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Describing Oscillation

Level 1 - Physics topic page in Oscillations.

Principle

Oscillation is motion that repeats about an equilibrium state.

Position, velocity, and acceleration describe motion. This page adds the timing language used for repeated motion.

Notation

\(x(t)\)
displacement from equilibrium
\(\mathrm{m}\)
\(A\)
amplitude
\(\mathrm{m}\)
\(T\)
period
\(\mathrm{s}\)
\(f\)
frequency
\(\mathrm{Hz}\)
\(\omega\)
angular frequency
\(\mathrm{rad\,s^{-1}}\)
\(\phi\)
phase constant
\(\mathrm{rad}\)

Method

Derivation 1: Identify the repeated quantity

Measure displacement from the equilibrium position. A periodic oscillator returns to the same displacement, velocity, and acceleration after one period.

Repeat displacement
\[x(t+T)=x(t)\]
Repeat velocity
\[v(t+T)=v(t)\]
Repeat acceleration
\[a(t+T)=a(t)\]
A repeated position alone is not enough if the motion reverses differently each time.

Derivation 2: Connect period, frequency, and phase

Frequency counts cycles per second. Angular frequency counts radians of phase per second, with one cycle equal to \(2\\pi\) radians.

Cycle rate
\[f=\frac{1}{T}\]
One cycle
\[\Delta\varphi=2\pi\]
Phase rate
\[\omega=\frac{\Delta\varphi}{T}=\frac{2\pi}{T}\]
Frequency form
\[\omega=2\pi f\]

Derivation 3: Build the sinusoidal model

A sinusoid is the simplest smooth periodic model. The amplitude scales the displacement and the phase constant sets the initial position in the cycle.

Start with phase
\[\varphi(t)=\omega t+\phi\]
Scale displacement
\[x(t)=A\cos\varphi(t)\]
Substitute phase
\[x(t)=A\cos(\omega t+\phi)\]

The graph below is a time history, not a path through space. Read the period horizontally and the amplitude vertically.

0123456-1.2-0.600.61.2txx(t)
The pattern repeats after one period; amplitude is measured from equilibrium.

Rules

These are the compact results from the derivations above.

Periodic condition
\[x(t+T)=x(t)\]
Frequency definition
\[f=\frac{1}{T}\]
Angular frequency
\[\omega=2\pi f=\frac{2\pi}{T}\]
Sinusoidal displacement
\[x(t)=A\cos(\omega t+\phi)\]

Examples

Question
An oscillator completes
\[12\]
cycles in
\[6.0\,\mathrm{s}\]
Find \(f\), \(T\), and \(\omega\).
Answer
\[f=12/6.0=2.0\,\mathrm{Hz}\]
\[T=1/f=0.50\,\mathrm{s}\]
\[\omega=2\pi f=4\pi\,\mathrm{rad\,s^{-1}}\]

Checks

  • Amplitude is measured from equilibrium, not peak to peak.
  • Period is time per cycle.
  • Frequency is cycles per second.
  • Phase must be in radians in calculus formulas.

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