Simple Harmonic Motion
A mass bobbing on a spring, a pendulum swinging gently, a guitar string vibrating after being plucked — wildly different setups, yet all governed by the same underlying equation. Simple harmonic motion is the single mathematical pattern tying these oscillations together, and this lesson pins down precisely what that pattern is.
What makes an oscillation "simple harmonic"
Plenty of things oscillate — swinging or moving back and forth, repeatedly passing through some central point. But not every oscillation is simple harmonic motion (SHM). SHM is defined by a precise mathematical condition linking an object's acceleration to its displacement:
An object performs simple harmonic motion if its acceleration is directly proportional to its displacement from a fixed equilibrium position, and is always directed towards that equilibrium position (i.e. it always acts to restore the object back towards the centre).
Written as an equation, with displacement $x$ measured from the equilibrium position:
where $a$ is acceleration, $x$ is displacement from equilibrium, and $\omega$ is a constant called the angular frequency (rad s⁻¹), setting how rapidly the oscillation proceeds.
Why the minus sign is essential
The negative sign is the entire physical content of "always directed towards equilibrium". When $x$ is positive, $a$ comes out negative — pointing back towards equilibrium. When $x$ is negative, $a$ comes out positive — again pointing back. This is a restoring effect: whichever side of equilibrium the object is on, acceleration always pushes it back towards the centre, and the further away it is, the stronger that pull (since $a \propto x$ in magnitude).
Why proportionality (not just "restoring") matters
Plenty of restoring forces aren't proportional to displacement, and those produce oscillations that aren't simple harmonic. SHM specifically requires a linear relationship between $a$ and $x$: double the displacement, and acceleration must exactly double too (oppositely directed). This precise proportionality gives SHM its distinctive behaviour — sinusoidal graphs of displacement, velocity and acceleration, and a period independent of amplitude. Acceleration proportional to $x^2$ or $x^3$ would still be periodic and restoring, but not simple harmonic.
Two classic examples
Mass on a spring
A mass attached to a spring, displaced and released, oscillates back and forth. By Hooke's law, the restoring force is $F = -kx$, where $k$ is the spring constant. Applying Newton's second law, $F=ma$, gives $ma = -kx$, so $a = -\dfrac{k}{m}x$ — directly proportional to $x$ and oppositely directed, exactly the SHM condition, with $\omega^2 = k/m$.
The simple pendulum (for small angles)
A pendulum swinging with small angular displacement from vertical also satisfies the SHM condition to a very good approximation — the restoring force (a component of gravity along the arc) is approximately proportional to displacement from vertical, provided the swing angle stays small (conventionally under about 10°). At larger angles, the restoring force curves away from strict proportionality, and the motion is still periodic but no longer simple harmonic.
Angular frequency $\omega$ (rad s⁻¹) plays the same role in SHM as angular velocity in circular motion — SHM can be thought of as the projection of uniform circular motion onto a single axis, which is why $\omega$ carries the same units and why sine and cosine describe it. $\omega$ relates to period $T$ and frequency $f$ via $\omega = \dfrac{2\pi}{T} = 2\pi f$, identical in form to the circular-motion relationship. Phase describes how far through a cycle an oscillation currently is, relative to a reference point — two oscillations with the same amplitude and frequency but different starting points differ in phase, important when comparing displacement, velocity and acceleration graphs in the next lesson.
Contrast: oscillatory motion that isn't SHM
Not every periodic motion satisfies $a=-\omega^2x$. A ball bouncing elastically on a hard floor is periodic, but its acceleration is a constant $-g$ throughout each flight, not proportional to height — periodic, but not SHM. A pendulum swinging through a large angle is still periodic but no longer strictly simple harmonic, since the true restoring force is proportional to $\sin\theta$ rather than $\theta$ itself, and $\sin\theta \approx \theta$ only for small angles.
Recognising the difference matters because every SHM-specific result from here on — the sinusoidal solutions for $x$, $v$ and $a$, the period formulae, the energy relationships — relies on $a=-\omega^2x$ holding exactly. Apply an SHM formula to a system that only oscillates but isn't truly simple harmonic, and the result will be wrong.
Key takeaways
- SHM is defined by $a = -\omega^2 x$: acceleration is directly proportional to displacement from equilibrium, and always directed back towards it (restoring).
- The minus sign encodes "restoring" (acceleration always opposes displacement); the direct proportionality (not just any restoring relationship) is what makes the motion specifically simple harmonic.
- A mass-spring system satisfies SHM exactly, with $\omega^2 = k/m$; a simple pendulum satisfies it only approximately, valid for small angles of swing.
- Angular frequency $\omega = 2\pi/T = 2\pi f$ sets the pace of the oscillation; phase describes position within a cycle relative to a reference point.
- Not all periodic motion is SHM — a bouncing ball and a large-angle pendulum are both periodic but not simple harmonic, since their acceleration isn't proportional to displacement throughout.