Simple harmonic motion

Key ideas

• Simple harmonic motion (SHM) is the back-and-forth motion where the acceleration is proportional to the displacement and always points back toward the centre:

$a = -\omega^2 y$

• The minus sign is the whole idea: it is a restoring acceleration — the further out the object, the harder it is pulled back. This condition defines SHM.
• Each variable, with sub-bullets:
  • $y$ — displacement from the equilibrium (centre) position (m)
  • $\omega$ — the angular frequency (rad s⁻¹), where $\omega = 2\pi f = \dfrac{2\pi}{T}$
  • $A$ — the amplitude, the maximum displacement (m)
• Starting from the centre, displacement, velocity and acceleration vary with time as:

$y = A\sin\omega t \qquad v = A\omega\cos\omega t \qquad a = -A\omega^2\sin\omega t$

• (If the object starts at the end of its swing instead, swap $\sin$ and $\cos$: $y = A\cos\omega t$.)
• The maximum speed is $v_{max} = A\omega$ (at the centre), and the maximum acceleration is $a_{max} = A\omega^2$ (at the ends).

The reference circle and phasors

• SHM is the shadow (projection) of uniform circular motion. A point going around a circle of radius $A$ at angular velocity $\omega$ casts a shadow that moves in SHM.
• This is why SHM uses $\omega$ and why $v_{max} = A\omega$ and $a_{max} = A\omega^2$ — they are the rim speed and centripetal acceleration of that reference circle.
• A phasor is the rotating arrow itself; its angle gives the phase of the motion.

Tips

• $\omega$ is angular frequency, not frequency. Convert with $\omega = 2\pi f$ before substituting; using $f$ in place of $\omega$ is the most common SHM slip.
• Decide where the object starts: at the centre use $y = A\sin\omega t$, at the end use $y = A\cos\omega t$. Getting this wrong flips your whole graph.