Define simple harmonic motion by its defining equation, and describe the variation of displacement, velocity and acceleration with time and with displacement

What this dot point is asking

SEAB wants you to define simple harmonic motion (SHM) through its defining equation $a = -\omega^2 x$, and to describe how displacement, velocity and acceleration vary both with time and with displacement. SHM is the model for any system with a linear restoring force, from a mass on a spring to a vibrating molecule.

The answer

The defining condition

A body performs simple harmonic motion if its acceleration is proportional to its displacement from a fixed point and always directed toward that point:

Here $x$ is the displacement from equilibrium and $\omega$ is the angular frequency. The negative sign is the heart of SHM: the acceleration (and the restoring force) always points back toward equilibrium, which is what sustains the oscillation.

Angular frequency, period and frequency

The angular frequency relates to the period $T$ and frequency $f$ by:

For SHM, the period is independent of the amplitude (this is called isochronism), a defining and useful property.

Variation with time

Taking $x = x_0$ at $t = 0$, the displacement varies as:

Differentiating gives the velocity and acceleration:

The velocity leads the displacement by a quarter period, and the acceleration is exactly out of phase with the displacement.

Variation with displacement

Eliminating time gives the velocity directly in terms of displacement:

This shows:

• maximum speed $v_{\max} = \omega x_0$ at the equilibrium position ($x = 0$),
• zero speed at the extremes ($x = \pm x_0$),
• maximum acceleration $a_{\max} = \omega^2 x_0$ at the extremes,
• zero acceleration at equilibrium.

Standard oscillators

Two systems on the syllabus obey SHM:

• A mass $m$ on a spring of constant $k$: $\omega = \sqrt{\dfrac{k}{m}}$, so $T = 2\pi\sqrt{\dfrac{m}{k}}$.
• A simple pendulum of length $L$ (small angles): $\omega = \sqrt{\dfrac{g}{L}}$, so $T = 2\pi\sqrt{\dfrac{L}{g}}$.

Examples in context

Example 1. A mass on a vertical spring. Hanging a mass on a spring and displacing it slightly produces SHM about the new equilibrium, with $T = 2\pi\sqrt{m/k}$. Plotting $T^2$ against $m$ gives a straight line of gradient $4\pi^2/k$, a standard way to measure the spring constant and confirm the SHM model.

Example 2. A tuning fork. The prongs of a struck tuning fork vibrate with SHM at a fixed frequency set by their stiffness and mass, independent of how hard the fork is struck (the amplitude). This isochronism is why tuning forks give a stable reference pitch.

Try this

Q1. State the defining equation of SHM and explain what each symbol means. [2 marks]

• Cue. $a = -\omega^2 x$: $a$ acceleration, $x$ displacement from equilibrium, $\omega$ angular frequency; the minus sign means acceleration is toward equilibrium.

Q2. A pendulum has length $0.99\ \text{m}$. Find its period ($g = 9.81\ \text{m s}^{-2}$). [2 marks]

• Cue. $T = 2\pi\sqrt{L/g} = 2\pi\sqrt{0.99/9.81} = 2.0\ \text{s}$.

Q3. Sketch how the velocity of a body in SHM varies with its displacement, and mark where speed is greatest. [3 marks]

• Cue. An ellipse-like curve $v = \pm\omega\sqrt{x_0^2 - x^2}$; speed greatest at $x = 0$ (equilibrium), zero at $x = \pm x_0$.