Not every oscillation is neat and tidy. But there’s one special kind — where the force always pulls back in exact proportion to how far you’ve strayed — that shows up everywhere: pendulums, springs, guitar strings, even atoms in a solid. It’s called simple harmonic motion, and because it follows one clean rule, we can describe it with a single elegant equation. Get this rule straight and the whole topic opens up.
📘 What you need to know
SHM is an oscillation where the acceleration is proportional to the displacement from equilibrium and always points back towards it
The two conditions: acceleration ∝ displacement, and acceleration is in the opposite direction to displacement
The defining equation is a = −ω²x, where the minus sign carries the “back towards equilibrium” idea
The restoring force is what pulls the object back — it grows with displacement
Acceleration is maximum at the extremes (where displacement is largest) and zero at equilibrium
The time period of SHM is independent of amplitude (for small swings) — this is called being isochronous
The two conditions for SHM
An oscillation only counts as simple harmonic motion if it obeys two rules at every point in the swing. Both are about the object’s acceleration — how quickly its velocity is changing.
✅ The SHM test — both must be true
Acceleration is proportional to displacement. The further the object is from equilibrium, the bigger its acceleration. Double the displacement and you double the acceleration.
Acceleration always points back towards equilibrium. Whichever side the object is on, the acceleration pulls it back to the centre — it’s always in the opposite direction to the displacement.
Put those two ideas together and you can write them as a single proportionality, using a for acceleration and x for displacement:
The SHM conditiona ∝ −x
The minus sign is doing real work here: it says the acceleration and the displacement always point in opposite directions. That’s the mathematical heart of “always pulled back to the middle.”
Here’s the trick to remembering it: imagine stretching a spring. The further you pull, the harder it yanks back. That “harder the further you go, always back towards home” behaviour is exactly what a ∝ −x means. If an oscillation doesn’t do that — like someone bouncing on a trampoline, where the force is just their constant weight — it’s not SHM.
The restoring force
What causes that acceleration? A restoring force. This is the force that’s always trying to drag the object back to equilibrium. Because of Newton’s second law (F = ma), if the acceleration is proportional to displacement and points back to the centre, then the force must be too.
Displaced to the right, the mass feels a restoring force (and acceleration) pointing left — back towards equilibrium. Push it the other way and both arrows flip.
The defining equation of SHM
We can turn the proportionality a ∝ −x into a proper equation by putting in a constant. That constant turns out to be the square of the angular frequency, ω², which gives the equation that defines SHM:
Defining equation of SHMa = −ω²x
Where a is acceleration (m s−2), ω is angular frequency (rad s−1), and x is displacement (m). Two things are worth pausing on:
The minus sign guarantees the acceleration is always opposite to the displacement — the restoring behaviour, built right in.
Acceleration is biggest when x is biggest (at the amplitude, x = x0) and zero at the centre (x = 0). So the object is flung back hardest exactly where it’s furthest out.
Straight line through the origin with a negative slope. The steepness is ω²; the downward tilt is the minus sign. That’s a = −ω²x drawn out.
WE 1
A trolley on a spring oscillates in SHM with a frequency of 2.0 Hz and an amplitude of 0.15 m. Calculate the maximum acceleration of the trolley.
Step 1 — find the angular frequencyω = 2πf = 2π × 2.0 = 12.57 rad s⁻¹Step 2 — max acceleration is at max displacement (x = x₀)
size of a = ω²x₀
a = (12.57)² × 0.15 = 157.9 × 0.15a = 23.7 m s⁻²We ignore the minus sign here because the question asks for the size of the maximum acceleration.
WE 2
An object in SHM has an acceleration of 3.2 m s⁻² when its displacement is 2.0 cm. What is its acceleration when the displacement is 5.0 cm?
Step 1 — in SHM, a is proportional to x
so a₂ / a₁ = x₂ / x₁
Step 2 — scale up by the ratio of displacementsa₂ = 3.2 × (5.0 / 2.0) = 3.2 × 2.5a = 8.0 m s⁻²No need for ω at all — proportionality does the work. 2.5× the displacement means 2.5× the acceleration.
WE 3
A pendulum bob oscillating in SHM has an acceleration of 0.64 m s⁻² directed towards the centre when its displacement from equilibrium is 4.0 cm. Find the angular frequency and the time period.
Step 1 — use a = ω²x (sizes), so ω² = a / xω² = 0.64 / 0.040 = 16 rad² s⁻²Step 2 — take the square rootω = √16 = 4.0 rad s⁻¹Step 3 — period from T = 2π/ωT = 2π / 4.0 = 1.57 sω = 4.0 rad s⁻¹, T = 1.6 sConvert cm to metres first (4.0 cm = 0.040 m) or the numbers come out wrong by a factor of 100.
Displacement, velocity and acceleration graphs
Because SHM is so regular, the displacement, velocity and acceleration all trace smooth sine or cosine curves against time — and they’re neatly linked. If the object starts at equilibrium, its displacement follows a sine curve. Velocity is the gradient of displacement, and acceleration is the gradient of velocity.
Displacement (sine), velocity (cosine, a quarter-cycle ahead) and acceleration (upside-down sine). Notice the acceleration curve is the displacement curve flipped — that’s the minus sign in a = −ω²x.
Look at the top and bottom graphs together: whenever displacement is at a positive peak, acceleration is at its most negative, and vice versa. They’re perfect mirror images. That’s the clearest picture of “acceleration is opposite to displacement” you’ll ever get — and examiners love asking you to spot it.
An example that is NOT SHM
To really understand SHM, it helps to see something that looks oscillatory but breaks the rule. A person bouncing on a trampoline is the classic example.
While they’re in the air, the only force on them is their weight — and weight is constant. It doesn’t get bigger the higher they jump, and it always points down rather than back towards some equilibrium. Since the restoring force isn’t proportional to displacement, the two SHM conditions fail. It’s a repeating motion, but it is not simple harmonic motion.
Quick check: for SHM you need a restoring force that grows with displacement and always points back to equilibrium. Constant forces (like weight on a trampoline) don’t qualify.
Isochronous motion
One beautiful feature of SHM: for small oscillations, the time period doesn’t depend on the amplitude. A pendulum given a big push and a gentle push takes the same time to complete each swing. Motion with a constant period like this is called isochronous — and it’s exactly why pendulums were used to keep time in clocks for centuries.
Examples of SHM
Oscillator
What provides the restoring force
A simple pendulum (small swings)
Component of gravity along the arc
A mass on a spring
The spring’s tension/compression (Hooke’s law)
A guitar or violin string
Tension in the string
A marble rolling in a bowl
Component of gravity along the curve
Atoms vibrating in a solid
Bonds acting like tiny springs
💡 Top tips
a = −ω²x is the equation to know cold — the minus sign is not optional, it defines SHM.
Acceleration is max at the extremes, zero at equilibrium. Velocity is the opposite: max at equilibrium, zero at the extremes.
To spot SHM in a graph, check the acceleration–displacement plot is a straight line through the origin with negative gradient.
If a question gives a and x, you can get ω straight away: ω² = size of a ÷ x.
Always convert cm to metres before substituting — a very common slip.
⚠ Common mistakes
Dropping the minus sign in a = −ω²x — it’s what makes the motion “restoring”
Thinking acceleration is greatest at the centre — it’s actually zero there and greatest at the extremes
Assuming a bigger amplitude means a longer period — for SHM the period is independent of amplitude
Calling any back-and-forth motion SHM — the restoring force must be proportional to displacement
Forgetting to convert cm or mm to metres before using the equation
Quick recap: SHM is an oscillation where acceleration is proportional to displacement and always points back to equilibrium: a = −ω²x. A restoring force (growing with displacement) drives it. Acceleration is maximum at the extremes and zero at the centre; the period is independent of amplitude (isochronous). The x, v and a graphs are sine and cosine curves, with acceleration the exact mirror of displacement.
Now that you know what makes an oscillation “simple harmonic,” the next step is to put real physics into that ω. For a mass on a spring, ω depends on the mass and the spring’s stiffness — which leads straight to the formula for the time period of a mass–spring system. That’s the next page.
Want SHM to finally click?
Book a free meeting and we’ll work through the defining equation, the graphs and past-paper SHM questions together.
