17.1 Simple harmonic 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.