Period Derivation for the Simple Harmonic Oscillator

(Derivation)

x = A sin(ωt + ϕ)

(1)

The sine function as a function of ωt is shown in figure 1.

Figure1. Sine function.

One can see that the sine function repeats every 2π. Periodicicty was defined in oscillations as a function that is equal to itself at t + T,

f(t + T) = f (t)

(2)

Therefore, the arguments to the sine function must be equal to itself again at 2π (plug in t + T in for t).

ω (t + T ) + ϕ = ωt + ϕ + 2π

(3)

expand paranthesis and subtract each side by ϕ

ωt + ωT = ωt + 2 π

ωt cancels to leave

ωT = 2 π

Finally, rearrange to get the eqaution for period

2π T = --- ω

(4)

[1] A. P. French, Vibrations and Waves, W. W. Norton, 1971.

[2] PhysicsLibrary, Oscillation.

"Period Derivation for the Simple Harmonic Oscillator" is owned by bloftin.
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Pending Errata and Addenda