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Waves, Signals & Oscillations

Level 5 · Physics & Physical Modeling · Prerequisites: Courses 1.2 and 2.1 · Connects to Electronics, Control and AI

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7 canonical classes · 4 laboratories · 4 module assessments · project · final

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MODULE 1

Oscillations & Periodic Motion

Build quantitative models of cycles, phase and harmonic response.

Dependencies: Courses 1.2 and 2.1; algebra, trigonometry, calculus, graphs and SI units.

5.3.1 · v1 · 90 minutes

Harmonic motion

Why it matters

Oscillations model vibration, timing, sensors and many physical systems while connecting mechanics to differential equations.

Prerequisites

Courses 1.2 and 2.1 and preceding canonical classes where applicable.

Concept and explanation

An oscillator moves about equilibrium. Period T is time per cycle; frequency f=1/T is cycles per second; angular frequency ω=2πf is radians per second. Simple harmonic motion is the ideal linear model x=A cos(ωt+φ), not a claim that every oscillator is sinusoidal.

Key terms

equilibrium; displacement; amplitude; period; frequency; angular frequency; phase; damping

Notation and reading strategy

A and x use length units; T in s; f in Hz; ω in rad/s; φ in rad. Read the cosine argument as dimensionless.

Physical / signal model

Ideal mass-spring oscillator with linear restoring force F=−kx, negligible damping and known initial state. Compare only within the small-deformation range.

Mathematical development

m x''+kx=0 gives ω₀=√(k/m). For x=A cos(ωt+φ), v=−Aω sin(ωt+φ), a=−ω²x; ideal total energy is (1/2)kA².

Learning objectives

  • Relate amplitude, period, frequency, angular frequency and phase.
  • Interpret position, velocity, acceleration and energy in simple harmonic motion.

Worked examples

Worked example 1

Physical/signal situation, given and goal: A system completes 12 cycles in 6.0 s. Find T, f and ω.

Representation, model, development, calculation and units: T=6/12=0.50 s; f=1/T=2.0 Hz; ω=2πf=4π≈12.57 rad/s.

Time/frequency interpretation, validation and limitations: fT=1 and ωT=2π. Cycle count and timing resolution limit precision.

Worked example 2

Physical/signal situation, given and goal: For x=0.040 cos(5t) m, find maximum speed and acceleration.

Representation, model, development, calculation and units: v_max=Aω=0.200 m/s; a_max=Aω²=1.00 m/s².

Time/frequency interpretation, validation and limitations: Units are m/s and m/s²; extrema occur at different phases.

Common mistake and counterexample

Incorrect: frequency and angular frequency are interchangeable. A 2 Hz oscillator has ω=4π rad/s, not 2 rad/s; convert with ω=2πf.

Guided practice

  1. Rebuild Worked Example 1 with every quantity and unit labeled.
  2. Change one input, predict the effect, calculate it, then validate independently.

Knowledge Check and Summary

  1. State the model and its assumptions.
  2. Carry units through every calculation.
  3. Interpret the result in time or frequency.
  4. Validate with an independent relationship or limiting case.

Mastery criterion: 4/4 correct with representation, units, interpretation and independent validation. Correct each miss and complete a fresh equivalent check.

Related Laboratory

Next class: Waves