Oscillations

Revision notes

Simple harmonic motion (SHM) occurs when acceleration is proportional to displacement from equilibrium and directed toward equilibrium: a = −ω²x. For ideal SHM, x can be written as A cos(ωt + φ), with period T = \(\dfrac{2\pi }{\omega }\) and frequency f = \(\dfrac{1}{T}\). Maximum speed is ωA.

In an ideal mass–spring oscillator, energy transfers continuously between kinetic energy and elastic potential energy while total mechanical energy remains constant. Speed is maximum at equilibrium and zero at the turning points. Real oscillators lose mechanical energy because of damping. Light damping gradually reduces amplitude; stronger damping can return a system to equilibrium more rapidly without sustained oscillation.

A periodic driving force produces forced oscillations. The response amplitude depends on driving frequency and damping. Near the natural frequency, energy transfer from the driver can be especially effective and the amplitude becomes large: this is resonance. Greater damping lowers and broadens the resonance peak.

Simple harmonic motion

Simple harmonic motion occurs when acceleration is proportional to displacement from equilibrium and directed towards equilibrium: a = -ω²x. The negative sign is essential because acceleration and displacement are opposite in direction.

Angular frequency is ω = 2πf and period T = \(\dfrac{1}{f}\). For ideal SHM, displacement can be represented sinusoidally. Velocity is maximum at equilibrium and zero at the turning points; acceleration is zero at equilibrium and maximum in magnitude at the turning points.

Energy in SHM

In an ideal oscillator, total mechanical energy is constant. For a mass-spring system, energy transfers between kinetic and elastic potential forms. At maximum displacement, kinetic energy is zero and potential energy is maximum. At equilibrium, potential energy relative to equilibrium is minimum and kinetic energy is maximum.

Energy relationships can be used to find speed at an intermediate displacement without explicitly using time.

Mass-spring and pendulum systems

For an ideal spring-mass oscillator, period depends on mass and spring constant. For a simple pendulum under the small-angle approximation, period depends on length and gravitational field strength but not on bob mass.

These formulae rely on model assumptions. Large pendulum amplitudes or non-linear spring behaviour invalidate the simplest SHM treatment.

Damping

Damping removes mechanical energy from an oscillator. Light damping causes amplitude to decrease gradually; heavy damping can suppress oscillation. The damping mechanism determines where the energy is transferred, for example to thermal energy in the surroundings.

Forced oscillations and resonance

A periodic driving force can produce forced oscillation. The response depends on driving frequency and damping. Near the system's natural frequency, amplitude can become large: resonance. Greater damping reduces the maximum resonant amplitude and broadens the response.

Resonance can be useful, as in tuning systems, or hazardous when structures are driven near a natural frequency.

Phase relationships

In SHM, acceleration is always in antiphase with displacement. Velocity is a quarter cycle out of phase with displacement in the ideal sinusoidal description. Phase diagrams or equations can help determine whether an oscillator is moving towards or away from equilibrium.

Exam technique

Use the defining equation a = -ω²x to identify SHM rather than simply saying motion repeats. At turning points, remember velocity is zero but acceleration is maximum. Check whether the question's model includes damping or forcing before assuming total mechanical energy is constant.

SHM equations

A mass–spring system has period T = 2π\(\sqrt{\dfrac{m}{k}}\).

A simple pendulum, for small angles, has period T = 2π\(\sqrt{\dfrac{l}{g}}\), independent of the mass and the amplitude.

The total energy of an oscillator is ½kA² (or ½mω²A²), shared between kinetic and potential energy.

Free oscillations and damping types

A free oscillation has no driving force and oscillates at its natural frequency. With light damping, the amplitude decays gradually over many cycles. Heavy damping returns it slowly without oscillating.