Oscillations & Simple Harmonic Motion

amplitude

The amplitude tells you how big the oscillation is: how far the object gets from the middle at its most extreme. On a swing it is how high you rise above the bottom; in sound it relates to how loud a note is; in a wobbling spring it is the greatest stretch from the resting point. It answers the question: how large is this back-and-forth?

Precisely, the amplitude A is the maximum displacement from the equilibrium position. In the SHM description x(t) = A cos(omega t + phi), the value A is the peak the position reaches, so the object moves between +A and -A. Amplitude is always taken as a positive number and carries units of length (metres) for a bob, or the relevant units for other oscillations. The total energy stored in an ideal oscillation is proportional to the square of the amplitude.

A key and sometimes surprising fact: for ideal simple harmonic motion the period does not depend on amplitude, but the energy very much does, since energy equals 1/2 k A^2 for a mass-spring system. So a louder sound or a wider swing carries more energy, yet takes the same time per cycle. In real, damped oscillations the amplitude does not stay fixed it shrinks over time as energy leaks away.

If a spring oscillator has amplitude A = 0.10 m and stiffness k = 200 N/m, its total energy is E = 1/2 k A^2 = 1/2 (200)(0.10)^2 = 1.0 J. Doubling the amplitude to 0.20 m would quadruple the energy to 4.0 J.

Energy grows with the square of amplitude, but the period stays the same.

Bigger amplitude means more energy (E ~ A^2), not a longer period; the two are independent for ideal SHM.

Also called
Apeak displacement