

It is known that, when we describe the spring pendulum, we are bound to use the formula
$T = 2\pi \sqrt{m/k}$, however, we can go further and set
$\omega = \frac{2\pi}{T}$
I ponder why is this substitution legal, because there is no angular motion in this pendulum!

Answer 1

In this equation $\omega$ does not refer to the speed of angular motion, but the frequency of oscillation when measured in angular terms (usually radians/sec, but it can be degrees/sec). Frequency is usually measured in cycles per second (Hertz), but it is sometimes more conveniently measured in angular terms, when it is called angular frequency.

The angle which is being measured here is the phase angle, which describes how far through the cycle the oscillation has gotten, as though the oscillation is moving round in a circle at a constant angular speed $\omega$.

You might well be confused because you cannot see any angular motion in the 'spring' pendulum. But it is even more confusing when dealing with the 'string' pendulum because here the displacement is measured by an angle $\theta$, but this angle is not the same as the phase angle (which is usually called $\phi$, the Greek phi for phase). The angular speed is $\large{\frac{d\theta}{dt}}$, which is normally called $\omega$, but this $\omega=\large{\frac{d\theta}{dt}}$ is not the same as the angular frequency $\omega=\large{\frac{d\phi}{dt}=\frac{2\pi}{T}}$. Angular speed varies during the oscillation, from $0$ at the extremes to the maximum as it passes through the vertical. But angular frequency does not change during the oscillation, it is a constant for ideal string or spring pendulums, depending on $\large{\frac Lg}$ or $\large{\frac km}$.