Oscillations

This guide covers Topic 17, Oscillations, in full — subtopics 17.1 Simple harmonic oscillations, 17.2 Energy in simple harmonic motion and 17.3 Damped and forced oscillations, resonance — from Cambridge International AS & A Level Physics 9702, 2025–2027 series. This is A Level content, and the first topic of the mechanics-and-waves strand at A Level following the thermal physics topics.

Before studying this

This resource assumes circular motion from Motion in a Circle, and wave concepts from Waves: Progressive Waves, the Doppler Effect and Polarisation.

Syllabus coverage

CAMBRIDGE INTERNATIONAL AS & A LEVEL PHYSICS 9702 — A Level, Topic 17

17.1 Simple harmonic oscillations — describing simple examples of free oscillations; understanding and using the terms period, frequency, amplitude, displacement and phase difference in the context of oscillations, and expressing the period in terms of both frequency and angular frequency; recalling and using a = −ω²x as the defining equation of simple harmonic motion; recognising and using solutions of the form x = x₀ sin ωt and x = x₀ cos ωt; recalling and using v = v₀ cos ωt, v = −v₀ sin ωt and v₀ = ωx₀; recalling and using v = ±ω√(x₀² − x²); sketching and interpreting graphs of displacement, velocity and acceleration against time for an oscillator undergoing simple harmonic motion.

17.2 Energy in simple harmonic motion — describing the interchange between kinetic and potential energy during simple harmonic motion; recalling and using E_K = ½mω²(x₀² − x²) for kinetic energy and the total energy of the system E = ½mω²x₀².

17.3 Damped and forced oscillations, resonance — understanding that oscillatory systems may be periodically forced, and understanding the concept of resonance, and describing graphically how the amplitude of a forced oscillation changes with frequency near to the natural frequency of the system; describing practical examples of damped oscillations, and distinguishing between free, damped and forced oscillations.

Simple harmonic motion

A system undergoes simple harmonic motion (SHM) if its acceleration is always directed toward a fixed equilibrium point and is proportional to its displacement from that point:

a = −ω²x

The negative sign shows the acceleration always acts to restore the displacement back toward equilibrium (a restoring acceleration), which is why SHM oscillates rather than moving off indefinitely. Solutions to this equation, depending on where timing starts, take the form:

x = x₀ sin ωt   or   x = x₀ cos ωt

where x₀ is the amplitude. Differentiating gives velocity:

v = v₀ cos ωt   or   v = −v₀ sin ωt,   with v₀ = ωx₀

Velocity can also be found directly from displacement, without reference to time, using:

v = ±ω√(x₀² − x²)

which confirms that speed is maximum (v = v₀ = ωx₀) at x = 0 and zero at x = ±x₀, the two extremes of the oscillation.

Worked example. A mass oscillates with SHM, amplitude 0.050 m and angular frequency 4.0 rad s⁻¹. Its maximum speed:

v₀ = ωx₀ = 4.0 × 0.050 = 0.20 m s⁻¹

Its speed when the displacement is 0.030 m:

v = ω√(x₀² − x²) = 4.0 × √(0.050² − 0.030²) = 4.0 × 0.040 = 0.16 m s⁻¹

A simple pendulum is the standard example of a free oscillation: for small angles of swing, its period is given by T = 2π√(l/g), where l is the pendulum’s length and g the gravitational field strength — independent of the mass on the end or the amplitude of swing, provided the angle stays small. This relationship is not listed in the standard 9702 data/formulae section — check a recent past paper or the published data booklet to see whether it would be supplied in a question. If not, it should be memorised.

Period and frequency

The period T is the time for one complete oscillation, and frequency f is the number of oscillations per second, related to angular frequency ω by:

ω = 2π/T = 2πf

Graphs of displacement, velocity and acceleration

For x = x₀ cos ωt, the three graphs against time are all sinusoidal but out of phase with each other:

• Displacement–time: x = x₀ cos ωt, oscillating between +x₀ and −x₀.
• Velocity–time: v = −v₀ sin ωt, a quarter-cycle (90°) ahead of displacement — velocity is zero when displacement is at a maximum, and maximum (v₀ = ωx₀) when displacement is zero.
• Acceleration–time: a = −ω²x, always in antiphase with displacement (a half-cycle, 180°, out of phase) — acceleration is maximum in magnitude at the extremes, where displacement is maximum, but points back toward equilibrium, and is zero when displacement is zero.

Energy in simple harmonic motion

As a system oscillates, its energy continually transforms between kinetic energy and potential energy (elastic, gravitational, or another form depending on the system), while the total mechanical energy remains constant in the absence of damping. Kinetic energy is maximum, and potential energy zero, at the equilibrium position (maximum speed, zero displacement); potential energy is maximum, and kinetic energy zero, at maximum displacement (zero speed, at the amplitude).

The kinetic energy at displacement x is:

E_K = ½mω²(x₀² − x²)

and the total energy of the system — constant throughout the oscillation, in the absence of damping — is the kinetic energy at x = 0 (equivalently, the potential energy at x = x₀):

E = ½mω²x₀²

so the potential energy at displacement x is E_P = E − E_K = ½mω²x².

Free, damped and forced oscillations

A free oscillation occurs at the system’s natural frequency with no external periodic driving force and, in an idealised case, no energy loss. A damped oscillation loses energy (commonly to resistive forces such as friction or air resistance), so its amplitude decreases over time. A forced oscillation occurs when a system is driven by an external periodic force at a chosen driving frequency, which may differ from its natural frequency.

Damping itself comes in degrees. Light damping lets amplitude decay only gradually over many oscillations, approaching zero but never quite reaching it. Heavy damping returns the system to equilibrium only slowly, without oscillating. Critical damping sits between the two, returning the system to equilibrium in the shortest possible time without any oscillation at all — the level of damping engineers choose whenever a system must settle quickly without overshooting, such as a car’s suspension after a bump.

Resonance

Resonance occurs when the driving frequency of a forced oscillation matches the natural frequency of the system, producing a maximum amplitude of oscillation. As driving frequency is increased from zero toward the natural frequency, amplitude rises sharply to a peak at resonance, then falls again as driving frequency increases further past it. The amount of damping present affects the sharpness and height of this peak: lighter damping gives a sharper, higher resonance peak; heavier damping gives a flatter, lower one.

Common mistakes

• Forgetting the negative sign in a = −ω²x, or treating it as an optional detail — the sign is what makes the motion a restoring, oscillatory one rather than an accelerating, runaway one.
• Confusing ω (angular frequency, rad s⁻¹) with f (frequency, Hz) — always check units and convert using ω = 2πf.
• Assuming amplitude stays constant for damped oscillations — by definition, damped oscillations lose amplitude over time.
• Thinking resonance only happens at exactly zero damping — resonance happens at any level of damping, but the peak becomes sharper and taller as damping decreases.

Quick revision checklist

• a = −ω²x as the defining equation of SHM
• x = x₀ sin ωt / x₀ cos ωt, and v = v₀ cos ωt / −v₀ sin ωt with v₀ = ωx₀
• v = ±ω√(x₀² − x²), the velocity–displacement relation
• x–t, v–t and a–t graphs: v leads x by 90°, a is in antiphase with x
• Energy interchange between kinetic and potential energy during SHM
• E_K = ½mω²(x₀² − x²) and total energy E = ½mω²x₀²
• Free vs damped vs forced oscillations, and the resonance amplitude peak

Related resources

• Thermodynamics — the previous A Level topic
• Electric Fields — the next A Level topic
• Cambridge AS & A Level Physics hub

Written against Cambridge International AS & A Level Physics 9702, 2025–2027 series. Always check the current syllabus for your examination year.