Plenty of things wobble back and forth, but only a special family of them counts as simple harmonic motion — the ones where the acceleration is always proportional to how far the object has been pushed from the middle, and always points back towards it.
📘 What you need to know
An oscillation is any repeated to-and-fro motion about an equilibrium (rest) position
Motion is simple harmonic when acceleration is proportional to displacement from equilibrium and directed back towards it
The defining equation is a = −ω2x — the minus sign is the whole point
The angular frequencyω links to period and frequency by ω = 2π ÷ T = 2πf
For SHM the period is independent of amplitude (isochronous)
Displacement, velocity and acceleration each vary as a sine/cosine wave, all 90° out of phase with one another
What Makes Motion “Simple Harmonic”?
Start with the language of oscillations. The displacementx is how far the object is from equilibrium at a given instant; the amplitudex0 is the largest displacement it reaches. One full there-and-back cycle takes a time periodT, and the number of cycles per second is the frequencyf.
An oscillation only qualifies as simple harmonic when two conditions hold at every point in the motion:
The acceleration is proportional to the displacement from equilibrium
The acceleration is always directed towards the equilibrium position — i.e. opposite to the displacement
Behind the acceleration sits a restoring force that does the same job: it grows with displacement and always pulls the object back towards the middle. On a horizontal mass–spring system, stretch the spring to the right and both the force and the acceleration point left.
Displaced right (grey), the restoring force F (violet) and acceleration a (teal) both point back left towards equilibrium — the further out, the bigger they get.
The Defining Equation of SHM
Both conditions are captured in one compact statement, where ω is the angular frequency of the motion:
Defining equation of SHMa = −ω2x
The size of ω2 tells you how much acceleration you get per unit displacement, and the minus sign guarantees the acceleration points opposite to the displacement — back towards equilibrium. Angular frequency itself is fixed by how quickly the object cycles:
Angular frequencyω = 2π ÷ T = 2πf
Because a is directly proportional to x, a graph of acceleration against displacement is a straight line through the origin with a negative gradient equal to −ω2. This is one of the quickest ways an exam can ask you to confirm SHM.
Acceleration against displacement is a straight line through the origin; its gradient is −ω2, and the negative slope shows a and x point opposite ways.
Displacement, Velocity and Acceleration Graphs
Track the three quantities against time and a neat pattern appears. If the object starts at equilibrium, displacement follows a sine curve. Velocity is the gradient of displacement, so it is a cosine curve — a quarter-cycle (90°) ahead. Acceleration is the gradient of velocity, another 90° on, which makes it a negative sine curve — the mirror image of displacement, exactly as a = −ω2x demands.
Displacement (sine, teal), velocity (cosine, blue) and acceleration (negative sine, red) for an object released from equilibrium — each shifted 90° from the next.
Everyday Examples of SHM
You will meet two SHM models in the SL course: a simple pendulum swinging through small angles, and a mass–spring system oscillating vertically or horizontally. Other good approximations include a mass bobbing on a spring, a tuning fork, and a trolley tethered between two springs. Their period formulas are the focus of the next two pages.
Watch out for motion that only looks like SHM. A person bouncing on a trampoline is not SHM: while airborne the only force is their constant weight, which does not grow with displacement, so the “proportional to x” condition fails.
🧭 Cracking an SHM calculation
Pin down what you’re told — is it a period, frequency, amplitude, or an acceleration-at-a-displacement pair?
Findω first, using ω = 2π ÷ T = 2πf, or by rearranging a = −ω2x
Apply the defining equation, remembering the maximum acceleration happens at the amplitude (x = x0)
Keep signs honest — a negative displacement gives a positive acceleration and vice versa
Set your calculator to radians whenever a sine or cosine of ωt is involved
Quick recap: SHM means a = −ω2x — acceleration proportional to displacement and always aimed back at equilibrium, giving sine/cosine graphs that are 90° out of phase.
WE 1
A particle oscillates with simple harmonic motion of period 0.80 s and amplitude 0.12 m.
(a) Calculate the angular frequency of the motion.
(b) Calculate the maximum acceleration of the particle.
Part (a)ω = 2π ÷ T = 2π ÷ 0.80ω ≈ 7.9 rad s⁻¹Part (b)
Acceleration is greatest at the amplitude, where x = x0amax = ω²x0 = 7.854² × 0.12amax ≈ 7.4 m s⁻²The minus sign in a = −ω²x just tells you the acceleration points back towards the middle.
WE 2
An object moving with SHM has an acceleration of 3.2 m s⁻² when its displacement from equilibrium is 5.0 cm.
(a) Determine the angular frequency of the oscillation.
(b) Hence find the period of the motion.
Part (a)
Using the size of a = −ω²x, so ω² = a ÷ x
ω² = 3.2 ÷ 0.050 = 64ω = 8.0 rad s⁻¹Part (b)T = 2π ÷ ω = 2π ÷ 8.0T ≈ 0.79 sNotice the period never used the mass or the amplitude — for SHM it depends only on ω.
💡 Top tips
Always find ω before anything else — almost every SHM answer flows from it
“Maximum acceleration” means evaluate a = −ω2x at the amplitude; “maximum speed” happens at equilibrium where a = 0
A straight line through the origin on an a–x graph is the signature of SHM; read −ω2 straight off its gradient
Keep the minus sign in your working — displacement and acceleration are vectors pointing opposite ways
⚠ Common mistakes
Dropping the minus sign in a = −ω2x and losing the direction of the acceleration
Confusing angular frequency ω (rad s⁻¹) with ordinary frequency f (Hz) — they differ by a factor of 2π
Thinking a bigger amplitude means a longer period; for SHM the period is independent of amplitude
Leaving the calculator in degrees when working with sin(ωt) or cos(ωt)
Assuming any back-and-forth motion is SHM — check that acceleration really is proportional to displacement first
Up next: now that the defining equation is nailed down, we put it to work on the first real system — the Time Period of a Mass–Spring System, where the spring constant sets the value of ω.
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