What is simple harmonic motion?

Simple harmonic motion is oscillation in which the acceleration is proportional to the displacement from equilibrium and always directed back towards it: a = −ω²x. A mass on a spring and a small-angle pendulum are the standard examples.

Why is the acceleration negative in a = −ω²x?

The minus sign means the acceleration always points opposite to the displacement — back towards equilibrium. When the mass is pulled right, the restoring force pushes it left, and vice versa. That restoring relationship is exactly what produces oscillation.

What is the phase difference between displacement and velocity in SHM?

Velocity leads displacement by a quarter of a cycle (90°, or π/2 radians). Velocity is greatest as the mass passes through equilibrium, where displacement is zero, and zero at the extremes, where displacement is greatest.

How does energy change during SHM?

Total energy stays constant (ignoring damping). It shifts continuously between kinetic and potential: all kinetic at the equilibrium point (maximum speed) and all potential at the extremes (momentarily at rest), summing to the same total everywhere.

Simple Harmonic Motion Explained (Interactive) — SHM, Phase & Energy

A mass on a spring, a swinging pendulum, a vibrating guitar string — all oscillate the same way. Simple harmonic motion (SHM) is defined by one clean condition, $a = -\omega^2 x$ , and from it flows the sine curves, the quarter-cycle phase shifts, and the endless swap between kinetic and potential energy.

What makes motion “harmonic”

Pull a mass on a spring and let go. The further you pull it, the harder the spring pulls back — and it always pulls towards the middle. That single property, a restoring force proportional to displacement, is what produces the smooth, repeating motion of SHM.

The system is always being pulled home, and the further it strays the harder it’s pulled. Overshoot the middle and the pull reverses — so it oscillates forever (until friction wins).

Set it oscillating

Play the oscillation and watch three graphs draw themselves in step. Displacement, velocity and acceleration are all sine curves of the same period — but shifted a quarter-cycle apart. The arrow on the mass is the restoring acceleration: notice it always points back towards the middle.

InteractiveSHM: mass, spring and the three graphs

Loading interactive…

Play it, or scrub the time slider. Velocity peaks as the mass flies through the middle; acceleration peaks at the extremes, pointing back to centre.

Text description ↓

An interactive simple-harmonic-motion display. A mass hangs on a spring and oscillates about an equilibrium line; an arrow on the mass shows the restoring acceleration, always directed back towards equilibrium. Alongside are three time graphs sharing one sweeping playhead: displacement x = A cos(ωt), velocity v = −Aω sin(ωt), and acceleration a = −Aω² cos(ωt). The velocity graph is a quarter-cycle (90°) ahead of displacement, and acceleration is exactly out of phase with displacement (a = −ω²x). Sliders set the amplitude and period; a time slider scrubs through one and a half cycles. Displacement is greatest at the extremes (where velocity is zero) and zero at the centre (where speed is greatest).

The defining equation

SHM is defined by the condition that acceleration is proportional to displacement and directed back towards equilibrium:

a = -\omega^2 x

The constant $\omega$ is the angular frequency, related to the period $T$ and frequency $f$ by $\omega = 2\pi f = 2\pi / T$ . The minus sign is the whole point — it’s what makes the force restoring. Solving this equation gives sinusoidal motion:

x = A\cos(\omega t), \qquad v = -A\omega\sin(\omega t), \qquad a = -A\omega^2\cos(\omega t).

Displacement, velocity, acceleration

Reading the three curves against each other is a favourite exam question. The key phase relationships:

Velocity leads displacement by 90°. The mass moves fastest as it passes through equilibrium, exactly where displacement is zero.
Acceleration is 180° out of phase with displacement. That’s the minus sign in $a=-\omega^2 x$ : when displacement is at a positive maximum, acceleration is at its most negative.
The maximum speed is $v_\text{max} = A\omega$ and the maximum acceleration is $a_\text{max} = A\omega^2$ .

Where the energy goes

In SHM the total energy stays constant (ignoring damping); it just sloshes back and forth between kinetic and potential.

At the extremes, the mass is momentarily at rest: all the energy is potential (stretched spring / raised pendulum).
At the centre, the mass moves fastest: all the energy is kinetic.

Everywhere in between the two share the fixed total, so the kinetic and potential energy curves are mirror images that always add to the same value.

Common mistakes

Your turn

A mass oscillates in SHM with amplitude $0.05\,\text{m}$ and period $0.40\,\text{s}$ . What is its maximum speed?

Show the answer ↓

$\omega = 2\pi/T = 2\pi/0.40 = 15.7\,\text{rad s}^{-1}$ , so $v_\text{max} = A\omega = 0.05 \times 15.7 = 0.79\,\text{m s}^{-1}$ .

What makes motion “harmonic”

The system is always being pulled home, and the further it strays the harder it’s pulled. Overshoot the middle and the pull reverses — so it oscillates forever (until friction wins).

Set it oscillating

InteractiveSHM: mass, spring and the three graphs

Loading interactive…

Play it, or scrub the time slider. Velocity peaks as the mass flies through the middle; acceleration peaks at the extremes, pointing back to centre.

Text description ↓

The defining equation

SHM is defined by the condition that acceleration is proportional to displacement and directed back towards equilibrium:

a = -\omega^2 x

x = A\cos(\omega t), \qquad v = -A\omega\sin(\omega t), \qquad a = -A\omega^2\cos(\omega t).

Displacement, velocity, acceleration

Reading the three curves against each other is a favourite exam question. The key phase relationships:

Velocity leads displacement by 90°. The mass moves fastest as it passes through equilibrium, exactly where displacement is zero.
Acceleration is 180° out of phase with displacement. That’s the minus sign in $a=-\omega^2 x$ : when displacement is at a positive maximum, acceleration is at its most negative.
The maximum speed is $v_\text{max} = A\omega$ and the maximum acceleration is $a_\text{max} = A\omega^2$ .

Where the energy goes

In SHM the total energy stays constant (ignoring damping); it just sloshes back and forth between kinetic and potential.

At the extremes, the mass is momentarily at rest: all the energy is potential (stretched spring / raised pendulum).
At the centre, the mass moves fastest: all the energy is kinetic.

Everywhere in between the two share the fixed total, so the kinetic and potential energy curves are mirror images that always add to the same value.

Common mistakes

Your turn

A mass oscillates in SHM with amplitude $0.05\,\text{m}$ and period $0.40\,\text{s}$ . What is its maximum speed?

Show the answer ↓

$\omega = 2\pi/T = 2\pi/0.40 = 15.7\,\text{rad s}^{-1}$ , so $v_\text{max} = A\omega = 0.05 \times 15.7 = 0.79\,\text{m s}^{-1}$ .

Simple Harmonic Motion, Felt

What makes motion “harmonic”

Set it oscillating

The defining equation

Displacement, velocity, acceleration

Where the energy goes

Common mistakes

Your turn

Frequently asked questions

Keep exploring

Want a tutor to walk you through it?

Simple Harmonic Motion, Felt

What makes motion “harmonic”

Set it oscillating

The defining equation

Displacement, velocity, acceleration

Where the energy goes

Common mistakes

Your turn

Frequently asked questions

Keep exploring

Want a tutor to walk you through it?