Simple harmonic motion, explained with examples

By The STEM-Tutor team · Updated October 2026

Simple harmonic motion (SHM) describes oscillations such as a mass bouncing on a spring or a pendulum swinging. It is one of the most mathematical topics in A level Physics, and it links mechanics to waves.

The defining condition

An object moves with simple harmonic motion when its acceleration is directly proportional to its displacement from the equilibrium position, and always directed towards that position. In symbols:

a = −ω²x

Here x is the displacement from equilibrium, ω is the angular frequency in rad/s, and the minus sign shows that the acceleration always points back towards equilibrium. The further the object is displaced, the harder it is pulled back. This restoring behaviour is what makes SHM so common in nature.

Key quantities

• Amplitude A: the maximum displacement from equilibrium.
• Period T: the time for one complete oscillation.
• Frequency f: the number of oscillations per second, f = 1/T.
• Angular frequency ω: ω = 2πf = 2π/T.

An important property of SHM is that the period does not depend on the amplitude. A pendulum swinging through a small arc or a slightly larger arc takes the same time per swing. Oscillators like this are called isochronous, which is why pendulums were used in clocks.

Displacement, velocity and acceleration

If timing starts at maximum displacement, the motion is described by:

• x = A cos(ωt)
• v = −Aω sin(ωt), with v = ±ω√(A² − x²) at any displacement
• a = −Aω² cos(ωt) = −ω²x

The maximum speed is v_max = ωA, reached as the object passes through equilibrium. The maximum acceleration is a_max = ω²A, at the extremes of the motion where the object is momentarily at rest. On graphs against time, velocity is a quarter of a cycle (π/2) ahead of displacement, and acceleration is exactly out of phase (π) with displacement.

The mass-spring system

For a mass m on a spring of spring constant k, T = 2π√(m/k). A heavier mass oscillates more slowly; a stiffer spring oscillates faster. This is true whether the spring is horizontal or vertical.

The simple pendulum

For a simple pendulum of length L, T = 2π√(L/g). This only holds for small angles, below about 10°, where sin θ ≈ θ in radians. The period does not depend on the mass of the bob. Measuring T for different lengths and plotting T² against L gives a straight line with gradient 4π²/g, a classic way to measure g.

Worked examples

Example 1: maximum speed and acceleration

A mass oscillates with amplitude 0.04 m and period 0.5 s. Find its maximum speed and maximum acceleration.

• ω = 2π/0.5 = 4π ≈ 12.6 rad/s.
• v_max = ωA = 12.6 × 0.04 ≈ 0.50 m/s.
• a_max = ω²A = 158 × 0.04 ≈ 6.3 m/s².

Example 2: speed at a displacement

For the same oscillator, find the speed when x = 0.02 m.

• v = ω√(A² − x²) = 12.6 × √(0.0016 − 0.0004) = 12.6 × 0.0346.
• v ≈ 0.44 m/s.

Example 3: a mass on a spring

A 0.25 kg mass hangs from a spring with k = 40 N/m. Find the period and frequency.

• T = 2π√(0.25/40) = 2π × 0.079 ≈ 0.50 s.
• f = 1/T ≈ 2.0 Hz.

Example 4: a pendulum

Find the length of a pendulum with a period of exactly 2 s. (g = 9.81 m/s²)

• L = gT²/(4π²) = 9.81 × 4/39.48.
• L ≈ 0.994 m, close to one metre.

Example 5: displacement at a time

An oscillator has A = 5 cm and f = 2 Hz and starts at maximum displacement. Find x at t = 0.1 s.

• ω = 4π. x = 5 cos(4π × 0.1) = 5 cos(1.257 rad).
• x ≈ 1.5 cm. Make sure your calculator is in radians.

Measuring the period accurately

In practical work, time 10 or 20 complete oscillations and divide, rather than timing one, because reaction time is then a much smaller fraction of the total. Start timing as the object passes through equilibrium, where it moves fastest and the moment is easiest to judge. Use a fiducial marker, such as a pin, at the equilibrium position. Repeat each measurement and take a mean.

Energy in SHM

In an undamped oscillator, the total energy stays constant, continually swapping between kinetic and potential energy. At equilibrium, kinetic energy is maximum and potential energy is zero. At maximum displacement, kinetic energy is zero and potential energy is maximum. The total energy is E = ½mω²A², so doubling the amplitude quadruples the energy. Graphs of kinetic and potential energy against displacement are parabolas; against time, each completes two cycles for every one oscillation.

Damping and resonance

In real systems, friction and air resistance remove energy, so the amplitude decreases over time. This is damping. Light damping lets the system oscillate with slowly decreasing amplitude. Critical damping returns the system to equilibrium in the shortest time without oscillating, as in car suspension. Heavy damping returns it very slowly.

When a system is driven by a periodic force, it oscillates at the driving frequency. If the driving frequency matches the system's natural frequency, the amplitude becomes very large: this is resonance. Resonance is useful in musical instruments and microwave ovens, but dangerous in bridges and buildings. More damping lowers and broadens the resonance peak.

Common mistakes

• Using degrees instead of radians in x = A cos(ωt).
• Thinking the period depends on amplitude.
• Placing maximum speed at the extremes instead of at equilibrium.
• Using T = 2π√(L/g) for large swings, where the small-angle approximation fails.

Practice questions

Try each one before opening the answer.

Frequently asked questions

Is SHM in AS or A level Physics?

In most specifications, including AQA, Edexcel and OCR, SHM is a second-year (A level) topic. Check your board.

Is SHM on the AP Physics exams?

Yes. Simple harmonic motion is in AP Physics 1 and AP Physics C: Mechanics, with mass-spring systems and pendulums.

Key takeaways

• SHM: a = −ω²x, acceleration proportional to displacement and towards equilibrium.
• v_max = ωA at equilibrium; a_max = ω²A at the extremes.
• Mass-spring: T = 2π√(m/k). Pendulum: T = 2π√(L/g).
• Period does not depend on amplitude.
• Damping reduces amplitude; resonance occurs at the natural frequency.

Related guides

• A level Physics formula sheet
• A level Physics waves
• A level Physics required practicals