17. Oscillations

Displacement x measures position from equilibrium; amplitude A is maximum displacement; period T is cycle time; frequency f=1/T; angular frequency ω=2πf.

Keep phase and sign in displacement, while amplitude is a positive maximum. Use radians for angular frequency relationships.

A 5 Hz oscillator has T=0.20 s and ω=10π rad s⁻¹; its displacement alternates between ±A.

Amplitude is not peak-to-peak displacement, and angular frequency is not ordinary frequency in hertz.

Simple harmonic motion occurs when a=−ω²x: acceleration is proportional to displacement x from equilibrium and opposite in direction.

The restoring condition must hold throughout the motion. At equilibrium x=0 and speed is greatest; at an extreme x=±A and acceleration magnitude is greatest.

A mass on an ideal spring oscillates about its equilibrium position because the spring force reverses as the mass crosses the centre.

Constant acceleration is not SHM; the acceleration changes sign and magnitude with displacement.

A sinusoidal solution x=x₀ sin(ωt+φ) satisfies a=−ω²x, with amplitude x₀, angular frequency ω and phase φ.

Use the initial phase to match the starting position and direction; differentiate to obtain velocity and acceleration.

If the oscillator starts at equilibrium moving positive, x=x₀ sinωt is a natural phase choice.

x₀ is amplitude, not an arbitrary displacement, and omitting phase can give the wrong initial condition.

For x=x₀sinωt, v=x₀ωcosωt and v²=ω²(x₀²−x²), with the sign of v set by direction of motion.

Use the time form for phase questions and the squared form when only position and speed magnitude are known.

At x=0 the speed is maximum v₀=ωx₀; at x=±x₀ the speed is zero.

The ± sign in v=±ω√(x₀²−x²) cannot be chosen without considering the direction of travel.

For sinusoidal SHM, velocity is 90° out of phase with displacement and acceleration is 180° out of phase with displacement.

At equilibrium displacement is zero while speed is maximum; at extremes speed is zero while acceleration points back toward equilibrium.

A displacement sine graph has a velocity cosine graph and an acceleration graph inverted relative to displacement.

The graphs do not all peak at the same time; phase relationships encode the restoring motion.

As an SHM oscillator moves, kinetic energy is greatest at equilibrium and elastic or other potential energy is greatest at the extremes; their sum is constant in the ideal model.

Use the position to infer the energy split and identify any damping or driving that would break constancy.

A spring mass has maximum elastic energy at x=±A and maximum kinetic energy at x=0.

Energy is not created at the centre; it has transferred from potential to kinetic.

For amplitude x₀ and angular frequency ω, total mechanical energy is E=½mω²x₀², equal to maximum kinetic or potential energy.

The formula assumes ideal SHM with no energy loss and uses amplitude, not instantaneous displacement.

Doubling amplitude quadruples total energy; doubling mass doubles it for fixed ω and amplitude.

Instantaneous kinetic energy is not always the total energy; it equals the total only at equilibrium.

A resistive force opposing motion transfers energy from an oscillating system to other stores, causing the amplitude to decrease with time.

The damping force may depend on speed; distinguish the ideal SHM frequency from the changed response of a damped system.

Air resistance makes a pendulum’s swings gradually smaller because mechanical energy becomes thermal energy in the air.

Damping does not necessarily stop oscillation immediately, and reduced amplitude is not the same as reduced equilibrium position.

Light damping allows oscillations with decreasing amplitude; critical damping returns to equilibrium fastest without oscillating; heavy damping returns more slowly without overshoot.

Sketch displacement against time with or without crossings of equilibrium and compare settling time, not just initial slope.

A door closer is designed near critical damping so the door settles promptly without repeated swinging.

Critical damping is not “maximum resistance” in every situation; too much damping can make return slower.

A driven oscillator resonates when the driving frequency is near a natural frequency, producing a large amplitude; damping limits and broadens the peak.

Identify the driving source, natural frequency and response amplitude. More damping usually lowers and broadens the resonance peak.

Pushing a swing at its natural period builds a larger motion than pushing randomly, even with the same average effort.

Resonance is not always destructive and does not require zero damping; it is a frequency-response phenomenon.