Unit 5: Oscillations

This guide covers sub-topic 5.5 Oscillations, the third of four sub-topics in Unit 5: Thermodynamics, Radiation, Oscillations and Cosmology, from the Pearson Edexcel International Advanced Level in Physics (YPH11), Issue 3 specification. This topic is commonly studied using applications such as the construction of buildings in earthquake zones.

Before studying this

This topic follows sub-topics 5.3 and 5.4, and assumes comfort with angular velocity (ω) from Unit 4’s circular motion content and with sin and cos functions.

Syllabus coverage

PEARSON EDEXCEL INTERNATIONAL A LEVEL PHYSICS (YPH11) — Sub-topic 5.5

Candidates will be assessed on their ability to: understand that the condition for simple harmonic motion is F = −kx, and hence understand how to identify situations in which simple harmonic motion will occur; use the equations a = −ω²x, x = Acosωt, v = −Aωsinωt, a = −Aω²cosωt, and T = 2π/ω = 1/f and ω = 2πf as applied to a simple harmonic oscillator; use the equations for a simple harmonic oscillator T = 2π√(m/k), and a simple pendulum T = 2π√(l/g); draw and interpret a displacement-time graph for an object oscillating and know that the gradient at a point gives the velocity at that point; draw and interpret a velocity-time graph for an oscillating object and know that the gradient at a point gives the acceleration at that point; understand what is meant by resonance; CORE PRACTICAL 16: determine the value of an unknown mass using the resonant frequencies of the oscillation of known masses; understand how to apply conservation of energy to damped and undamped oscillating systems; understand the distinction between free and forced oscillations; understand how the amplitude of a forced oscillation changes at and around the natural frequency of a system and know, qualitatively, how damping affects resonance; and understand how damping and the plastic deformation of ductile materials reduce the amplitude of oscillation.

Simple harmonic motion

Simple harmonic motion (SHM) occurs whenever the restoring force on an object is directly proportional to its displacement from equilibrium and directed towards it:

F = −kx

This condition identifies which physical systems undergo SHM (e.g. a mass on a spring, a simple pendulum for small angles). The corresponding equations of motion are:

a = −ω²x
x = A cos ωt
v = −Aω sin ωt
a = −Aω² cos ωt

with T = 2π/ω = 1/f and ω = 2πf, where A is amplitude, ω is angular frequency, T is period and f is frequency.

Two specific SHM systems have standard period equations: a mass-spring system, T = 2π√(m/k), and a simple pendulum (for small angles), T = 2π√(l/g). A displacement-time graph for an oscillating object has a gradient at any point equal to velocity; a velocity-time graph’s gradient at any point gives acceleration — matching the general displacement-velocity-acceleration graph relationship used throughout the mechanics units.

Energy, free and forced oscillations, resonance

In an undamped oscillating system, total mechanical energy (kinetic plus potential) is conserved, continuously exchanging between kinetic and potential forms as the object oscillates. In a damped system, energy is progressively transferred out (e.g. to heat via friction or air resistance), so amplitude decreases over time — conservation of energy still applies overall, once this energy loss is accounted for.

A free oscillation occurs at a system’s own natural frequency, with no external periodic driving force; a forced oscillation occurs when an external periodic force drives the system, potentially at a different frequency. Resonance occurs when the driving frequency of a forced oscillation matches (or is very close to) the natural frequency of the system, producing a large increase in amplitude. CORE PRACTICAL 16 uses this relationship to determine an unknown mass from the resonant frequencies of oscillation of known masses.

As driving frequency approaches the natural frequency, amplitude of a forced oscillation increases sharply, peaking at (or near) resonance; damping reduces the peak amplitude at resonance and broadens the resonance curve, qualitatively flattening the sharp peak that would otherwise occur. Damping, together with the plastic deformation of ductile materials within a real oscillating structure, dissipates energy and reduces the amplitude of oscillation over successive cycles.

Worked example. A mass of 0.40 kg oscillates on a spring of spring constant 25 N/m. Find (a) the period of oscillation, and (b) the maximum acceleration if the amplitude is 0.050 m.

(a) T = 2π√(m/k) = 2π√(0.40/25) = 2π√(0.016) = 2π(0.1265) = 0.795 s

(b) ω² = k/m = 25/0.40 = 62.5 s⁻² (so ω = 7.91 rad/s)

Maximum acceleration occurs at maximum displacement (x = A):

a_max = ω²A = 62.5 × 0.050 = 3.13 m/s² (3.125 m/s², to 3 s.f.)

Using ω² = k/m directly avoids the rounding error you get by squaring a rounded ω taken from T.

Worked example 2 (same oscillator). The mass is released from rest at x = +0.050 m at t = 0. Find (a) its maximum speed, (b) its displacement and velocity at t = 0.10 s, and (c) check (a) using energy.

(a) ω = √62.5 = 7.906 rad/s, so v_max = Aω = 0.050 × 7.906 = 0.395 m/s.

(b) Released from x = A at t = 0, so x = A cos ωt. With the calculator in radians, ωt = 7.906 × 0.10 = 0.7906 rad:

x = 0.050 × cos(0.7906) = 0.050 × 0.7034 = 0.0352 m

v = −Aω sin ωt = −0.395 × sin(0.7906) = −0.281 m/s (moving back towards equilibrium)

(c) Energy stored in the (horizontal) spring at maximum displacement = ½kA² = ½ × 25 × 0.050² = 0.03125 J. At equilibrium this is all kinetic: ½ × 0.40 × v_max² = 0.03125, so v_max² = 0.15625 and v_max = 0.395 m/s, as in (a).

A common slip here is leaving the calculator in degrees: ωt is an angle in radians.

Common mistakes

Assuming any oscillation is simple harmonic — SHM specifically requires F = −kx (or equivalently a = −ω²x); a pendulum only approximates SHM for small angles. Confusing natural frequency (a property of the system itself) with driving frequency (the frequency of an external periodic force) — resonance is specifically when these two coincide. Forgetting the negative sign in a = −ω²x, which shows acceleration is always directed back towards equilibrium, opposite to displacement. Assuming damping increases resonant amplitude — damping always reduces the peak amplitude at resonance.

Quick revision checklist

• State the SHM condition F = −kx and identify systems undergoing SHM.
• Use a = −ω²x, x = Acosωt, v = −Aωsinωt, a = −Aω²cosωt, T = 2π/ω = 1/f.
• Use T = 2π√(m/k) for a mass-spring system and T = 2π√(l/g) for a simple pendulum.
• Interpret displacement-time and velocity-time graphs for oscillating objects.
• Explain resonance, and describe CORE PRACTICAL 16.
• Apply conservation of energy to damped and undamped oscillations.
• Distinguish free from forced oscillations, and explain how damping affects the resonance curve.

Related resources

• Unit 5: Nuclear Decay — the previous sub-topic
• Unit 5: Astrophysics and Cosmology — the next and final sub-topic
• Pearson Edexcel International A Level Physics hub

This guide is intended to support, not replace, engagement with the official Pearson Edexcel specification and your own teacher’s guidance. Always check the current version of the specification for authoritative detail.