The condition that defines it
Simple harmonic motion — Motion in which the acceleration is proportional to the displacement from a fixed point and always directed towards that point: a = −ω²x.
Not every repeating motion is simple harmonic. SHM is a specific and rather narrow case, defined by one condition on the acceleration: it must be proportional to the displacement from equilibrium, and directed back towards that equilibrium.
The minus sign in a = −ω²x carries the entire physical meaning. It says that whenever the object is displaced one way, the acceleration points the other. Remove the minus and you have exponential runaway, not oscillation.
Both parts of the condition are examined. "Acceleration proportional to displacement" alone is not enough for the marks — the direction must be stated too.
- ω
- angular frequencyin rad s⁻¹, equal to 2πf
- x₀
- the amplitudethe maximum displacement from equilibrium
- T
- the periodindependent of amplitude, which is the striking feature
Change the amplitude and watch the period stay exactly the same — that independence is what makes a pendulum useful as a clock. Then change the mass and the stiffness, which do alter it.
Where the speed and the acceleration are largest
The two quantities peak at opposite ends of the motion, and getting this the right way round is worth several marks across a paper.
At maximum displacement the restoring force is largest, so the acceleration is maximum — but the object is momentarily stationary, so the speed is zero. At the equilibrium position there is no net force at all, so the acceleration is zero, yet this is exactly where the object is moving fastest.
The period of SHM does not depend on the amplitude. Pull a pendulum back further and it travels further, but proportionately faster, so it takes the same time. This property is called isochronism and it is why pendulums were used as clocks for three centuries.
| Position | Displacement | Velocity | Acceleration | Energy |
|---|---|---|---|---|
| equilibrium | 0 | maximum | 0 | all kinetic |
| maximum displacement | x₀ | 0 | maximum | all potential |
| halfway (x = x₀/2) | x₀/2 | 0.87 v_max | half of max | mostly kinetic |
- E_total
- the total energyconstant if there is no damping
- k
- the spring constantstiffer spring means shorter period
- L
- the pendulum lengththe only thing you can change to alter its period
Energy goes as the amplitude squared
Doubling the amplitude does not double the energy — it quadruples it, since E ∝ x₀². The same squared relationship appears for the maximum speed being proportional to amplitude, so the kinetic energy at the centre scales with the square. Questions exploit this regularly.
Damping and resonance
Real oscillators lose energy to friction and air resistance, so the amplitude decays. This is damping, and how much of it is present changes the behaviour qualitatively rather than just quantitatively.
Light damping lets the system oscillate many times with a slowly shrinking amplitude. Critical damping returns it to equilibrium in the shortest possible time without overshooting — the behaviour a car suspension or a door closer is designed for. Heavy damping returns it slowly, without oscillating at all.
When a system is driven by a periodic force, the amplitude of its response depends on the driving frequency. As that frequency approaches the system's own natural frequency the amplitude rises sharply: this is resonance. Damping reduces the height of the resonance peak and broadens it.
| Damping | Behaviour | Example |
|---|---|---|
| none | amplitude constant forever | an idealisation only |
| light | many oscillations, slowly decaying | a swinging pendulum |
| critical | returns fastest with no overshoot | car suspension, door closers |
| heavy | slow return, no oscillation | a pendulum in treacle |
What resonance requires, and what damping does to it
- Resonance occurs when the driving frequency equals the natural frequency.
- At resonance the amplitude is a maximum and energy transfer is most efficient.
- More damping gives a lower and broader resonance peak.
- Heavy damping also shifts the peak slightly below the natural frequency.
- Useful resonance: musical instruments, radio tuning, MRI scanners.
- Unwanted resonance: bridges under rhythmic loading, vibrating machinery.
Reading the three graphs against each other
Displacement, velocity and acceleration all vary sinusoidally with time, but they peak at different moments — and the relationships between the three graphs are examined more often than the equations themselves.
Velocity is the gradient of the displacement graph, so it is a quarter of a cycle ahead: where displacement is at a maximum its gradient is zero, and where displacement crosses zero its gradient is steepest. Acceleration is the gradient of velocity, putting it another quarter cycle ahead — which makes it exactly half a cycle out of step with displacement.
That half-cycle relationship is simply a = −ω²x drawn out. Whenever displacement is positive the acceleration is negative, so the acceleration graph is the displacement graph turned upside down and scaled by ω².
| At this point | Displacement | Velocity | Acceleration |
|---|---|---|---|
| equilibrium, moving right | 0 | +maximum | 0 |
| maximum right | +x₀ | 0 | −maximum |
| equilibrium, moving left | 0 | −maximum | 0 |
| maximum left | −x₀ | 0 | +maximum |
Energy has twice the frequency
Kinetic energy is at a maximum every time the object passes through the centre — which happens twice per oscillation, once in each direction. So the energy graphs complete two full cycles for every one cycle of displacement. Sketching an energy-time graph with the same period as the displacement is a common and costly error.
Two systems, and what actually sets the period
The syllabus needs two concrete oscillators, and the useful thing about them is what each period formula does not contain.
A mass on a spring has period T = 2π√(m/k). A heavier mass has more inertia and swings more slowly; a stiffer spring pulls harder and swings faster. Notably, gravity does not appear at all — the same spring oscillates at the same rate on the Moon, because gravity only shifts the equilibrium position rather than changing the restoring force about it.
A simple pendulum has period T = 2π√(L/g). Here the mass is absent instead, for the same reason a heavier object does not fall faster: a larger mass has proportionally more weight pulling it back, and the two cancel. Only the length and the local gravitational field strength matter, which is why a pendulum can be used to measure g.
| System | Period | Depends on | Does NOT depend on |
|---|---|---|---|
| mass on a spring | T = 2π√(m/k) | mass and stiffness | gravity, amplitude |
| simple pendulum | T = 2π√(L/g) | length and g | mass, amplitude |
The pendulum formula assumes a small angle
T = 2π√(L/g) holds only for small oscillations, roughly under 10°. The derivation replaces sin θ with θ, which is accurate only for small angles in radians. At larger amplitudes the motion is still periodic but no longer simple harmonic, and the period grows slightly — so isochronism itself quietly fails once the swing gets wide.