# Does a cycle (in Simple Harmonic Motion) have to equal 2π?

So, I search for the definition of cycle and I get this in Wikipedia:

A turn is a unit of angle measurement equal to 360° or 2π radians (or ...). A turn is also referred to as a revolution or complete rotation or full circle or cycle or rev or rot.

When I look for motion in pendulum, I get:-

The time for one complete cycle, a left swing and a right swing, is called the period.

So, in the first quote, it says that one cycle has to be 2π radians, but in the second quote it says "one complete cycle" is "a left swing and a right swing" (i.e the full cycle doesn't have to be equal to "2π radians").

The $2\pi$ refers to the phase angle ($\phi$), whereas in a pendulum, we have an additional, physical angle (angle of inclination $\theta$). In a pendulum of small oscillations of maximum angle of inclination $\theta_0$, the two are related by $\theta=\theta_0\sin\phi$. Note that $\phi=\phi_0+\omega t$ is a time-dependent quantity.

Simply put, this phase angle denotes the state of a periodic system in one variable. The system repeats every time $\phi$ increments by $2\pi$. An angle is used because simple harmonic oscillators are sinusoidal and can be represented on a circle. In this imaginary circle, one 'turn' is $2\pi$ change in $\phi$. It is simultaneously a cycle in the pendulum.

Remember that definitions die when taken out of context ;)

• Something is wrong with your LaTeX formatting. Commented Mar 11, 2012 at 17:58
• Fixed;thanks. For some reason the edit button had gone missing. After bruteforcing the link from the system, the errant $seemed to be in place. Software glitch? Commented Mar 11, 2012 at 18:30 • I'm guessing you meant to write$\theta = \theta_0\cos\phi$instead of$\theta = \theta\$? Commented Mar 11, 2012 at 20:10
• Aah there was a percentage sign in mathjax as well. Commented Mar 12, 2012 at 0:32

The two quoted definitions are both right but they apply to different contexts.

A general definition of a "cycle" is the minimal time after which the system returns to the same configuration. For a rotating object (e.g., a pendulum turning in one direction only) , the cycle is a full turn. For an oscillating pendulum, the cycle is a single "there-and-back" tour.

The way you interpreted the "cycle" is wrong. One cycle need not to be 2π radians physically. We just REPRESENT an cycle or a periodic motion as if we are varying the angle in a circle and measuring the quantities like time period and frequency etc. As you know if we vary an angle θ to θ+2π then also the value repeats.